The size of a set from the size of its countable power set

William Chan

Vol. 344 (2026), No. 1, 219–239

DOI: 10.2140/pjm.2026.344.219

Abstract

Let ${[{\omega }_1]}^{\omega }$ and ${[{\omega }_1]}^{<{\omega }_1}$ be the set of all increasing $\omega$-length and countable-length sequences of countable ordinals, respectively. Assume ${\omega }_1{\to }_{\ast }{({\omega }_1)}_2^{{\omega }_1}$. For any sequence $⟨Y_n:n\in \omega ⟩$, if $\left|{[{\omega }_1]}^{<{\omega }_1}\right|\le \left|{\prod <!--FUNCTION\; APPLICATION-->}_{n\in \omega }{Y}_n\right|$, then there exists an $\overline{n}\in \omega$ such that $\left|{[{\omega }_1]}^{\omega }\right|\le \left|Y_{\overline{n}}\right|$.

Assume Woodin’s extension ${AD}^{+}$ of the axiom of determinacy. For any set $X$, let $\mathcal{𝒫}(X)$ be the set of subsets of $X$ and let ${\mathcal{𝒫}}_{{\omega }_1}(X)$ be the set of countable subsets of $X$. For any $X$ which is a surjective image of $ℝ$, if $|\mathcal{𝒫}({\omega }_1)|\le |{\mathcal{𝒫}}_{{\omega }_1}(X)|$, then $\left|{[{\omega }_1]}^{\omega }\right|<\left|X\right|$.

Keywords

partition relations, cardinalities

Mathematical Subject Classification

Primary: 03E02, 03E60

Milestones

Received: 13 July 2025

Revised: 23 March 2026

Accepted: 27 May 2026

Published: 10 August 2026