Editing Talk:2610: Assigning Numbers
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:::I'm not entirely sure what you mean by "paradox"; to my knowledge, that word doesn't have a formal mathematical definition. I assume you mean a non-true non-false statement? (feel free to correct me) In which case, Gödel did not consider this because he was working within classical logic, wherein statements can either be "true" or "false" and there is no third value. The reason he chose classical logic is because mathematics is currently performed using classical logic. And although most proofs of "the Gödel sentence is true" are a bit wishy-woshy, you can actually formalise a proof within ZFC set theory (a theory based on classical logic) that the Gödel sentence is true for the standard natural numbers (see my comment above). Of course, you could reject ZFC (and base mathematics on something like [https://en.wikipedia.org/wiki/Paraconsistent_logic paraconsistent logic]) but you'll probably have a hard time convincing mathematicians. Regardless, was more concerned with the incompleteness of the system than with the truth of the Gödel sentence, and doesn't mention truth at all in Theorem VI (the First Incompleteness Theorem) of his original paper.--[[User:Underbase|Underbase]] ([[User talk:Underbase|talk]]) 10:43, 28 April 2022 (UTC) | :::I'm not entirely sure what you mean by "paradox"; to my knowledge, that word doesn't have a formal mathematical definition. I assume you mean a non-true non-false statement? (feel free to correct me) In which case, Gödel did not consider this because he was working within classical logic, wherein statements can either be "true" or "false" and there is no third value. The reason he chose classical logic is because mathematics is currently performed using classical logic. And although most proofs of "the Gödel sentence is true" are a bit wishy-woshy, you can actually formalise a proof within ZFC set theory (a theory based on classical logic) that the Gödel sentence is true for the standard natural numbers (see my comment above). Of course, you could reject ZFC (and base mathematics on something like [https://en.wikipedia.org/wiki/Paraconsistent_logic paraconsistent logic]) but you'll probably have a hard time convincing mathematicians. Regardless, was more concerned with the incompleteness of the system than with the truth of the Gödel sentence, and doesn't mention truth at all in Theorem VI (the First Incompleteness Theorem) of his original paper.--[[User:Underbase|Underbase]] ([[User talk:Underbase|talk]]) 10:43, 28 April 2022 (UTC) | ||
::::I won't argue with that. (I'll also back off to "non-true non-false," since I'm unsure how to understand other definitions.). "Incompleteness" (rather than "inconsistency") is still the missing piece. One claim in the above explanation: "David Hilbert's famous proclamation "We must know, we will know" is simply incorrect," Ignores this qualification -- making it a misapplication of what Gödel actually proved. Maybe we can eventually know truth -- but the limited tools constituting Gödel's proof were simply not up to that task.--[[Special:Contributions/172.69.33.83|172.69.33.83]] 20:04, 28 April 2022 (UTC) -edited --[[Special:Contributions/172.70.214.81|172.70.214.81]] 21:26, 28 April 2022 (UTC) | ::::I won't argue with that. (I'll also back off to "non-true non-false," since I'm unsure how to understand other definitions.). "Incompleteness" (rather than "inconsistency") is still the missing piece. One claim in the above explanation: "David Hilbert's famous proclamation "We must know, we will know" is simply incorrect," Ignores this qualification -- making it a misapplication of what Gödel actually proved. Maybe we can eventually know truth -- but the limited tools constituting Gödel's proof were simply not up to that task.--[[Special:Contributions/172.69.33.83|172.69.33.83]] 20:04, 28 April 2022 (UTC) -edited --[[Special:Contributions/172.70.214.81|172.70.214.81]] 21:26, 28 April 2022 (UTC) | ||
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I, for one, am very pleased with the current compromise. The use of ellipsis and the inclusion of "(ironically)" has totally sold me on it. Also, if anyone knows how to make those notes where you have the little number you can click on to see the full explanation, I think the proof by contradiction part could benefit from having the parenthetical statements moved to notes. I'm going to look up how to do it, and I'll try, but if it all goes horribly wrong...[[Special:Contributions/108.162.221.101|108.162.221.101]] 20:27, 4 May 2022 (UTC) | I, for one, am very pleased with the current compromise. The use of ellipsis and the inclusion of "(ironically)" has totally sold me on it. Also, if anyone knows how to make those notes where you have the little number you can click on to see the full explanation, I think the proof by contradiction part could benefit from having the parenthetical statements moved to notes. I'm going to look up how to do it, and I'll try, but if it all goes horribly wrong...[[Special:Contributions/108.162.221.101|108.162.221.101]] 20:27, 4 May 2022 (UTC) |