ZF+DC+AX$_4$

TL;DR

This paper explores a set theory combining ZF, DC, and a well-ordering condition for certain power sets, demonstrating theorems about cofinalities, covering numbers, successor cardinals, and applications to Abelian groups.

Contribution

It introduces a reasonable set theory framework and proves the pcf theorem with modifications, establishing the existence of covering numbers and successor cardinals, and applying these to Abelian groups.

Findings

Proved pcf theorem with true cofinalities in the new set theory.

Established existence of covering numbers and successor cardinals.

Applied results to diagonalization arguments in Abelian groups.

Abstract

We consider mainly the following version of set theory:"ZF + DC and for every $\lambda,\lambda^{\aleph_0}$ is well ordered", our thesis is that this is a reasonable set theory, e.g. much can be said. In particular, we prove that for a sequence $\bar\delta = \langle \delta_s:s \in Y\rangle,\textrm{cf}(\delta_s)$ large enough compared to $Y$, we can prove the pcf theorem with minor changes (using true cofinalities not the pseudo one). We then deduce the existence of covering numbers and define and prove existence of truly successor cardinals. From this we show that some diagonalization arguments (more specifically some black boxes and consequence) on Abelian groups. We end by showing some such consequences hold in ZF above.