February 23
Corey Switzer,
University of Vienna
Independence in PA: The Method of $(\mathcal L, n)$-Models
The purpose of this talk is to exposit a method for proving independence over PA of 'mathematical' statements (whatever that means). The method uses the concept of an $(\mathcal L, n)$-model: a finite sequence of finite $\mathcal L$-structures for some first order $\mathcal L$ extending the language of arithmetic. The idea is that this finite sequence is intended to represent increasing approximations of a potentially infinite structure and the machinery developed allows one to translate (meta-mathematical) compactness type statements, which are easily seen to be independent of PA, into statements about finite combinatorics, which have 'mathematical content'. $(\mathcal L, n)$-models were introduced by Shelah in the 70's in his alternative proof of the Paris-Harrington Theorem and also appears (implicitly) in his example of a true, unprovable $\Pi^0_1$ statement of some 'mathematical' content. A similar idea was discovered independently by Kripke (unpublished). In this talk we will flesh out the details of this method and extend the general theory. This will allow us to present, in a fairly systematic fashion, proofs of the Paris-Harrington Theorem and the independence over PA of several, similar, Ramsey Theoretic statements including some which are $\Pi^0_1$.