LST Numbers for $Q^{\text{e.c.}}$ and $I$ style quantifiers

TL;DR

This paper introduces two new quantifier schemes related to regular cardinals with small Cantor-Bendixson rank, analyzing their L"owenheim-Skolem-Tarski numbers and establishing exact bounds under supercompact assumptions.

Contribution

It defines new quantifiers analogous to $I$ and $Q^{e.c.}$, and determines their LST number bounds assuming supercompact cardinals.

Findings

Exact lower bounds for LST numbers of the new quantifiers.

Interaction analysis between the LST numbers of different quantifiers.

Results depend on the consistency of supercompact cardinals.

Abstract

We introduce two schemes of quantifiers analogous to $I$ and $Q^\text{e.c.}$, which tell us about regular cardinals of small Cantor-Bendixson rank. We examine how the L\"owenheim-Skolem-Tarski numbers of these quantifiers interact with one another, and with those of $I$ and $Q^{\text{e.c.}}$. We then find the exact lower bound for each of the LST numbers, assuming the consistency of supercompacts.