http://www.explainxkcd.com/wiki/index.php?title=Special:RecentChangesLinked/Category:Math&feed=atom&target=Category%3AMathexplain xkcd - Changes related to "Category:Math" [en]2016-08-29T23:25:36ZRelated changesMediaWiki 1.19.17//www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=125902&oldid=1258891724: Proofs2016-08-29T19:03:09Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Category:Comics featuring Cueball]]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Category:Comics featuring Cueball]]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Category:Math]]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Category:Math]]</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">[[Category: Axiom of Choice]]</ins></div></td></tr>
</table>162.158.203.144//www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=125889&oldid=1258611724: Proofs2016-08-29T14:29:24Z<p>no, it's wrong summary</p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Although Miss Lenhart did retire a year ago after [[1519: Venus]], she seems to have returned here for a math course at university level, but continues the trend she finished with in her prior class.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Although Miss Lenhart did retire a year ago after [[1519: Venus]], she seems to have returned here for a math course at university level, but continues the trend she finished with in her prior class.</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">'''TL;DR''' Cueball suspect Ms Lenhart already made-up an answer for a made-up mathematical function (hence ''magic''), which is confirmed at the last panel.</del></div></td><td colspan="2"> </td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Transcript==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Transcript==</div></td></tr>
</table>162.158.133.138//www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=125861&oldid=1258291724: Proofs2016-08-29T07:23:59Z<p><span dir="auto"><span class="autocomment">Explanation: </span> </span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Although Miss Lenhart did retire a year ago after [[1519: Venus]], she seems to have returned here for a math course at university level, but continues the trend she finished with in her prior class.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Although Miss Lenhart did retire a year ago after [[1519: Venus]], she seems to have returned here for a math course at university level, but continues the trend she finished with in her prior class.</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">'''TL;DR''' Cueball suspect Ms Lenhart already made-up an answer for a made-up mathematical function (hence ''magic''), which is confirmed at the last panel.</ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Transcript==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Transcript==</div></td></tr>
</table>162.158.167.35//www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=125829&oldid=1258141724: Proofs2016-08-27T07:18:59Z<p>removing "standard" </p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Alternatively, instead of a proof by contradiction the setup could be for a one way function. For example, it is relatively easy to test that a solution to a differential equation is valid but choosing the correct solution to test can seem like black magic to students.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Alternatively, instead of a proof by contradiction the setup could be for a one way function. For example, it is relatively easy to test that a solution to a differential equation is valid but choosing the correct solution to test can seem like black magic to students.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>The way that Ms Lenhart's proof refers to the act of doing math itself, is characteristic of metamathematical proofs, for example {{w|Gödel's incompleteness theorems}}, which, at first sight, may indeed look like black magic, even if in the end they must be a "perfectly sensible chain of reasoning" like the rest of good mathematics. While <del class="diffchange diffchange-inline">standard </del>mathematical theorems and their proofs deal with <del class="diffchange diffchange-inline">standard </del>mathematical objects<del class="diffchange diffchange-inline">, like </del>numbers, functions, points or lines, the metamathematical theorems treat other theorems as objects of interest. In this way you can propose and prove theorems about possibility of proving other theorems. For example, in 1931 {{w|Kurt Gödel}} was able to prove that any mathematical system based on arithmetics (that is using numbers) has statements that are true, but can be neither proved nor disproved. This kind of metamathematical reasoning is especially useful in {{w|set theory}}, where many statements become impossible to prove and disprove if the {{w|axiom of choice}} is not taken as a part of the axiomatic system.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>The way that Ms Lenhart's proof refers to the act of doing math itself, is characteristic of metamathematical proofs, for example {{w|Gödel's incompleteness theorems}}, which, at first sight, may indeed look like black magic, even if in the end they must be a "perfectly sensible chain of reasoning" like the rest of good mathematics. While <ins class="diffchange diffchange-inline">typical </ins>mathematical theorems and their proofs deal with <ins class="diffchange diffchange-inline">such </ins>mathematical objects <ins class="diffchange diffchange-inline">as </ins>numbers, functions, points or lines, the metamathematical theorems treat other theorems as objects of interest. In this way you can propose and prove theorems about possibility of proving other theorems. For example, in 1931 {{w|Kurt Gödel}} was able to prove that any mathematical system based on arithmetics (that is using numbers) has statements that are true, but can be neither proved nor disproved. This kind of metamathematical reasoning is especially useful in {{w|set theory}}, where many statements become impossible to prove and disprove if the {{w|axiom of choice}} is not taken as a part of the axiomatic system.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Using a position on the blackboard as a part of the proof is a joke, but it bears a resemblance to {{w|Cantor's diagonal argument}} where a position in a sequence of digits of a real number was a tool in a proof that not all infinite sets have the same {{w|cardinality}} (rough equivalent of the number of elements). This "diagonal method" is also often used in metamathematical proofs.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Using a position on the blackboard as a part of the proof is a joke, but it bears a resemblance to {{w|Cantor's diagonal argument}} where a position in a sequence of digits of a real number was a tool in a proof that not all infinite sets have the same {{w|cardinality}} (rough equivalent of the number of elements). This "diagonal method" is also often used in metamathematical proofs.</div></td></tr>
</table>162.158.85.249//www.explainxkcd.com/wiki/index.php?title=1017:_Backward_in_Time&diff=125822&oldid=1227121017: Backward in Time2016-08-27T00:13:45Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[940|(Also, the workout website, Fitocracy has been mentioned previously in xkcd.)]]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[940|(Also, the workout website, Fitocracy has been mentioned previously in xkcd.)]]</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Note that as of the time that this page was last cached, the comic was uploaded at {{#expr:100*sqrt((ln(({{#time:U}}-1329195600)/31536000+e^3)-3)/20.3444)}}% progress.</ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Transcript==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Transcript==</div></td></tr>
</table>LegionMammal978//www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=125814&oldid=1257971724: Proofs2016-08-26T19:44:08Z<p><span dir="auto"><span class="autocomment">Explanation: </span> </span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Miss Lenhart]] is back teaching a math class. She begins a proof when one of her students ([[Cueball]]) interrupts her asking if this is one of those dark-magic (unclear, incomprehensible) proofs. She says no, but it soon turns out that it is; Cueball exclaims that he just knew it would be.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Miss Lenhart]] is back teaching a math class. She begins a proof when one of her students ([[Cueball]]) interrupts her asking if this is one of those dark-magic (unclear, incomprehensible) proofs. She says no, but it soon turns out that it is; Cueball exclaims that he just knew it would be.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>The proof she starts setting up resembles a {{w|proof by contradiction}}. This kind of proof assumes that a particular <del class="diffchange diffchange-inline">theorem </del>is true, and shows that this assumption leads to a contradiction, which disproves the initial assumption. For example assumption that √2 is a {{w|rational number}} means that, for some natural ''a'' and ''b'', √2=''a/b'', where ''a/b'' is an {{w|irreducible fraction}}. Yet, multiplying this equation by itself, we get 2=''a²/b²'' which in turn rearranges to 2''b²''=''a²''. Therefore ''a²'' is even (as any integer multiplied by 2 is even), which means that ''a'' is an even number, as an even number squared is always even and an odd number squared is always odd. This means, that ''a=2k'' and ''2b²=(2k)²=4k²'', meaning ''b²''=2''k²'', so ''b'' must be even too. But if both ''a'' and ''b'' are even, ''a/b'' cannot be irreducible. Contradiction means that the initial assumption is false, and √2 cannot be a rational number.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>The proof she starts setting up resembles a {{w|proof by contradiction}}. This kind of proof assumes that a particular <ins class="diffchange diffchange-inline">condition </ins>is true, and shows that this assumption leads to a contradiction, which disproves the initial assumption. For example assumption that √2 is a {{w|rational number}} means that, for some natural ''a'' and ''b'', √2=''a/b'', where ''a/b'' is an {{w|irreducible fraction}}. Yet, multiplying this equation by itself, we get 2=''a²/b²'' which in turn rearranges to 2''b²''=''a²''. Therefore ''a²'' is even (as any integer multiplied by 2 is even), which means that ''a'' is an even number, as an even number squared is always even and an odd number squared is always odd. This means, that ''a=2k'' and ''2b²=(2k)²=4k²'', meaning ''b²''=2''k²'', so ''b'' must be even too. But if both ''a'' and ''b'' are even, ''a/b'' cannot be irreducible. Contradiction means that the initial assumption is false, and √2 cannot be a rational number.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Alternatively, instead of a proof by contradiction the setup could be for a one way function. For example, it is relatively easy to test that a solution to a differential equation is valid but choosing the correct solution to test can seem like black magic to students.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Alternatively, instead of a proof by contradiction the setup could be for a one way function. For example, it is relatively easy to test that a solution to a differential equation is valid but choosing the correct solution to test can seem like black magic to students.</div></td></tr>
