Local sentences and Mahlo cardinals

Local sentences and Mahlo cardinals

Local sentences were introduced by Ressayre in [Res88] 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 lef...

Automorphisms, Mahlo Cardinals, and NFU

Wellordering Proofs for Metapredicative Mahlo

In this article we provide wellordering proofs for metapredicative systems of explicit mathematics and admissible set theory featuring suitable axioms about the Mahloness of the underlying universe of discourse. In particular, it is shown that in the corresponding theories EMA of explicit mathematics and KPm of admissible set theory, transfinite induction along initial segments of the ordinal φ...

On decidability properties of local sentences

Local (first order) sentences, introduced by Ressayre, enjoy very nice decidability properties, following from some stretching theorems stating some remarkable links between the finite and the infinite model theory of these sentences [Res88]. Another stretching theorem of Finkel and Ressayre implies that one can decide, for a given local sentence φ and an ordinal α < ωω, whether φ has a model o...

Jónsson Cardinals, Erdös Cardinals, and The Core Model

We show that if there is no inner model with a Woodin cardinal and the Steel core model K exists, then every Jónsson cardinal is Ramsey in K, and every δ-Jónsson cardinal is δ-Erdős in K. In the absence of the Steel core model K we prove the same conclusion for any model L[E ] such that either V = L[E ] is the minimal model for a Woodin cardinal, or there is no inner model with a Woodin cardina...