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Proofs
Next, let's assume the decision of whether to take the Axiom of Choice is made by a deterministic process ...
Title text: Next, let's assume the decision of whether to take the Axiom of Choice is made by a deterministic process ...

Explanation

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.

The proof she starts setting up resembles a 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 rational number means that, for some natural a and b, √2=a/b, where a/b is an irreducible fraction. Yet, multiplying this equation by itself, we get 2=a²/b² which in turn rearranges to 2=, therefor 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 =2, 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.

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.

The way, Ms Lenhart's proof refers to the act of doing math itself, is characteristic to metamathematical proofs, for example 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 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 set theory, where many statements become impossible to prove and disprove if the axiom of choice is not taken as a part of the axiomatic system.

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 axiom of determinacy of the set theory. It is, however, incompatible with the axiom of choice, which builds another layer of the joke. The axiom of choice was mentioned earlier in 804: Pumpkin Carving.

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.

Transcript

[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.]
Miss Lenhart: ... Let's assume there exists some function F(a,b,c...) which produces the correct answer-
Cueball (off-panel): Hang on.
[In a frame-less panel Cueball is sitting on a chair at a desk with a pen in his hand taking notes.]
Cueball: This is going to be one of those weird, dark magic proofs, isn't it? I can tell.
[Miss Lenhart has turned right towards Cueball, who is again speaking off-panel. The white board is also off-panel.]
Miss Lenhart: What? No, no, it's a perfectly sensible chain of reasoning.
Cueball (off-panel): All right...
[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.]
Miss Lenhart: Now, let's assume that the correct answer will eventually be written on the board at the coordinates (x, y). If we—
Cueball (off-panel): I knew it!


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