</table>108.162.219.68//www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=125797&oldid=1257811724: Proofs2016-08-26T13:52:20Z<p><span dir="auto"><span class="autocomment">Explanation: </span> rm "the"</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Miss Lenhart]] is back teaching a math class. She begins a proof when one of her students ([[Cueball]]) interrupts her asking if this is one of those dark-magic (unclear, incomprehensible) proofs. She says no, but it soon turns out that it is; Cueball exclaims that he just knew it would be.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Miss Lenhart]] is back teaching a math class. She begins a proof when one of her students ([[Cueball]]) interrupts her asking if this is one of those dark-magic (unclear, incomprehensible) proofs. She says no, but it soon turns out that it is; Cueball exclaims that he just knew it would be.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>The proof she starts setting up resembles a {{w|proof by contradiction}}. This kind of proof assumes that a particular theorem is true, and shows that this assumption leads to a contradiction, which disproves the initial assumption. For example assumption that √2 is a {{w|rational number}} means that, for some natural ''a'' and ''b'', √2=''a/b'', where ''a/b'' is an {{w|irreducible fraction}}. Yet, multiplying this equation by itself, we get 2=''a²/b²'' which in turn rearranges to 2''b²''=''a²''<del class="diffchange diffchange-inline">, therefor </del>''a²'' is even (as any integer multiplied by 2 is even), which means that ''a'' is an even number, as an even number squared is always even and an odd number squared is always odd. This means, that ''a=2k'' and ''2b²=(2k)²=4k²'', meaning ''b²''=2''k²'', so ''b'' must be even too. But if both ''a'' and ''b'' are even, ''a/b'' cannot be irreducible. Contradiction means that the initial assumption is false, and √2 cannot be a rational number.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>The proof she starts setting up resembles a {{w|proof by contradiction}}. This kind of proof assumes that a particular theorem is true, and shows that this assumption leads to a contradiction, which disproves the initial assumption. For example assumption that √2 is a {{w|rational number}} means that, for some natural ''a'' and ''b'', √2=''a/b'', where ''a/b'' is an {{w|irreducible fraction}}. Yet, multiplying this equation by itself, we get 2=''a²/b²'' which in turn rearranges to 2''b²''=''a²''<ins class="diffchange diffchange-inline">. Therefore </ins>''a²'' is even (as any integer multiplied by 2 is even), which means that ''a'' is an even number, as an even number squared is always even and an odd number squared is always odd. This means, that ''a=2k'' and ''2b²=(2k)²=4k²'', meaning ''b²''=2''k²'', so ''b'' must be even too. But if both ''a'' and ''b'' are even, ''a/b'' cannot be irreducible. Contradiction means that the initial assumption is false, and √2 cannot be a rational number.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Alternatively, instead of a proof by contradiction the setup could be for a one way function. For example, it is relatively easy to test that a solution to a differential equation is valid but choosing the correct solution to test can seem like black magic to students.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Alternatively, instead of a proof by contradiction the setup could be for a one way function. For example, it is relatively easy to test that a solution to a differential equation is valid but choosing the correct solution to test can seem like black magic to students.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>The way<del class="diffchange diffchange-inline">, </del>Ms Lenhart's proof refers to the act of doing math itself, is characteristic <del class="diffchange diffchange-inline">to </del>metamathematical proofs, for example {{w|Gödel's incompleteness theorems}}, which, at first sight, may indeed look like black magic, even if in the end they must be a "perfectly sensible chain of reasoning" like the rest of good mathematics. While standard mathematical theorems and their proofs deal with standard mathematical objects, like numbers, functions, points or lines, the metamathematical theorems treat other theorems as objects of interest. In this way you can propose and prove theorems about possibility of proving other theorems. For example, in 1931 {{w|Kurt Gödel}} was able to prove that any mathematical system based on arithmetics (that is using numbers) has statements that are true, but can be neither proved nor disproved. This kind of metamathematical reasoning is especially useful in <del class="diffchange diffchange-inline">the </del>{{w|set theory}}, where many statements become impossible to prove and disprove if the {{w|axiom of choice}} is not taken as a part of the axiomatic system.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>The way <ins class="diffchange diffchange-inline">that </ins>Ms Lenhart's proof refers to the act of doing math itself, is characteristic <ins class="diffchange diffchange-inline">of </ins>metamathematical proofs, for example {{w|Gödel's incompleteness theorems}}, which, at first sight, may indeed look like black magic, even if in the end they must be a "perfectly sensible chain of reasoning" like the rest of good mathematics. While standard mathematical theorems and their proofs deal with standard mathematical objects, like numbers, functions, points or lines, the metamathematical theorems treat other theorems as objects of interest. In this way you can propose and prove theorems about possibility of proving other theorems. For example, in 1931 {{w|Kurt Gödel}} was able to prove that any mathematical system based on arithmetics (that is using numbers) has statements that are true, but can be neither proved nor disproved. This kind of metamathematical reasoning is especially useful in {{w|set theory}}, where many statements become impossible to prove and disprove if the {{w|axiom of choice}} is not taken as a part of the axiomatic system.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Using a position on the blackboard as a part of the proof is a joke, but it bears a resemblance to <del class="diffchange diffchange-inline">the </del>{{w|Cantor's diagonal argument}} where a position in a sequence of digits of a real number was a tool in a proof that not all infinite sets have the same {{w|cardinality}} (rough equivalent of the number of elements). This "diagonal method" is also often used in metamathematical proofs.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Using a position on the blackboard as a part of the proof is a joke, but it bears a resemblance to {{w|Cantor's diagonal argument}} where a position in a sequence of digits of a real number was a tool in a proof that not all infinite sets have the same {{w|cardinality}} (rough equivalent of the number of elements). This "diagonal method" is also often used in metamathematical proofs.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The axiom of choice itself states that for every collection of nonempty sets, you can have a function that draws one element from each set of the collection. This axiom, once considered controversial, was added relatively late to the axiomatic set theory, and even contemporary mathematicians still study which theorems really require its inclusion. In the title text the decision of whether to take the axiom of choice is made by a deterministic process, that is a process which future states can be developed with no randomness involved. {{w|Determinacy}} of infinite games is used as a tool in the set theory, however the deterministic process is rather a term of the {{w|stochastic process|stochastic processes theory}}, and the {{w|dynamical systems theory}}, branches of mathematics far from the abstract set theory, which makes the proof even more exotic. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]].</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The axiom of choice itself states that for every collection of nonempty sets, you can have a function that draws one element from each set of the collection. This axiom, once considered controversial, was added relatively late to the axiomatic set theory, and even contemporary mathematicians still study which theorems really require its inclusion. In the title text the decision of whether to take the axiom of choice is made by a deterministic process, that is a process which future states can be developed with no randomness involved. {{w|Determinacy}} of infinite games is used as a tool in the set theory, however the deterministic process is rather a term of the {{w|stochastic process|stochastic processes theory}}, and the {{w|dynamical systems theory}}, branches of mathematics far from the abstract set theory, which makes the proof even more exotic. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]].</div></td></tr>
</table>Slashme//www.explainxkcd.com/wiki/index.php?title=925:_Cell_Phones&diff=125791&oldid=125768925: Cell Phones2016-08-26T13:25:19Z<p><span dir="auto"><span class="autocomment">Explanation: </span> I don't know how that lasted that long</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>After hearing about the "Cell Phones Don't Cause Cancer" study, which refutes a claim made by the ''{{w|World Health Organization}}'' (just Google the debate, the comic doesn't focus much on it), [[Black Hat]] plots "Total Cancer Incidence" per 100,000 and "Cell Phone Users" per 100 on the same graph. The graph in frame 3 shows an exponential rise in cancer followed by an exponential rise in cell phone usage, which makes Black Hat comically come to the conclusion that cancer causes cell phones.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>After hearing about the "Cell Phones Don't Cause Cancer" study, which refutes a claim made by the ''{{w|World Health Organization}}'' (just Google the debate, the comic doesn't focus much on it), [[Black Hat]] plots "Total Cancer Incidence" per 100,000 and "Cell Phone Users" per 100 on the same graph. The graph in frame 3 shows an exponential rise in cancer followed by an exponential rise in cell phone usage, which makes Black Hat comically come to the conclusion that cancer causes cell phones.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>The comic highlights a well-known fallacy known as ''{{w|post hoc ergo propter hoc}}'', often shortened to simply ''post hoc.'' The Latin translates to "after this, therefore because of this," referring to the common mistake that because two events happen in chronological order, the former event must have caused the latter event. The fallacy is often the root cause of many superstitions (e.g., a person noticing he/she wore a special bracelet before getting a good test score thinks the bracelet was the source of his/her good fortune <del class="diffchange diffchange-inline">when it was more likely to be her socks.</del>), but it often crosses into more serious areas of thinking. In this case, the scientific research community, which often prides itself on its intellectual aptitude, is gently mocked for being nonetheless prone to such poor reasoning all too often. The different possibilities are generally known as causation, when one thing is proven to cause another, or correlation, when changes in one thing are aligned with changes in another, but there is no proof that they are actually related.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>The comic highlights a well-known fallacy known as ''{{w|post hoc ergo propter hoc}}'', often shortened to simply ''post hoc.'' The Latin translates to "after this, therefore because of this," referring to the common mistake that because two events happen in chronological order, the former event must have caused the latter event. The fallacy is often the root cause of many superstitions (e.g., a person noticing he/she wore a special bracelet before getting a good test score thinks the bracelet was the source of his/her good fortune), but it often crosses into more serious areas of thinking. In this case, the scientific research community, which often prides itself on its intellectual aptitude, is gently mocked for being nonetheless prone to such poor reasoning all too often. The different possibilities are generally known as causation, when one thing is proven to cause another, or correlation, when changes in one thing are aligned with changes in another, but there is no proof that they are actually related.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The title text refers to the way Black Hat holds the laptop in panel 2. Being that Cueball (and Randall, for that matter) are quite into computers, the potential damage to a laptop screen either from the weight of its lower body or the pressure of the user's fingers on the LCD screen is enough to make him squirm in discomfort.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The title text refers to the way Black Hat holds the laptop in panel 2. Being that Cueball (and Randall, for that matter) are quite into computers, the potential damage to a laptop screen either from the weight of its lower body or the pressure of the user's fingers on the LCD screen is enough to make him squirm in discomfort.</div></td></tr>
