New inner models from second order logics

TL;DR

This paper introduces a new inner model C2(omega) based on a fragment of second order logic, compares it to C(aa), and explores its properties and implications under large cardinal assumptions.

Contribution

It defines the inner model C2(omega), compares it to C(aa), and investigates its size and large cardinal content under various assumptions.

Findings

C2(omega) appears larger than C(aa) under large cardinal assumptions

C2(omega) contains inner models with n Woodin cardinals for all n

Omega_1 is Mahlo in C2(omega) assuming a Woodin limit of Woodin cardinals

Abstract

We define a new inner model C2(omega) based on the fragment of second order logic in which second order variables range over countable subsets of the domain. We compare C2(omega) to the previously studied inner model C(aa). We argue that C2(omega) appears to be a much bigger inner model than C(aa), although this cannot be literally true in ZFC alone. However, we conjecture that it follows from large cardinal assumptions. For example, assuming large cardinals, C2(omega) contains, for every n, an inner model with n Woodin cardinals, while C(aa) contains, under the same assumption, no inner model with a Woodin cardinal. As to large cardinals in C2(omega), we show that, assuming a Woodin limit of Woodin cardinals, the cardinal omega_1 of V is Mahlo in C2(omega). A stronger result is proved for the combination C2(omega, aa) of C(aa) and C2(omega). We also show that the question whether HOD1, a variant of HOD, is the same as HOD cannot be decided on the basis of ZFC even if we add the assumption that there are supercompact cardinals.