Large transitive models in local ZFC

Abstract

This paper is a sequel to Tzouvaras (Arch Math Log 49(5):571–601, 2010), where a local version of ZFC, LZFC, was introduced and examined and transitive models of ZFC with properties that resemble large cardinal properties, namely Mahlo and ${\Pi_1^1}$-indescribable models, were considered. By analogy we refer to such models as “large models”, and the properties in question as “large model properties”. Continuing here in the same spirit we consider further large model properties, that resemble stronger large cardinals, namely, “elementarily embeddable”, “extendible” and “strongly extendible”, “critical” and “strongly critical”, “self-critical” and “strongly self-critical”, the definitions of which involve elementary embeddings. Each large model property ϕ gives rise to a localization axiom Loc ^ϕ(ZFC) saying that every set belongs to a transitive model of ZFC satisfying ϕ. The theories LZFC^ϕ = LZFC + Loc ^ϕ(ZFC) are local analogues of the theories ZFC+“there is a proper class of large cardinals ψ”, where ψ is a large cardinal property. If sext(x) is the property of strong extendibility, it is shown that LZFC^sext proves Powerset and Σ₁-Collection. In order to refute V = L over LZFC, we combine the existence of strongly critical models with an axiom of different flavor, the Tall Model Axiom (TMA). V = L can also be refuted by TMA plus the axiom GC saying that “there is a greatest cardinal”, although it is not known if TMA + GC is consistent over LZFC. Finally Vopěnka’s Principle (V P) and its impact on LZFC are examined. It is shown that LZFC^sext + V P proves Powerset and Replacement, i.e., ZFC is fully recovered. The same is true for some weaker variants of LZFC^sext. Moreover the theories LZFC^sext+V P and ZFC+V P are shown to be identical.