Compactness for small cardinals in mathematics: principles, consequences, and limitations
Radek Honzik
TL;DR
The paper surveys compactness principles at small cardinals, distinguishing logical forms tied to infinitary logics from mathematical forms governing graphs, groups, and topological spaces. It analyzes indestructibility and preservation under forcing, showing many compactness consequences are robust yet fail to decide certain independent statements, and highlights the tension between forcing axioms like MM/PFA and principles such as Rado's Conjecture. By examining equivalences in $L$ and the role of stationary reflection, it connects two avenues—large-cardinal-inspired logic and concrete mathematical structures—into a cohesive framework. The work discusses standard model constructions (e.g., Mitchell forcing) that realize compactness at small cardinals, compares unifications (e.g., Laver-generic large cardinals vs MM), and evaluates whether these principles are viable as axioms, emphasizing their potential and limitations for guiding foundational mathematics.
Abstract
We discuss some well-known compactness principles for uncountable structures of small regular sizes ($ω_n$ for $2 \le n<ω$, $\aleph_{ω+1}$, $\aleph_{ω^2+1}$, 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, $Δ$-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 $ω_2$. Additionally, we observe that Rado's Conjecture plus $2^ω= ω_2$ 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).