Local Sentences and Mahlo Cardinals

TL;DR

This paper establishes the exact consistency strength of certain stretching theorems for local sentences, showing they are equivalent to the existence of n-Mahlo cardinals, thus linking model theory with large cardinal axioms.

Contribution

It proves that the stretching principles for local sentences are equivalent to the existence of n-Mahlo cardinals, resolving an open problem in the field.

Findings

Stretching principles are equivalent to n-Mahlo cardinals.

Constructed local sentences with models of all non-Mahlo order types.

Connected model-theoretic properties with large cardinal hypotheses.

Abstract

Local sentences were introduced by J.-P. Ressayre who proved certain remarkable stretching theorems establishing the equivalence between the existence of finite models for these sentences and the existence of some infinite well ordered models. Two of these stretching theorems were only proved under certain large cardinal axioms but the question of their exact (consistency) strength was left open in [O. Finkel and J.-P. Ressayre, Stretchings, Journal of Symbolic Logic, Volume 61 (2), 1996, p. 563-585 ]. Here, we solve this problem, using a combinatorial result of J. H. Schmerl. In fact, we show that the stretching principles are equivalent to the existence of n-Mahlo cardinals for appropriate integers n. This is done by proving first that for each integer n, there is a local sentence phi_n which has well ordered models of order type alpha, for every infinite ordinal alpha > omega which is not an n-Mahlo cardinal.