# What mathematics cannot be reduced to pigeonhole?

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?

• Pythagoras' theorem Commented Nov 1, 2018 at 5:46
• Please post with explanation. Is Euler characteristic of planar graph provable? Commented Nov 1, 2018 at 5:50
• @edm explicit and implicit are same if principle exists in proof. Commented Nov 1, 2018 at 6:06
• 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)? Commented Nov 1, 2018 at 14:29
• 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. Commented Nov 1, 2018 at 14:30

The infinite pigeonhole principle is equivalent to B$$\Sigma^0_2$$ in the sense of reverse mathematics.