Compactness for small cardinals in mathematics: principles, consequences, and limitations

TL;DR

This paper explores compactness principles for uncountable structures at small regular cardinals, analyzing their consistency, preservation under forcing, and implications for set-theoretic conjectures and axioms.

Contribution

It provides a detailed analysis of the consistency and indestructibility of various compactness principles at small regular cardinals and their implications for major set-theoretic conjectures.

Findings

Many classical problems are independent from strong forms of compactness at 2.

Rado's Conjecture plus 2 is consistent with negative solutions of some conjectures.

Compactness principles may serve as potential axioms with significant consequences.

Abstract

We discuss some well-known compactness principles for uncountable structures of small regular sizes ($\omega_n$ for $2 \le n<\omega$, $\aleph_{\omega+1}$, $\aleph_{\omega^2+1}$, etc.), consistent from weakly compact (the size-restricted versions) or strongly compact or supercompact cardinals (the unrestricted versions). We divide the principles into logical principles (various tree properties) and mathematical principles, which directly postulate compactness for structures like groups, graphs, or topological spaces (for instance, countable chromatic and color compactness of graphs, compactness of abelian groups, $\Delta$-reflection, Fodor-type reflection principle, and Rado's Conjecture). We focus on indestructibility, or preservation, of these principles in forcing extensions. Using the existing preservation results we observe that many traditional problems such as Suslin Hypothesis, Whitehead's Conjecture, Kaplansky's Conjecture, and Baumagartner's Axiom, are independent from some of the strongest forms of compactness at $\omega_2$. Additionally, we observe that Rado's Conjecture plus $2^\omega = \omega_2$ is consistent with the negative solutions of some of these conjectures (as they hold in $V = L$), verifying that they hold in suitable Mitchell models. Finally, we comment on whether the compactness principles under discussion are good candidates for axioms. We consider their consequences and the existence or non-existence of convincing unifications (such as Martin's Maximum or Rado's Conjecture). This part is a modest follow-up to the articles by Foreman ``Generic large cardinals: new axioms for mathematics?'' (1998) and Feferman et al. ``Does mathematics need new axioms?'' (2000).