Several proofs of PA-unprovability

In this article we give unprovability proofs of several combinatorial statements using the method of indiscernibles. Our first two statements are strong versions of KM (the Kanamori-McAloon principle) and KM(2), (the Kanamori-McAloon principle for pairs) that both allow unexpectedly simple proofs. We proceed to discuss results by A.Weiermann and G.Lee on IΣkprovability/unprovability of the principles PH (k+1) log(n) and KM (k+1) log(n) for different n ∈ N. We contrast a result by G.Lee on provability of KM log(n) for n ≥ k with the following result. If the condition “for all x, y in H, x < y implies 2x < y” is imposed on the min-homogeneous set H then the modified statement KM (k+1) log(n) becomes IΣk-unprovable for all n, k ∈ N. For k ≥ 2, the restriction |H| > 2c, where c is the second element of H also makes the modified KM log(n) unprovable in IΣk for all n ∈ N. The article is directed at a broad audience and is intended to be suitable for expository purposes. Here, we present several model-theoretic proofs of unprovability (in Peano Arithmetic, PA, and in its fragments IΣk, the theories of induction for formulas containing not more than k quantifiers) usinig the method of indiscernibles. We give full proofs and intend this article to be suitable for expository purposes and accessible to mathematicians interested in “how is it possible to prove that a concrete statement about natural numbers is unprovable?”.