On forcing axioms and weakenings of the Axiom of Choice
Diego Lima Bomfim, Charles Morgan, Samuel Gomes da Silva
TL;DR
This work identifies forcing-axiom equivalents for weakenings of the Axiom of Choice, notably $H_\kappa$ and $DC_\kappa$ for regular $\kappa$, and introduces a new forcing-axiom template that fixes, for each forcing, a particular family of dense sets. It establishes precise equivalences between these choice weakenings and forcing-axiom principles over carefully chosen forcing classes (e.g., $\mathrm{Coll}(\kappa,X)$, $\mathcal T_\kappa$, $\Lambda$), including the novel $DC^*_\kappa$ formulation and a special case $DC$ when $\kappa=\omega$, which corresponds to $\mathrm{FA}_\omega(\Lambda)$. The results connect the Hartogs-type trichotomy framework with forcing axioms and provide new forcing-axiom formulations of the Axiom of Choice, giving a template to derive AC from tailored forcing notions. Altogether, the paper advances the understanding of how non-constructive principles like $DC_\kappa$ and $H_\kappa$ can be characterized by forcing axioms with restricted dense sets, offering a pathway to reformulate AC in forcing-axiom terms.
Abstract
We prove forcing axiom equivalents of two families of weakenings of the axiom of choice: a trichotomy principle for cardinals isolated by Lévy, ${\rm H\hskip0.05pt}_κ$, and ${\rm DC}_κ$, the principle of dependent choices generalized to cardinals $κ$, for regular cardinals $κ$. Using these equivalents we obtain new forcing axiom formulations of the axiom of choice. A point of interest is that we use a new template for forcing axioms. For the class of forcings to which we asks that the axioms apply, we do not ask that they apply to all collections of dense sets of a certain cardinality, but rather only for each particular forcing to a specific family of dense sets of the cardinality in question.