</table>Lackadaisical//www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=125781&oldid=1257761724: Proofs2016-08-26T12:27:36Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The way, Ms Lenhart's proof refers to the act of doing math itself, is characteristic to metamathematical proofs, for example {{w|Gödel's incompleteness theorems}}, which, at first sight, may indeed look like black magic, even if in the end they must be a "perfectly sensible chain of reasoning" like the rest of good mathematics. While standard mathematical theorems and their proofs deal with standard mathematical objects, like numbers, functions, points or lines, the metamathematical theorems treat other theorems as objects of interest. In this way you can propose and prove theorems about possibility of proving other theorems. For example, in 1931 {{w|Kurt Gödel}} was able to prove that any mathematical system based on arithmetics (that is using numbers) has statements that are true, but can be neither proved nor disproved. This kind of metamathematical reasoning is especially useful in the {{w|set theory}}, where many statements become impossible to prove and disprove if the {{w|axiom of choice}} is not taken as a part of the axiomatic system.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The way, Ms Lenhart's proof refers to the act of doing math itself, is characteristic to metamathematical proofs, for example {{w|Gödel's incompleteness theorems}}, which, at first sight, may indeed look like black magic, even if in the end they must be a "perfectly sensible chain of reasoning" like the rest of good mathematics. While standard mathematical theorems and their proofs deal with standard mathematical objects, like numbers, functions, points or lines, the metamathematical theorems treat other theorems as objects of interest. In this way you can propose and prove theorems about possibility of proving other theorems. For example, in 1931 {{w|Kurt Gödel}} was able to prove that any mathematical system based on arithmetics (that is using numbers) has statements that are true, but can be neither proved nor disproved. This kind of metamathematical reasoning is especially useful in the {{w|set theory}}, where many statements become impossible to prove and disprove if the {{w|axiom of choice}} is not taken as a part of the axiomatic system.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>In the title text the decision of whether to take the axiom of choice is made by a deterministic process, that is a process which future states can be developed with no randomness involved. <del class="diffchange diffchange-inline">The axiom </del>of <del class="diffchange diffchange-inline">choice itself </del>is a <del class="diffchange diffchange-inline">non-constructive axiom </del>in <del class="diffchange diffchange-inline">mathematics</del>, <del class="diffchange diffchange-inline">it asserts </del>the <del class="diffchange diffchange-inline">existence </del>of <del class="diffchange diffchange-inline">objects</del>, <del class="diffchange diffchange-inline">without providing a method </del>of <del class="diffchange diffchange-inline">constructing them</del>, <del class="diffchange diffchange-inline">in particular there is no deterministic process by </del>which <del class="diffchange diffchange-inline">we can define objects whose existence can only be proved using </del>the <del class="diffchange diffchange-inline">axiom of choice</del>. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]].</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">Using a position on the blackboard as a part of the proof is a joke, but it bears a resemblance to the {{w|Cantor's diagonal argument}} where a position in a sequence of digits of a real number was a tool in a proof that not all infinite sets have the same {{w|cardinality}} (rough equivalent of the number of elements). This "diagonal method" is also often used in metamathematical proofs.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">The axiom of choice itself states that for every collection of nonempty sets, you can have a function that draws one element from each set of the collection. This axiom, once considered controversial, was added relatively late to the axiomatic set theory, and even contemporary mathematicians still study which theorems really require its inclusion. </ins>In the title text the decision of whether to take the axiom of choice is made by a deterministic process, that is a process which future states can be developed with no randomness involved. <ins class="diffchange diffchange-inline">{{w|Determinacy}} </ins>of <ins class="diffchange diffchange-inline">infinite games </ins>is <ins class="diffchange diffchange-inline">used as </ins>a <ins class="diffchange diffchange-inline">tool </ins>in <ins class="diffchange diffchange-inline">the set theory</ins>, <ins class="diffchange diffchange-inline">however </ins>the <ins class="diffchange diffchange-inline">deterministic process is rather a term </ins>of <ins class="diffchange diffchange-inline">the {{w|stochastic process|stochastic processes theory}}</ins>, <ins class="diffchange diffchange-inline">and the {{w|dynamical systems theory}}, branches </ins>of <ins class="diffchange diffchange-inline">mathematics far from the abstract set theory</ins>, which <ins class="diffchange diffchange-inline">makes </ins>the <ins class="diffchange diffchange-inline">proof even more exotic</ins>. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]].</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Although Miss Lenhart did retire a year ago after [[1519: Venus]], she seems to have returned here for a math course at university level, but continues the trend she finished with in her prior class.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Although Miss Lenhart did retire a year ago after [[1519: Venus]], she seems to have returned here for a math course at university level, but continues the trend she finished with in her prior class.</div></td></tr>
</table>162.158.133.138//www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=125776&oldid=1257741724: Proofs2016-08-26T08:33:35Z<p>The explanation for the hover text was completely off. It has nothing to do with Determinacy.</p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The way, Ms Lenhart's proof refers to the act of doing math itself, is characteristic to metamathematical proofs, for example {{w|Gödel's incompleteness theorems}}, which, at first sight, may indeed look like black magic, even if in the end they must be a "perfectly sensible chain of reasoning" like the rest of good mathematics. While standard mathematical theorems and their proofs deal with standard mathematical objects, like numbers, functions, points or lines, the metamathematical theorems treat other theorems as objects of interest. In this way you can propose and prove theorems about possibility of proving other theorems. For example, in 1931 {{w|Kurt Gödel}} was able to prove that any mathematical system based on arithmetics (that is using numbers) has statements that are true, but can be neither proved nor disproved. This kind of metamathematical reasoning is especially useful in the {{w|set theory}}, where many statements become impossible to prove and disprove if the {{w|axiom of choice}} is not taken as a part of the axiomatic system.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The way, Ms Lenhart's proof refers to the act of doing math itself, is characteristic to metamathematical proofs, for example {{w|Gödel's incompleteness theorems}}, which, at first sight, may indeed look like black magic, even if in the end they must be a "perfectly sensible chain of reasoning" like the rest of good mathematics. While standard mathematical theorems and their proofs deal with standard mathematical objects, like numbers, functions, points or lines, the metamathematical theorems treat other theorems as objects of interest. In this way you can propose and prove theorems about possibility of proving other theorems. For example, in 1931 {{w|Kurt Gödel}} was able to prove that any mathematical system based on arithmetics (that is using numbers) has statements that are true, but can be neither proved nor disproved. This kind of metamathematical reasoning is especially useful in the {{w|set theory}}, where many statements become impossible to prove and disprove if the {{w|axiom of choice}} is not taken as a part of the axiomatic system.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>In the title text the decision of whether to take the axiom of choice is made by a deterministic process, that is a process which future states can be developed with no randomness involved. <del class="diffchange diffchange-inline">It may be an allusion to the proposed {{w|</del>axiom of <del class="diffchange diffchange-inline">determinacy}} of the set theory. It </del>is, <del class="diffchange diffchange-inline">however, {{w|Axiom_of_determinacy#Incompatibility_of_the_axiom_of_determinacy_with_the_axiom_of_choice|incompatible}} with </del>the <del class="diffchange diffchange-inline">axiom </del>of <del class="diffchange diffchange-inline">choice</del>, <del class="diffchange diffchange-inline">which builds another layer </del>of the <del class="diffchange diffchange-inline">joke</del>. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]].</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>In the title text the decision of whether to take the axiom of choice is made by a deterministic process, that is a process which future states can be developed with no randomness involved. <ins class="diffchange diffchange-inline">The </ins>axiom of <ins class="diffchange diffchange-inline">choice itself </ins>is <ins class="diffchange diffchange-inline">a non-constructive axiom in mathematics</ins>, <ins class="diffchange diffchange-inline">it asserts </ins>the <ins class="diffchange diffchange-inline">existence </ins>of <ins class="diffchange diffchange-inline">objects</ins>, <ins class="diffchange diffchange-inline">without providing a method </ins>of <ins class="diffchange diffchange-inline">constructing them, in particular there is no deterministic process by which we can define objects whose existence can only be proved using </ins>the <ins class="diffchange diffchange-inline">axiom of choice</ins>. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]].</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Although Miss Lenhart did retire a year ago after [[1519: Venus]], she seems to have returned here for a math course at university level, but continues the trend she finished with in her prior class.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Although Miss Lenhart did retire a year ago after [[1519: Venus]], she seems to have returned here for a math course at university level, but continues the trend she finished with in her prior class.</div></td></tr>
</table>162.158.86.173//www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=125774&oldid=1257721724: Proofs2016-08-26T02:30:26Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The proof she starts setting up resembles a {{w|proof by contradiction}}. This kind of proof assumes that a particular theorem is true, and shows that this assumption leads to a contradiction, which disproves the initial assumption. For example assumption that √2 is a {{w|rational number}} means that, for some natural ''a'' and ''b'', √2=''a/b'', where ''a/b'' is an {{w|irreducible fraction}}. Yet, multiplying this equation by itself, we get 2=''a²/b²'' which in turn rearranges to 2''b²''=''a²'', therefor ''a²'' is even (as any integer multiplied by 2 is even), which means that ''a'' is an even number, as an even number squared is always even and an odd number squared is always odd. This means, that ''a=2k'' and ''2b²=(2k)²=4k²'', meaning ''b²''=2''k²'', so ''b'' must be even too. But if both ''a'' and ''b'' are even, ''a/b'' cannot be irreducible. Contradiction means that the initial assumption is false, and √2 cannot be a rational number.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The proof she starts setting up resembles a {{w|proof by contradiction}}. This kind of proof assumes that a particular theorem is true, and shows that this assumption leads to a contradiction, which disproves the initial assumption. For example assumption that √2 is a {{w|rational number}} means that, for some natural ''a'' and ''b'', √2=''a/b'', where ''a/b'' is an {{w|irreducible fraction}}. Yet, multiplying this equation by itself, we get 2=''a²/b²'' which in turn rearranges to 2''b²''=''a²'', therefor ''a²'' is even (as any integer multiplied by 2 is even), which means that ''a'' is an even number, as an even number squared is always even and an odd number squared is always odd. This means, that ''a=2k'' and ''2b²=(2k)²=4k²'', meaning ''b²''=2''k²'', so ''b'' must be even too. But if both ''a'' and ''b'' are even, ''a/b'' cannot be irreducible. Contradiction means that the initial assumption is false, and √2 cannot be a rational number.</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Alternatively, instead of a proof by contradiction the setup could be for a one way function. For example, it is relatively easy to test that a solution to a differential equation is valid but choosing the correct solution to test can seem like black magic to students.</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The way, Ms Lenhart's proof refers to the act of doing math itself, is characteristic to metamathematical proofs, for example {{w|Gödel's incompleteness theorems}}, which, at first sight, may indeed look like black magic, even if in the end they must be a "perfectly sensible chain of reasoning" like the rest of good mathematics. While standard mathematical theorems and their proofs deal with standard mathematical objects, like numbers, functions, points or lines, the metamathematical theorems treat other theorems as objects of interest. In this way you can propose and prove theorems about possibility of proving other theorems. For example, in 1931 {{w|Kurt Gödel}} was able to prove that any mathematical system based on arithmetics (that is using numbers) has statements that are true, but can be neither proved nor disproved. This kind of metamathematical reasoning is especially useful in the {{w|set theory}}, where many statements become impossible to prove and disprove if the {{w|axiom of choice}} is not taken as a part of the axiomatic system.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The way, Ms Lenhart's proof refers to the act of doing math itself, is characteristic to metamathematical proofs, for example {{w|Gödel's incompleteness theorems}}, which, at first sight, may indeed look like black magic, even if in the end they must be a "perfectly sensible chain of reasoning" like the rest of good mathematics. While standard mathematical theorems and their proofs deal with standard mathematical objects, like numbers, functions, points or lines, the metamathematical theorems treat other theorems as objects of interest. In this way you can propose and prove theorems about possibility of proving other theorems. For example, in 1931 {{w|Kurt Gödel}} was able to prove that any mathematical system based on arithmetics (that is using numbers) has statements that are true, but can be neither proved nor disproved. This kind of metamathematical reasoning is especially useful in the {{w|set theory}}, where many statements become impossible to prove and disprove if the {{w|axiom of choice}} is not taken as a part of the axiomatic system.</div></td></tr>
