Combinatorial Unprovability Proofs and Their Model-Theoretic Counterparts
Mojtaba Aghaei, Amir Khamseh · Notre Dame Journal of Formal Logic · 2014
For a function f with domain [X]n, where X⊆N, we say that H⊆X is canonical for f if there is a υ⊆n such that for any x0,…,xn−1 and y0,…,yn−1 in H, f(x0,…,xn−1)=f(y0,…,yn−1) iff xi=yi for all i∈υ. The canonical Ramsey theorem is the statement that for any n∈N, if f:[N]n→N, then there is an infinite H⊆N canonical for f. This paper is concerned with a model-theoretic study of a finite version of the canonical Ramsey theorem with a largeness condition and also a version of the Kanamori–McAloon principle. As a consequence, we produce new indicators for cuts satisfying PA.