The van der Waerden theorem is a theorem in the branch of mathematics called Ramsey theory which states that for any given positive integers r and k, there is some number N such that if the integers {1, 2, ..., N} are colored, each with one of r different colors, then there are at least k integers in arithmetic progression whose elements are of the same color

This theorem applies to "classical" mathematics (which is basically mathematics based on classical logic). But are there any mathematical systems based on different types of logic (e.g non classical logics) where this theorem would not hold? Since there are logics where fundamental principles of classical logic do not hold (e.g the law of excluded middle in intuitionistic logic), wouldn't it be the same for this theorem?

  • Van Der Waender's theorem is a theorem of intuitionistic logic, although its proof deviates from the classical proofs. See SEP. So one would need to look further afield to find a logic where it is not a theorem. – Nick Nov 26 '20 at 21:53
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    Why the van der Waerden theorem specifically? That there are infinitely many primes is also derived using classical logic. However, classical logic is part of what makes what we are talking about the standard arithmetic. One can change even the non-logical axioms to make either of those false, but it is unclear that we will still be talking about the same thing rather than, say, a ring different from integers. As Quine put it, "Here, evidently, is the deviant logician's predicament: when he tries to deny the doctrine he only changes the subject". – Conifold Nov 26 '20 at 22:30
  • @Nick can you think of any particular logic where Van Der Waerden's theorem would not hold? – vengaq Nov 27 '20 at 13:44
  • Why are you specifically interested in van der Waerden's theorem? – Noah Schweber Nov 28 '20 at 0:41

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