</table>173.245.52.76//www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=125772&oldid=1257701724: Proofs2016-08-25T22:57:47Z<p><span dir="auto"><span class="autocomment">Explanation: </span> </span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Miss Lenhart]] is back teaching a math class. She begins a proof when one of her students ([[Cueball]]) interrupts her asking if this is one of those dark-magic (unclear, incomprehensible) proofs. She says no, but it soon turns out that it is; Cueball exclaims that he just knew it would be.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Miss Lenhart]] is back teaching a math class. She begins a proof when one of her students ([[Cueball]]) interrupts her asking if this is one of those dark-magic (unclear, incomprehensible) proofs. She says no, but it soon turns out that it is; Cueball exclaims that he just knew it would be.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>The proof she starts setting up resembles a {{w|proof by contradiction}}. This kind of proof assumes that a particular theorem is true, and shows that this assumption leads to a contradiction, which disproves the initial assumption. For example assumption that √2 is a {{w|rational number}} means that, for some natural ''a'' and ''b'', √2=''a/b'', where ''a/b'' is an {{w|irreducible fraction}}. Yet, multiplying this equation by itself, we get 2=''a²/b²'', which means that ''a'' is an even number. This means, that ''a=2k'' and ''2b²=(2k)²=4k²'', so ''b'' must be even too. But if both ''a'' and ''b'' are even, ''a/b'' cannot be irreducible. Contradiction means that the initial assumption is false, and √2 cannot be a rational number.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>The proof she starts setting up resembles a {{w|proof by contradiction}}. This kind of proof assumes that a particular theorem is true, and shows that this assumption leads to a contradiction, which disproves the initial assumption. For example assumption that √2 is a {{w|rational number}} means that, for some natural ''a'' and ''b'', √2=''a/b'', where ''a/b'' is an {{w|irreducible fraction}}. Yet, multiplying this equation by itself, we get 2=''a²/b²'' <ins class="diffchange diffchange-inline">which in turn rearranges to 2''b²''=''a²'', therefor ''a²'' is even (as any integer multiplied by 2 is even)</ins>, which means that ''a'' is an even number<ins class="diffchange diffchange-inline">, as an even number squared is always even and an odd number squared is always odd</ins>. This means, that ''a=2k'' and ''2b²=(2k)²=4k²<ins class="diffchange diffchange-inline">'', meaning ''b²''=2''k²</ins>'', so ''b'' must be even too. But if both ''a'' and ''b'' are even, ''a/b'' cannot be irreducible. Contradiction means that the initial assumption is false, and √2 cannot be a rational number.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>The way, Ms Lenhart's proof refers to the act of doing math itself, is characteristic to metamathematical proofs, for example {{w|Gödel's incompleteness theorems}}, which, at first sight, may indeed look like black magic, even if in the end they must be a "perfectly sensible chain of <del class="diffchange diffchange-inline">resoning</del>" like the rest of good mathematics. While standard mathematical theorems and their proofs deal with standard mathematical objects, like numbers, functions, points or lines, the metamathematical theorems treat other theorems as objects of interest. In this way you can propose and prove theorems about possibility of proving other theorems. For example, in 1931 {{w|Kurt Gödel}} was able to prove that any mathematical system based on arithmetics (that is using numbers) has statements that are true, but can be neither proved nor disproved. This kind of metamathematical reasoning is especially useful in the {{w|set theory}}, where many statements become impossible to prove and disprove if the {{w|axiom of choice}} is not taken as a part of the axiomatic system.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>The way, Ms Lenhart's proof refers to the act of doing math itself, is characteristic to metamathematical proofs, for example {{w|Gödel's incompleteness theorems}}, which, at first sight, may indeed look like black magic, even if in the end they must be a "perfectly sensible chain of <ins class="diffchange diffchange-inline">reasoning</ins>" like the rest of good mathematics. While standard mathematical theorems and their proofs deal with standard mathematical objects, like numbers, functions, points or lines, the metamathematical theorems treat other theorems as objects of interest. In this way you can propose and prove theorems about possibility of proving other theorems. For example, in 1931 {{w|Kurt Gödel}} was able to prove that any mathematical system based on arithmetics (that is using numbers) has statements that are true, but can be neither proved nor disproved. This kind of metamathematical reasoning is especially useful in the {{w|set theory}}, where many statements become impossible to prove and disprove if the {{w|axiom of choice}} is not taken as a part of the axiomatic system.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>In the title text the decision of whether to take the axiom of choice is made by a deterministic process, that is a process which future states can be developed with no randomness involved. It may be an allusion to the proposed {{w|axiom of determinacy}} of the set theory. It is, however, {{w|Axiom_of_determinacy#Incompatibility_of_the_axiom_of_determinacy_with_the_axiom_of_choice|incompatible}} with the axiom of choice, which builds another layer of the joke. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]].</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>In the title text the decision of whether to take the axiom of choice is made by a deterministic process, that is a process which future states can be developed with no randomness involved. It may be an allusion to the proposed {{w|axiom of determinacy}} of the set theory. It is, however, {{w|Axiom_of_determinacy#Incompatibility_of_the_axiom_of_determinacy_with_the_axiom_of_choice|incompatible}} with the axiom of choice, which builds another layer of the joke. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]].</div></td></tr>
</table>Hppavilion1//www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=125770&oldid=1257671724: Proofs2016-08-25T22:09:10Z<p><span dir="auto"><span class="autocomment">Explanation: </span> removed 'undergraduate,' no indication this is a graduate class</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>In the title text the decision of whether to take the axiom of choice is made by a deterministic process, that is a process which future states can be developed with no randomness involved. It may be an allusion to the proposed {{w|axiom of determinacy}} of the set theory. It is, however, {{w|Axiom_of_determinacy#Incompatibility_of_the_axiom_of_determinacy_with_the_axiom_of_choice|incompatible}} with the axiom of choice, which builds another layer of the joke. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]].</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>In the title text the decision of whether to take the axiom of choice is made by a deterministic process, that is a process which future states can be developed with no randomness involved. It may be an allusion to the proposed {{w|axiom of determinacy}} of the set theory. It is, however, {{w|Axiom_of_determinacy#Incompatibility_of_the_axiom_of_determinacy_with_the_axiom_of_choice|incompatible}} with the axiom of choice, which builds another layer of the joke. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]].</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Although Miss Lenhart did retire a year ago after [[1519: Venus]], she seems to have returned here for a math course at university level, but continues the trend she finished with in <del class="diffchange diffchange-inline">the </del>her <del class="diffchange diffchange-inline">undergraduate </del>class.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Although Miss Lenhart did retire a year ago after [[1519: Venus]], she seems to have returned here for a math course at university level, but continues the trend she finished with in her <ins class="diffchange diffchange-inline">prior </ins>class.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Transcript==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Transcript==</div></td></tr>
</table>Miamiclay//www.explainxkcd.com/wiki/index.php?title=925:_Cell_Phones&diff=125768&oldid=112833925: Cell Phones2016-08-25T16:44:45Z<p><span dir="auto"><span class="autocomment">Explanation: </span> </span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>This comic is a good explanation of the correlation/causation fallacy, where one party states two unrelated events and posits that they must have influenced each other.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>This comic is a good explanation of the correlation/causation fallacy, where one party states two unrelated events and posits that they must have influenced each other.</div></td></tr>
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<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>After hearing about the "Cell Phones Don't Cause Cancer" study, which refutes a claim made by the World Health Organization (just Google the debate, the comic doesn't focus much on it), [[Black Hat]] plots "Total Cancer Incidence" per 100,000 and "Cell Phone Users" per 100 on the same graph. The graph in frame 3 shows an exponential rise in cancer followed by an exponential rise in cell phone usage, which makes Black Hat comically come to the conclusion that cancer causes cell phones.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>After hearing about the "Cell Phones Don't Cause Cancer" study, which refutes a claim made by the <ins class="diffchange diffchange-inline">''{{w|</ins>World Health Organization<ins class="diffchange diffchange-inline">}}'' </ins>(just Google the debate, the comic doesn't focus much on it), [[Black Hat]] plots "Total Cancer Incidence" per 100,000 and "Cell Phone Users" per 100 on the same graph. The graph in frame 3 shows an exponential rise in cancer followed by an exponential rise in cell phone usage, which makes Black Hat comically come to the conclusion that cancer causes cell phones.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The comic highlights a well-known fallacy known as ''{{w|post hoc ergo propter hoc}}'', often shortened to simply ''post hoc.'' The Latin translates to "after this, therefore because of this," referring to the common mistake that because two events happen in chronological order, the former event must have caused the latter event. The fallacy is often the root cause of many superstitions (e.g., a person noticing he/she wore a special bracelet before getting a good test score thinks the bracelet was the source of his/her good fortune when it was more likely to be her socks.), but it often crosses into more serious areas of thinking. In this case, the scientific research community, which often prides itself on its intellectual aptitude, is gently mocked for being nonetheless prone to such poor reasoning all too often. The different possibilities are generally known as causation, when one thing is proven to cause another, or correlation, when changes in one thing are aligned with changes in another, but there is no proof that they are actually related.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The comic highlights a well-known fallacy known as ''{{w|post hoc ergo propter hoc}}'', often shortened to simply ''post hoc.'' The Latin translates to "after this, therefore because of this," referring to the common mistake that because two events happen in chronological order, the former event must have caused the latter event. The fallacy is often the root cause of many superstitions (e.g., a person noticing he/she wore a special bracelet before getting a good test score thinks the bracelet was the source of his/her good fortune when it was more likely to be her socks.), but it often crosses into more serious areas of thinking. In this case, the scientific research community, which often prides itself on its intellectual aptitude, is gently mocked for being nonetheless prone to such poor reasoning all too often. The different possibilities are generally known as causation, when one thing is proven to cause another, or correlation, when changes in one thing are aligned with changes in another, but there is no proof that they are actually related.</div></td></tr>
