When cardinals determine the power set: inner models and Härtig 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 $\delta$.