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| − | :1) I'm not sure what you mean by "paradoxical". If you mean something like "true and false" or "neither true nor false", that fails classical logic. Gödel (with a bit of help from Rosser) proved that we can write down a sentence G of Peano arithmetic, then prove (in PA) that G is equivalent to "no integer encodes a proof in PA of G unless a smaller one encodes a proof in PA of not G". He then pointed out that if G was provable in PA, there was also a proof of not G (basically, work out what integer encodes that proof of G, then for each smaller integer, try to decode it into a proof of not G; if you succeed, you have a proof of not G; if you fail for all, you have proved by exhaustion that your integer encodes a proof of G and no smaller integer encodes a proof of not G; all this is a proof of not G). Thus, if PA is consistent, there is no proof in PA of G. Now assume there is a proof in PA of not G. Encode this proof into an integer N. We shall now prove either G or "every integer less than N does not encode a proof in PA of G". We thus work through every integer less than N, checking to see if it encodes a proof in PA of G. If it does we have proved G; if no integers less than N encode a proof of G then we have proved "for all n < N, n does not encode a proof in PA of G". In the latter case, we have proved that every integer encoding a proof in PA of G is greater than N, which is an integer encoding a proof in PA of not G; this implies G! As such, we started with a proof in PA of not G (NOTE: THIS IS DIFFERENT FROM MERELY ASSUMING not G), and produced a proof in PA of G. So if PA is consistent, there is no proof in PA of not G either. Hence PA is either inconsistent (as if PA proves either G or not G, it proves the other and hence false) or incomplete (proving neither).
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| − | :2) He proved that either PA proves false, or there is a statement such that PA proves neither the statement nor its negation. The first includes paradoxicality. (His second incompleteness theorem was essentially: "By the argument above, PA proves that if PA is consistent then G has no proof in PA, which easily implies that PA proves "If PA is consistent, then G". Now suppose PA proves that PA is consistent. Then by modus ponens, PA proves G, and therefore PA is inconsistent. So if PA proves that PA is consistent, then PA is inconsistent.") (It ''is'' possible for a consistent system to prove its own inconsistency.)
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| − | :3) Most mathematicians assume that ZFC is consistent, even augmented by some pretty strong large cardinal hypotheses. [[Special:Contributions/172.70.35.72|172.70.35.72]] 17:11, 27 April 2022 (UTC)
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| | :The short answer to your questions is that Godel's method was rigorous. Godel numbering is much more precise than natural language ever could be. The longer answer is that there's a reason Godel's theorem is considered a work of genius; though the overall concept is fairly easy to grasp intuitively, making it into an actual theorem takes a lot of work and cleverness. There are multiple long Wikipedia pages about it just outlining the generals. The proof itself is rock solid, but far beyond the scope of this page. And the pithy answer is "Do you really think you're the first person to think of that? Mathematicians spent decades analyzing the theorems with uncharitable eyes."[[Special:Contributions/108.162.221.119|108.162.221.119]] 04:12, 27 April 2022 (UTC) | | :The short answer to your questions is that Godel's method was rigorous. Godel numbering is much more precise than natural language ever could be. The longer answer is that there's a reason Godel's theorem is considered a work of genius; though the overall concept is fairly easy to grasp intuitively, making it into an actual theorem takes a lot of work and cleverness. There are multiple long Wikipedia pages about it just outlining the generals. The proof itself is rock solid, but far beyond the scope of this page. And the pithy answer is "Do you really think you're the first person to think of that? Mathematicians spent decades analyzing the theorems with uncharitable eyes."[[Special:Contributions/108.162.221.119|108.162.221.119]] 04:12, 27 April 2022 (UTC) |