</table>162.158.85.123//www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=125767&oldid=1257531724: Proofs2016-08-25T15:29:28Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Explanation==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Explanation==</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">{{incomplete|More on the match, especially the title text.}}</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>[[Miss Lenhart]] is back teaching a math class. She begins a proof when one of her students ([[Cueball]]) interrupts her asking if this is one of those <ins class="diffchange diffchange-inline">dark-magic </ins>(<ins class="diffchange diffchange-inline">unclear, incomprehensible</ins>) proofs. She says no, but it soon turns out that it is; Cueball exclaims that he just knew it would be.</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>[[Miss Lenhart]] is back teaching a math class. She begins a proof when one of her students ([[Cueball]]) interrupts her asking if this is one of those <del class="diffchange diffchange-inline">{{w|Magic_</del>(<del class="diffchange diffchange-inline">programming</del>)<del class="diffchange diffchange-inline">#Variants|Dark Magic}} </del>proofs. She says no, but it soon turns out that it is; Cueball exclaims that he just knew it would be.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">If this actually refers to the </del>proof <del class="diffchange diffchange-inline">being magical</del>, <del class="diffchange diffchange-inline">or just </del>to the <del class="diffchange diffchange-inline">fact </del>that <del class="diffchange diffchange-inline">many students often feel like the resulting proof just appeared without any reason</del>, <del class="diffchange diffchange-inline">i.e</del>. <del class="diffchange diffchange-inline">either the teacher did not do it clearly</del>, <del class="diffchange diffchange-inline">or the student </del>is <del class="diffchange diffchange-inline">not up to the task of understanding proofs of </del>that <del class="diffchange diffchange-inline">complexity</del>, is <del class="diffchange diffchange-inline">not clear</del>.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">The </ins>proof <ins class="diffchange diffchange-inline">she starts setting up resembles a {{w|proof by contradiction}}. This kind of proof assumes that a particular theorem is true</ins>, <ins class="diffchange diffchange-inline">and shows that this assumption leads </ins>to <ins class="diffchange diffchange-inline">a contradiction, which disproves </ins>the <ins class="diffchange diffchange-inline">initial assumption. For example assumption that √2 is a {{w|rational number}} means </ins>that, <ins class="diffchange diffchange-inline">for some natural ''a'' and ''b'', √2=''a/b'', where ''a/b'' is an {{w|irreducible fraction}}</ins>. <ins class="diffchange diffchange-inline">Yet</ins>, <ins class="diffchange diffchange-inline">multiplying this equation by itself, we get 2=''a²/b²'', which means that ''a'' </ins>is <ins class="diffchange diffchange-inline">an even number. This means, </ins>that <ins class="diffchange diffchange-inline">''a=2k'' and ''2b²=(2k)²=4k²''</ins>, <ins class="diffchange diffchange-inline">so ''b'' must be even too. But if both ''a'' and ''b'' are even, ''a/b'' cannot be irreducible. Contradiction means that the initial assumption </ins>is <ins class="diffchange diffchange-inline">false, and √2 cannot be a rational number</ins>.</div></td></tr>
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<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>The proof <del class="diffchange diffchange-inline">she starts setting up resembles a </del>{{w|<del class="diffchange diffchange-inline">proof by contradiction</del>}}<del class="diffchange diffchange-inline">. These often involve making an assumption that there exists some formula or figure that fulfills the requirements given and plucking that answer out of abstract mathematics</del>, <del class="diffchange diffchange-inline">much </del>like <del class="diffchange diffchange-inline">summoning of demons is associated with </del>black magic<del class="diffchange diffchange-inline">. This is usually done by relying on knowledge </del>of the <del class="diffchange diffchange-inline">constraints </del>of <del class="diffchange diffchange-inline">the form (for example</del>, <del class="diffchange diffchange-inline">having </del>the <del class="diffchange diffchange-inline">square root </del>of <del class="diffchange diffchange-inline">2 be ''a/b'' where ''a'' </del>and <del class="diffchange diffchange-inline">''b'' are both integers and have no common factors when </del>proving <del class="diffchange diffchange-inline">that the square root of 2 is irrational)</del>. <del class="diffchange diffchange-inline">This common usage is then shown to be not the case </del>in <del class="diffchange diffchange-inline">the comic as the proof then goes </del>to <del class="diffchange diffchange-inline">claim </del>that <del class="diffchange diffchange-inline">the answer will be written in a specific place </del>(<del class="diffchange diffchange-inline">though this could be taken as indicating </del>that <del class="diffchange diffchange-inline">the result </del>is <del class="diffchange diffchange-inline">finite or </del>has <del class="diffchange diffchange-inline">a simple algorithm for continuing it). This may also be a reference to proof by induction</del>, <del class="diffchange diffchange-inline">which </del>can be <del class="diffchange diffchange-inline">thought </del>of as a <del class="diffchange diffchange-inline">proof </del>of the <del class="diffchange diffchange-inline">existence of an infinite number of more specific proofs</del>.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>The <ins class="diffchange diffchange-inline">way, Ms Lenhart's </ins>proof <ins class="diffchange diffchange-inline">refers to the act of doing math itself, is characteristic to metamathematical proofs, for example </ins>{{w|<ins class="diffchange diffchange-inline">Gödel's incompleteness theorems</ins>}}, <ins class="diffchange diffchange-inline">which, at first sight, may indeed look </ins>like black magic<ins class="diffchange diffchange-inline">, even if in the end they must be a "perfectly sensible chain </ins>of <ins class="diffchange diffchange-inline">resoning" like </ins>the <ins class="diffchange diffchange-inline">rest </ins>of <ins class="diffchange diffchange-inline">good mathematics. While standard mathematical theorems and their proofs deal with standard mathematical objects, like numbers, functions, points or lines</ins>, the <ins class="diffchange diffchange-inline">metamathematical theorems treat other theorems as objects </ins>of <ins class="diffchange diffchange-inline">interest. In this way you can propose </ins>and <ins class="diffchange diffchange-inline">prove theorems about possibility of </ins>proving <ins class="diffchange diffchange-inline">other theorems</ins>. <ins class="diffchange diffchange-inline">For example, </ins>in <ins class="diffchange diffchange-inline">1931 {{w|Kurt Gödel}} was able </ins>to <ins class="diffchange diffchange-inline">prove </ins>that <ins class="diffchange diffchange-inline">any mathematical system based on arithmetics </ins>(that is <ins class="diffchange diffchange-inline">using numbers) </ins>has <ins class="diffchange diffchange-inline">statements that are true</ins>, <ins class="diffchange diffchange-inline">but </ins>can be <ins class="diffchange diffchange-inline">neither proved nor disproved. This kind </ins>of <ins class="diffchange diffchange-inline">metamathematical reasoning is especially useful in the {{w|set theory}}, where many statements become impossible to prove and disprove if the {{w|axiom of choice}} is not taken </ins>as a <ins class="diffchange diffchange-inline">part </ins>of the <ins class="diffchange diffchange-inline">axiomatic system</ins>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>In the title text the decision of whether to take the <del class="diffchange diffchange-inline">{{w|</del>axiom of choice<del class="diffchange diffchange-inline">}} </del>is made by a deterministic process. <del class="diffchange diffchange-inline">The </del>{{w|axiom of determinacy}} is {{w|Axiom_of_determinacy#Incompatibility_of_the_axiom_of_determinacy_with_the_axiom_of_choice|incompatible}} with the <del class="diffchange diffchange-inline">{{w|</del>axiom of choice<del class="diffchange diffchange-inline">}}</del>, which <del class="diffchange diffchange-inline">is the continuation </del>of the joke <del class="diffchange diffchange-inline">of these dark magic proofs</del>. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]].</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>In the title text the decision of whether to take the axiom of choice is made by a deterministic process<ins class="diffchange diffchange-inline">, that is a process which future states can be developed with no randomness involved</ins>. <ins class="diffchange diffchange-inline">It may be an allusion to the proposed </ins>{{w|axiom of determinacy}} <ins class="diffchange diffchange-inline">of the set theory. It </ins>is<ins class="diffchange diffchange-inline">, however, </ins>{{w|Axiom_of_determinacy#Incompatibility_of_the_axiom_of_determinacy_with_the_axiom_of_choice|incompatible}} with the axiom of choice, which <ins class="diffchange diffchange-inline">builds another layer </ins>of the joke. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]].</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Although Miss Lenhart did retire a year ago after [[1519: Venus]], she seems to have returned here for a math course at university level, but continues the trend she finished with in the her undergraduate class.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Although Miss Lenhart did retire a year ago after [[1519: Venus]], she seems to have returned here for a math course at university level, but continues the trend she finished with in the her undergraduate class.</div></td></tr>
</table>162.158.133.138//www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=125753&oldid=1257471724: Proofs2016-08-25T11:29:31Z<p><span dir="auto"><span class="autocomment">Explanation: </span> </span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>In the title text the decision of whether to take the {{w|axiom of choice}} is made by a deterministic process. The {{w|axiom of determinacy}} is {{w|Axiom_of_determinacy#Incompatibility_of_the_axiom_of_determinacy_with_the_axiom_of_choice|incompatible}} with the {{w|axiom of choice}}, which is the continuation of the joke of these dark magic proofs. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]].</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>In the title text the decision of whether to take the {{w|axiom of choice}} is made by a deterministic process. The {{w|axiom of determinacy}} is {{w|Axiom_of_determinacy#Incompatibility_of_the_axiom_of_determinacy_with_the_axiom_of_choice|incompatible}} with the {{w|axiom of choice}}, which is the continuation of the joke of these dark magic proofs. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]].</div></td></tr>
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<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Although Miss Lenhart did retire a year ago after [[1519: Venus]], she seems to have returned here for a math course at university level, but continues the trend she finished with in the her undergraduate class<del class="diffchange diffchange-inline">..</del>.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Although Miss Lenhart did retire a year ago after [[1519: Venus]], she seems to have returned here for a math course at university level, but continues the trend she finished with in the her undergraduate class.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Transcript==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Transcript==</div></td></tr>
