Pigeonhole is a fundamental principle without which state of mathematics will be much different. However what examples of good mathematics has not yet been proved and cannot be proved with pigeonhole alone?
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1$\begingroup$ Pythagoras' theorem $\endgroup$– Hagen von EitzenNov 1, 2018 at 5:46
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$\begingroup$ Please post with explanation. Is Euler characteristic of planar graph provable? $\endgroup$– TurboNov 1, 2018 at 5:50
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$\begingroup$ @edm explicit and implicit are same if principle exists in proof. $\endgroup$– TurboNov 1, 2018 at 6:06
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1$\begingroup$ Establishing that a given result/technique isn't sufficient to prove a given theorem is generally extremely hard to do, mostly because precisely defining what that means is extremely subtle. For example, the reverse-mathematical approach Bjorn mentions below takes as "trivial" a large amount of non-trivial mathematics (e.g. the fundamental theorem of algebra). But there are many basic results for which the pigeonhole principle seems completely irrelevant; e.g. how would you use it to prove the Pythagorean theorem (per Hagen's comment above)? $\endgroup$– Noah SchweberNov 1, 2018 at 14:29
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1$\begingroup$ So your starting instinct here should definitely be that the pigeonhole principle - or anything else specific, really - will not be sufficient to tackle most problems you'll come across. $\endgroup$– Noah SchweberNov 1, 2018 at 14:30
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1 Answer
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The infinite pigeonhole principle is equivalent to B$\Sigma^0_2$ in the sense of reverse mathematics.
This implies that many theorems such as Heine-Borel, Konig's lemma, Ramsey's theorem are not provable from this Infinite Pigeonhole Principle.
answered Nov 1, 2018 at 6:03
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2$\begingroup$ @Brout That's a pretty explicit list of applications. Are you familiar with the theorems listed? $\endgroup$ Nov 1, 2018 at 14:26
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$\begingroup$ @NoahSchweber Except for Ramsey's theorem No. $\endgroup$– TurboNov 1, 2018 at 17:28