When cardinals determine the power set: inner models and H\"artig quantifier logic

TL;DR

This paper explores the relationship between inner models, cardinal characteristics, and the logical strength of the H"artig quantifier, revealing connections to large cardinals and the structure of the core model.

Contribution

It demonstrates that under certain assumptions, the power set operation is definable in a way that links second-order logic validity to H"artig quantifier logic, and analyzes the L"owenheim number in this context.

Findings

If no $L[E]$ model has a strong enough cardinal, then $ ext{l}_I$ is a limit of measurable cardinals.

The L"owenheim number $ ext{l}_I$ is less than the least weakly inaccessible $oldsymbol{ ightarrow}$ the Weak Covering Lemma holds at $oldsymbol{ ightarrow}$.

The validity of second-order logic reduces to the set of validities of H"artig quantifier logic.

Abstract

We make use of some observations on the core model, for example assuming $V=L [ E ]$, and that there is no inner model with a Woodin cardinal, and $M$ is an inner model with the same cardinals as $V$, then $V=M$. We conclude in this latter situation that "$x=\mathcal{P} ( y )$" is $\Sigma_{1} ( Card )$ where $Card$ is a predicate true of just the infinite cardinals. It is known that this implies the validities of second order logic are reducible to $V_I$ the set of validities of the H\"artig quantifier logic. We draw some further conclusions on the L\"owenheim number, $\ell_{I}$ of the latter logic: that if no $L[E]$ model has a cardinal strong up to an $\aleph$-fixed point, and $\ell_{I}$ is less than the least weakly inaccessible $\delta$, then (i) $\ell_I$ is a limit of measurable cardinals of $K$; (ii) the Weak Covering Lemma holds at $\delta$.