</table>RChandra//www.explainxkcd.com/wiki/index.php?title=804:_Pumpkin_Carving&diff=125749&oldid=116870804: Pumpkin Carving2016-08-25T06:36:39Z<p><span dir="auto"><span class="autocomment">Explanation: </span> </span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>In the 4th frame, [[Cueball]] is referencing the {{w|Banach-Tarski paradox}}, a theorem which states that it is possible to carve a three-dimensional ball, in this case a pumpkin, into a finite number of "pieces," and then reassemble the "pieces" into two different balls identical to the original. This paradox has been proven for just about anything theoretically, but requires infinitely complicated pieces, which are impossible for anything made of physical {{w|atomic theory|atoms}} rather than mathematical {{w|point (geometry)|points}}. The person off-screen in that frame references the {{w|Axiom of Choice}}, which says that given a set of buckets or bins, each containing one or more objects, it is possible to select exactly one object from each bucket. The Banach-Tarski rests on several axioms which are fairly well respected, but also requires the Axiom of Choice to work correctly. So a person who does not believe in the Axiom of Choice would not have been able to do what [[Cueball]] managed to do.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>In the 4th frame, [[Cueball]] is referencing the {{w|Banach-Tarski paradox}}, a theorem which states that it is possible to carve a three-dimensional ball, in this case a pumpkin, into a finite number of "pieces," and then reassemble the "pieces" into two different balls identical to the original. This paradox has been proven for just about anything theoretically, but requires infinitely complicated pieces, which are impossible for anything made of physical {{w|atomic theory|atoms}} rather than mathematical {{w|point (geometry)|points}}. The person off-screen in that frame references the {{w|Axiom of Choice}}, which says that given a set of buckets or bins, each containing one or more objects, it is possible to select exactly one object from each bucket. The Banach-Tarski rests on several axioms which are fairly well respected, but also requires the Axiom of Choice to work correctly. So a person who does not believe in the Axiom of Choice would not have been able to do what [[Cueball]] managed to do.</div></td></tr>
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<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>The title text says that {{w|Solomon|King Solomon}} developed the Banach-Tarski theorem first. This is a reference to the story of two women being brought before him. Both were arguing that a particular child was their own. Solomon said that the solution was to cut the baby in half and give each woman one of the halves. One of the two women said that the other should have the baby whole. Solomon then knew she was the true mother, and gave her the child. The joke is that Solomon may <del class="diffchange diffchange-inline">not </del>have intended to <del class="diffchange diffchange-inline">kill </del>the child, but, believing that two whole children could be made from the one, intended give a baby to each woman, and the Banach-Tarski paradox states that, were the baby infinitely divisible, it should <del class="diffchange diffchange-inline">be </del>possible.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>The title text says that {{w|Solomon|King Solomon}} developed the Banach-Tarski theorem first. This is a reference to the story of two women being brought before him. Both were arguing that a particular child was their own. Solomon said that the solution was to cut the baby in half and give each woman one of the halves. One of the two women said that the other should have the baby whole. Solomon then knew she was the true mother, and gave her the child. The joke is that Solomon may have <ins class="diffchange diffchange-inline">actually </ins>intended to <ins class="diffchange diffchange-inline">cut </ins>the child, but, believing that two whole children could be made from the one, intended <ins class="diffchange diffchange-inline">to </ins>give a baby to each woman, and the Banach-Tarski paradox states that, were the baby infinitely divisible, it should <ins class="diffchange diffchange-inline">have been </ins>possible.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Transcript==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Transcript==</div></td></tr>
</table>162.158.202.150//www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=125747&oldid=1257461724: Proofs2016-08-25T03:07:42Z<p>Comment about proof by induction.</p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>If this actually refers to the proof being magical, or just to the fact that many students often feel like the resulting proof just appeared without any reason, i.e. either the teacher did not do it clearly, or the student is not up to the task of understanding proofs of that complexity, is not clear.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>If this actually refers to the proof being magical, or just to the fact that many students often feel like the resulting proof just appeared without any reason, i.e. either the teacher did not do it clearly, or the student is not up to the task of understanding proofs of that complexity, is not clear.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>The proof she starts setting up resembles a {{w|proof by contradiction}}. These often involve making an assumption that there exists some formula or figure that fulfills the requirements given and plucking that answer out of abstract mathematics, much like summoning of demons is associated with black magic. This is usually done by relying on knowledge of the constraints of the form (for example, having the square root of 2 be ''a/b'' where ''a'' and ''b'' are both integers and have no common factors when proving that the square root of 2 is irrational). This common usage is then shown to be not the case in the comic as the proof then goes to claim that the answer will be written in a specific place (though this could be taken as indicating that the result is finite or has a simple algorithm for continuing it).</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>The proof she starts setting up resembles a {{w|proof by contradiction}}. These often involve making an assumption that there exists some formula or figure that fulfills the requirements given and plucking that answer out of abstract mathematics, much like summoning of demons is associated with black magic. This is usually done by relying on knowledge of the constraints of the form (for example, having the square root of 2 be ''a/b'' where ''a'' and ''b'' are both integers and have no common factors when proving that the square root of 2 is irrational). This common usage is then shown to be not the case in the comic as the proof then goes to claim that the answer will be written in a specific place (though this could be taken as indicating that the result is finite or has a simple algorithm for continuing it)<ins class="diffchange diffchange-inline">. This may also be a reference to proof by induction, which can be thought of as a proof of the existence of an infinite number of more specific proofs</ins>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>In the title text the decision of whether to take the {{w|axiom of choice}} is made by a deterministic process. The {{w|axiom of determinacy}} is {{w|Axiom_of_determinacy#Incompatibility_of_the_axiom_of_determinacy_with_the_axiom_of_choice|incompatible}} with the {{w|axiom of choice}}, which is the continuation of the joke of these dark magic proofs. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]].</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>In the title text the decision of whether to take the {{w|axiom of choice}} is made by a deterministic process. The {{w|axiom of determinacy}} is {{w|Axiom_of_determinacy#Incompatibility_of_the_axiom_of_determinacy_with_the_axiom_of_choice|incompatible}} with the {{w|axiom of choice}}, which is the continuation of the joke of these dark magic proofs. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]].</div></td></tr>
</table>108.162.245.109//www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=125746&oldid=1257281724: Proofs2016-08-25T00:42:53Z<p>italics</p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>If this actually refers to the proof being magical, or just to the fact that many students often feel like the resulting proof just appeared without any reason, i.e. either the teacher did not do it clearly, or the student is not up to the task of understanding proofs of that complexity, is not clear.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>If this actually refers to the proof being magical, or just to the fact that many students often feel like the resulting proof just appeared without any reason, i.e. either the teacher did not do it clearly, or the student is not up to the task of understanding proofs of that complexity, is not clear.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>The proof she starts setting up resembles a {{w|proof by contradiction}}. These often involve making an assumption that there exists some formula or figure that fulfills the requirements given and plucking that answer out of abstract mathematics, much like summoning of demons is associated with black magic. This is usually done by relying on knowledge of the constraints of the form (for example, having the square root of 2 be a/b where a and b are both integers and have no common factors when proving that the square root of 2 is irrational). This common usage is then shown to be not the case in the comic as the proof then goes to claim that the answer will be written in a specific place (though this could be taken as indicating that the result is finite or has a simple algorithm for continuing it).</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>The proof she starts setting up resembles a {{w|proof by contradiction}}. These often involve making an assumption that there exists some formula or figure that fulfills the requirements given and plucking that answer out of abstract mathematics, much like summoning of demons is associated with black magic. This is usually done by relying on knowledge of the constraints of the form (for example, having the square root of 2 be <ins class="diffchange diffchange-inline">''</ins>a/b<ins class="diffchange diffchange-inline">'' </ins>where <ins class="diffchange diffchange-inline">''</ins>a<ins class="diffchange diffchange-inline">'' </ins>and <ins class="diffchange diffchange-inline">''</ins>b<ins class="diffchange diffchange-inline">'' </ins>are both integers and have no common factors when proving that the square root of 2 is irrational). This common usage is then shown to be not the case in the comic as the proof then goes to claim that the answer will be written in a specific place (though this could be taken as indicating that the result is finite or has a simple algorithm for continuing it).</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>In the title text the decision of whether to take the {{w|axiom of choice}} is made by a deterministic process. The {{w|axiom of determinacy}} is {{w|Axiom_of_determinacy#Incompatibility_of_the_axiom_of_determinacy_with_the_axiom_of_choice|incompatible}} with the {{w|axiom of choice}}, which is the continuation of the joke of these dark magic proofs. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]].</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>In the title text the decision of whether to take the {{w|axiom of choice}} is made by a deterministic process. The {{w|axiom of determinacy}} is {{w|Axiom_of_determinacy#Incompatibility_of_the_axiom_of_determinacy_with_the_axiom_of_choice|incompatible}} with the {{w|axiom of choice}}, which is the continuation of the joke of these dark magic proofs. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]].</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Transcript==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Transcript==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:[Miss Lenhart is standing facing left in front of a whiteboard writing on it. Eleven left aligned lines of writing is shown as unreadable scribbles. A voice interrupts her from off-panel right.]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:[Miss Lenhart is standing facing left in front of a whiteboard writing on it. Eleven left aligned lines of writing is shown as unreadable scribbles. A voice interrupts her from off-panel right.]</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>:Miss Lenhart: ... Let's assume there exists some function ''F''(a,b,c...) which produces the correct answer-</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>:Miss Lenhart: ... Let's assume there exists some function ''F''(<ins class="diffchange diffchange-inline">''</ins>a,b,c<ins class="diffchange diffchange-inline">''</ins>...) which produces the correct answer-</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:Cueball (off-panel): Hang on.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:Cueball (off-panel): Hang on.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:[Miss Lenhart is facing the whiteboard again writing more scribbles behind some of the lines from before (the first line has disappeared). The lines that have more text added are now number three and five (four and six before). Cueball again speaks off-panel.]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:[Miss Lenhart is facing the whiteboard again writing more scribbles behind some of the lines from before (the first line has disappeared). The lines that have more text added are now number three and five (four and six before). Cueball again speaks off-panel.]</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>:Miss Lenhart: Now, let's assume that the correct answer will eventually be written on the board at the coordinates (x, y). If <del class="diffchange diffchange-inline">we-</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>:Miss Lenhart: Now, let's assume that the correct answer will eventually be written on the board at the coordinates (<ins class="diffchange diffchange-inline">''</ins>x, y<ins class="diffchange diffchange-inline">''</ins>). If <ins class="diffchange diffchange-inline">we—</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:Cueball (off-panel): I ''knew'' it!</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:Cueball (off-panel): I ''knew'' it!</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
