$\mu$-Clubs OF $P_\kappa (\lambda)$ : Paradise on earth

TL;DR

This paper investigates the isomorphism of $$-club filters on certain power sets of cardinals, demonstrating new cases of such isomorphisms in ZFC beyond previous assumptions.

Contribution

It extends the understanding of $$-club filter isomorphisms to many new triples of cardinals within ZFC, surpassing prior results limited to $V=L$.

Findings

Many triples $(, , )$ have isomorphic $$-club filters in ZFC.

The isomorphism occurs even when $u(, ) > $, broadening previous conditions.

The results hold without assuming $V=L$, unlike earlier work.

Abstract

If $V = L$, and $\mu$, $\kappa$ and $\lambda$ are three infinite cardinals with $\mu = {\rm cf} (\mu) < \kappa = {\rm cf}(\kappa) \leq \lambda$, then, as shown in \cite{Heaven}, the $\mu$-club filters on $P_\kappa (\lambda)$ and $P_\kappa (\lambda^{< \kappa})$ are isomorphic if and only if ${\rm cf} (\lambda) \not= \mu$. Now in $L$, $\lambda^{< \kappa}$ equals $u (\kappa, \lambda)$ (the least size of a cofinal subset in $(P_\kappa (\lambda), \subseteq)$) equals $\lambda$ if ${\rm cf} (\lambda) \geq \kappa$, and $\lambda^+$ otherwise. We show that, in ZFC, there are many triples $(\mu, \kappa, \lambda)$ for which ($u (\kappa, \lambda) > \lambda$ and) the $\mu$-club filters on $P_\kappa (\lambda)$ and $P_\kappa (u (\kappa, \lambda))$ are isomorphic.