</table>Wwoods//www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=125728&oldid=1257271724: Proofs2016-08-24T16:16:55Z<p><span dir="auto"><span class="autocomment">Explanation: </span> rewording</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Explanation==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Explanation==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{{incomplete|More on the match, especially the title text.}}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{{incomplete|More on the match, especially the title text.}}</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>[[Miss Lenhart]] is back teaching a math class. She begins a proof when one of her students ([[Cueball]]) interrupts her asking if this is one of those {{w|Magic_(programming)#Variants|Dark Magic}} proofs. She <del class="diffchange diffchange-inline">denies that </del>but it soon turns out that it <del class="diffchange diffchange-inline">will be, and </del>Cueball exclaims that he just knew it would be <del class="diffchange diffchange-inline">one of those</del>.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>[[Miss Lenhart]] is back teaching a math class. She begins a proof when one of her students ([[Cueball]]) interrupts her asking if this is one of those {{w|Magic_(programming)#Variants|Dark Magic}} proofs. She <ins class="diffchange diffchange-inline">says no, </ins>but it soon turns out that it <ins class="diffchange diffchange-inline">is; </ins>Cueball exclaims that he just knew it would be.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>If this actually refers to the proof being magical, or just to the fact that many students often feel like the resulting proof just appeared without any reason, i.e. either the teacher did not do it clearly, or the student is not up to the task of understanding proofs of that complexity, is not clear.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>If this actually refers to the proof being magical, or just to the fact that many students often feel like the resulting proof just appeared without any reason, i.e. either the teacher did not do it clearly, or the student is not up to the task of understanding proofs of that complexity, is not clear.</div></td></tr>
</table>Garik//www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=125727&oldid=1257221724: Proofs2016-08-24T15:59:15Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Explanation==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Explanation==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{{incomplete|More on the match, especially the title text.}}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{{incomplete|More on the match, especially the title text.}}</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>[[Miss Lenhart]] is back teaching a math class. She begins a proof when one of her students ([[Cueball]]) interrupts her asking if this is one of those {{w|Dark Magic}} proofs. She denies that but it soon turns out that it will be, and Cueball exclaims that he just knew it would be one of those.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>[[Miss Lenhart]] is back teaching a math class. She begins a proof when one of her students ([[Cueball]]) interrupts her asking if this is one of those {{w<ins class="diffchange diffchange-inline">|Magic_(programming)#Variants</ins>|Dark Magic}} proofs. She denies that but it soon turns out that it will be, and Cueball exclaims that he just knew it would be one of those.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>If this actually refers to the proof being magical, or just to the fact that many students often feel like the resulting proof just appeared without any reason, i.e. either the teacher did not do it clearly, or the student is not up to the task of understanding proofs of that complexity, is not clear.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>If this actually refers to the proof being magical, or just to the fact that many students often feel like the resulting proof just appeared without any reason, i.e. either the teacher did not do it clearly, or the student is not up to the task of understanding proofs of that complexity, is not clear.</div></td></tr>
</table>Zorlax the Mighty//www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=125722&oldid=1257191724: Proofs2016-08-24T15:03:03Z<p><span dir="auto"><span class="autocomment">Transcript: </span> </span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:[Miss Lenhart is facing the whiteboard again writing more scribbles behind some of the lines from before (the first line has disappeared). The lines that have more text added are now number three and five (four and six before). Cueball again speaks off-panel.]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:[Miss Lenhart is facing the whiteboard again writing more scribbles behind some of the lines from before (the first line has disappeared). The lines that have more text added are now number three and five (four and six before). Cueball again speaks off-panel.]</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>:Miss Lenhart: Now, let's assume that the correct answer will eventually be written on the board at (x, y). If we-</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>:Miss Lenhart: Now, let's assume that the correct answer will eventually be written on the board at <ins class="diffchange diffchange-inline">the coordinates </ins>(x, y). If we-</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:Cueball (off-panel): I ''knew'' it!</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:Cueball (off-panel): I ''knew'' it!</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
</table>N0lqu//www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=125719&oldid=1257181724: Proofs2016-08-24T12:54:18Z<p><span dir="auto"><span class="autocomment">Explanation: </span> </span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The proof she starts setting up resembles a {{w|proof by contradiction}}. These often involve making an assumption that there exists some formula or figure that fulfills the requirements given and plucking that answer out of abstract mathematics, much like summoning of demons is associated with black magic. This is usually done by relying on knowledge of the constraints of the form (for example, having the square root of 2 be a/b where a and b are both integers and have no common factors when proving that the square root of 2 is irrational). This common usage is then shown to be not the case in the comic as the proof then goes to claim that the answer will be written in a specific place (though this could be taken as indicating that the result is finite or has a simple algorithm for continuing it).</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The proof she starts setting up resembles a {{w|proof by contradiction}}. These often involve making an assumption that there exists some formula or figure that fulfills the requirements given and plucking that answer out of abstract mathematics, much like summoning of demons is associated with black magic. This is usually done by relying on knowledge of the constraints of the form (for example, having the square root of 2 be a/b where a and b are both integers and have no common factors when proving that the square root of 2 is irrational). This common usage is then shown to be not the case in the comic as the proof then goes to claim that the answer will be written in a specific place (though this could be taken as indicating that the result is finite or has a simple algorithm for continuing it).</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>In the title text the decision of whether to take the {{w|axiom of choice}} is made by a deterministic process. The {{w|axiom of determinacy}} is {{w|Axiom_of_determinacy#Incompatibility_of_the_axiom_of_determinacy_with_the_axiom_of_choice|<del class="diffchange diffchange-inline">incompatibility</del>}} with the {{w|axiom of choice}}, which is the continuation of the joke of these dark magic proofs. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]].</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>In the title text the decision of whether to take the {{w|axiom of choice}} is made by a deterministic process. The {{w|axiom of determinacy}} is {{w|Axiom_of_determinacy#Incompatibility_of_the_axiom_of_determinacy_with_the_axiom_of_choice|<ins class="diffchange diffchange-inline">incompatible</ins>}} with the {{w|axiom of choice}}, which is the continuation of the joke of these dark magic proofs. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]].</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Although Miss Lenhart did retire a year ago after [[1519: Venus]], she seems to have returned here for a math course at university level, but continues the trend she finished with in the her undergraduate class...</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Although Miss Lenhart did retire a year ago after [[1519: Venus]], she seems to have returned here for a math course at university level, but continues the trend she finished with in the her undergraduate class...</div></td></tr>
</table>J-beda//www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=125718&oldid=1257111724: Proofs2016-08-24T12:51:11Z<p>Axiom of choice</p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The proof she starts setting up resembles a {{w|proof by contradiction}}. These often involve making an assumption that there exists some formula or figure that fulfills the requirements given and plucking that answer out of abstract mathematics, much like summoning of demons is associated with black magic. This is usually done by relying on knowledge of the constraints of the form (for example, having the square root of 2 be a/b where a and b are both integers and have no common factors when proving that the square root of 2 is irrational). This common usage is then shown to be not the case in the comic as the proof then goes to claim that the answer will be written in a specific place (though this could be taken as indicating that the result is finite or has a simple algorithm for continuing it).</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The proof she starts setting up resembles a {{w|proof by contradiction}}. These often involve making an assumption that there exists some formula or figure that fulfills the requirements given and plucking that answer out of abstract mathematics, much like summoning of demons is associated with black magic. This is usually done by relying on knowledge of the constraints of the form (for example, having the square root of 2 be a/b where a and b are both integers and have no common factors when proving that the square root of 2 is irrational). This common usage is then shown to be not the case in the comic as the proof then goes to claim that the answer will be written in a specific place (though this could be taken as indicating that the result is finite or has a simple algorithm for continuing it).</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>In the title text the decision of whether to take the {{w|axiom of choice}} is made by a deterministic process. The {{w|axiom of determinacy}} is {{w|Axiom_of_determinacy#Incompatibility_of_the_axiom_of_determinacy_with_the_axiom_of_choice|incompatibility with the axiom of choice}}, which is the continuation of the joke of these dark magic proofs.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>In the title text the decision of whether to take the {{w|axiom of choice}} is made by a deterministic process. The {{w|axiom of determinacy}} is {{w|Axiom_of_determinacy#Incompatibility_of_the_axiom_of_determinacy_with_the_axiom_of_choice|incompatibility<ins class="diffchange diffchange-inline">}} </ins>with the <ins class="diffchange diffchange-inline">{{w|</ins>axiom of choice}}, which is the continuation of the joke of these dark magic proofs<ins class="diffchange diffchange-inline">. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]]</ins>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Although Miss Lenhart did retire a year ago after [[1519: Venus]], she seems to have returned here for a math course at university level, but continues the trend she finished with in the her undergraduate class...</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Although Miss Lenhart did retire a year ago after [[1519: Venus]], she seems to have returned here for a math course at university level, but continues the trend she finished with in the her undergraduate class...</div></td></tr>
</table>Condor70//www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=125711&oldid=1257101724: Proofs2016-08-24T12:12:03Z<p><span dir="auto"><span class="autocomment">Explanation: </span> spelling</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Explanation==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Explanation==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{{incomplete|More on the match, especially the title text.}}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{{incomplete|More on the match, especially the title text.}}</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>[[Miss Lenhart]] is back teaching a math class. She begins a proof when one of her students ([[Cueball]]) interrupts her asking if this is one of those {{w|Dark Magic}} proofs. She <del class="diffchange diffchange-inline">denyies </del>that but it soon turns out that it will be, and Cueball exclaims that he just knew it would be one of those.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>[[Miss Lenhart]] is back teaching a math class. She begins a proof when one of her students ([[Cueball]]) interrupts her asking if this is one of those {{w|Dark Magic}} proofs. She <ins class="diffchange diffchange-inline">denies </ins>that but it soon turns out that it will be, and Cueball exclaims that he just knew it would be one of those.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>If this actually refers to the proof being magical, or just to the fact that many students often feel like the resulting proof just appeared without any reason, i.e. either the teacher did not do it clearly, or the student is not up to the task of understanding proofs of that complexity, is not clear.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>If this actually refers to the proof being magical, or just to the fact that many students often feel like the resulting proof just appeared without any reason, i.e. either the teacher did not do it clearly, or the student is not up to the task of understanding proofs of that complexity, is not clear.</div></td></tr>
</table>108.162.221.13//www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=125710&oldid=1256991724: Proofs2016-08-24T12:11:09Z<p><span dir="auto"><span class="autocomment">Explanation: </span> Decline means refuse. Here she is denying</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Explanation==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Explanation==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{{incomplete|More on the match, especially the title text.}}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{{incomplete|More on the match, especially the title text.}}</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>[[Miss Lenhart]] is back teaching a math class. She begins a proof when one of her students ([[Cueball]]) interrupts her asking if this is one of those {{w|Dark Magic}} proofs. She <del class="diffchange diffchange-inline">declines </del>that but it soon turns out that it will be, and Cueball exclaims that he just knew it would be one of those.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>[[Miss Lenhart]] is back teaching a math class. She begins a proof when one of her students ([[Cueball]]) interrupts her asking if this is one of those {{w|Dark Magic}} proofs. She <ins class="diffchange diffchange-inline">denyies </ins>that but it soon turns out that it will be, and Cueball exclaims that he just knew it would be one of those.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>If this actually refers to the proof being magical, or just to the fact that many students often feel like the resulting proof just appeared without any reason, i.e. either the teacher did not do it clearly, or the student is not up to the task of understanding proofs of that complexity, is not clear.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>If this actually refers to the proof being magical, or just to the fact that many students often feel like the resulting proof just appeared without any reason, i.e. either the teacher did not do it clearly, or the student is not up to the task of understanding proofs of that complexity, is not clear.</div></td></tr>
</table>173.245.50.29//www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=125699&oldid=1256921724: Proofs2016-08-24T11:47:39Z<p><span dir="auto"><span class="autocomment">Explanation: </span> typo</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Miss Lenhart]] is back teaching a math class. She begins a proof when one of her students ([[Cueball]]) interrupts her asking if this is one of those {{w|Dark Magic}} proofs. She declines that but it soon turns out that it will be, and Cueball exclaims that he just knew it would be one of those.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Miss Lenhart]] is back teaching a math class. She begins a proof when one of her students ([[Cueball]]) interrupts her asking if this is one of those {{w|Dark Magic}} proofs. She declines that but it soon turns out that it will be, and Cueball exclaims that he just knew it would be one of those.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>If this actually <del class="diffchange diffchange-inline">reefers </del>to the proof being magical, or just to the fact that many students often feel like the resulting proof just appeared without any reason, i.e. either the teacher did not do it clearly, or the student is not up to the task of understanding proofs of that complexity, is not clear.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>If this actually <ins class="diffchange diffchange-inline">refers </ins>to the proof being magical, or just to the fact that many students often feel like the resulting proof just appeared without any reason, i.e. either the teacher did not do it clearly, or the student is not up to the task of understanding proofs of that complexity, is not clear.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The proof she starts setting up resembles a {{w|proof by contradiction}}. These often involve making an assumption that there exists some formula or figure that fulfills the requirements given and plucking that answer out of abstract mathematics, much like summoning of demons is associated with black magic. This is usually done by relying on knowledge of the constraints of the form (for example, having the square root of 2 be a/b where a and b are both integers and have no common factors when proving that the square root of 2 is irrational). This common usage is then shown to be not the case in the comic as the proof then goes to claim that the answer will be written in a specific place (though this could be taken as indicating that the result is finite or has a simple algorithm for continuing it).</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The proof she starts setting up resembles a {{w|proof by contradiction}}. These often involve making an assumption that there exists some formula or figure that fulfills the requirements given and plucking that answer out of abstract mathematics, much like summoning of demons is associated with black magic. This is usually done by relying on knowledge of the constraints of the form (for example, having the square root of 2 be a/b where a and b are both integers and have no common factors when proving that the square root of 2 is irrational). This common usage is then shown to be not the case in the comic as the proof then goes to claim that the answer will be written in a specific place (though this could be taken as indicating that the result is finite or has a simple algorithm for continuing it).</div></td></tr>
</table>141.101.98.62//www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=125692&oldid=1256901724: Proofs2016-08-24T11:26:00Z<p><span dir="auto"><span class="autocomment">Explanation: </span> </span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Miss Lenhart]] is back teaching a math class. She begins a proof when one of her students ([[Cueball]]) interrupts her asking if this is one of those {{w|Dark Magic}} proofs. She declines that but it soon turns out that it will be, and Cueball exclaims that he just knew it would be one of those.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Miss Lenhart]] is back teaching a math class. She begins a proof when one of her students ([[Cueball]]) interrupts her asking if this is one of those {{w|Dark Magic}} proofs. She declines that but it soon turns out that it will be, and Cueball exclaims that he just knew it would be one of those.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>The proof she starts setting up resembles a {{w|<del class="diffchange diffchange-inline">Proof </del>by <del class="diffchange diffchange-inline">Contradiction</del>}}. These often involve making an assumption that there exists some formula or figure that fulfills the requirements given and plucking that answer out of abstract mathematics, much like summoning of demons is associated with black magic. This is usually done by relying on knowledge of the constraints of the form (for example, having the square root of 2 be a/b where a and b are both integers and have no common factors when proving that the square root of 2 is irrational). This common usage is then shown to be not the case in the comic as the proof then goes to claim that the answer will be written in a specific place (though this could be taken as indicating that the result is finite or has a simple algorithm for continuing it).</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">If this actually reefers to the proof being magical, or just to the fact that many students often feel like the resulting proof just appeared without any reason, i.e. either the teacher did not do it clearly, or the student is not up to the task of understanding proofs of that complexity, is not clear.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>The proof she starts setting up resembles a {{w|<ins class="diffchange diffchange-inline">proof </ins>by <ins class="diffchange diffchange-inline">contradiction</ins>}}. These often involve making an assumption that there exists some formula or figure that fulfills the requirements given and plucking that answer out of abstract mathematics, much like summoning of demons is associated with black magic. This is usually done by relying on knowledge of the constraints of the form (for example, having the square root of 2 be a/b where a and b are both integers and have no common factors when proving that the square root of 2 is irrational). This common usage is then shown to be not the case in the comic as the proof then goes to claim that the answer will be written in a specific place (though this could be taken as indicating that the result is finite or has a simple algorithm for continuing it).</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>In the title text the decision of whether to take the {{w|axiom of choice}} is made by a deterministic process. The {{w|axiom of determinacy}} is {{w|Axiom_of_determinacy#Incompatibility_of_the_axiom_of_determinacy_with_the_axiom_of_choice|incompatibility with the axiom of choice}}, which is the continuation of the joke of these dark magic proofs.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>In the title text the decision of whether to take the {{w|axiom of choice}} is made by a deterministic process. The {{w|axiom of determinacy}} is {{w|Axiom_of_determinacy#Incompatibility_of_the_axiom_of_determinacy_with_the_axiom_of_choice|incompatibility with the axiom of choice}}, which is the continuation of the joke of these dark magic proofs.</div></td></tr>
</table>Kynde