Class choice and the surprising weakness of Kelley-Morse set theory
Victoria Gitman, Joel David Hamkins, Thomas A. Johnstone
TL;DR
The paper investigates Kelley–Morse (KM) set theory and reveals that KM does not prove the class choice scheme, nor Łoś-type preservation for internal second-order ultrapowers, nor invariance of $ ext{Σ}^1_n$ under first-order quantification. It develops forcing-based constructions showing precise weaknesses of KM (even for simple instances) and demonstrates how augmenting KM with the class choice scheme (KM$^+$) remedies these issues, aligning KM$^+$ with robust foundations akin to ZFC$^-$ plus an inaccessible cardinal. The results are achieved through a blend of symmetric-model techniques and large-cardinal assumptions, plus a detour to second-order arithmetic for intuition. The work thus positions KM$^+$ as a stronger, more reliable framework for second-order set theory and clarifies the landscape of class-choice principles and their separations.
Abstract
Kelley-Morse set theory KM is weaker than generally supposed and fails to prove several principles that may be desirable in a foundational second-order set theory. Even though KM includes the global choice principle, for example, (i) KM does not prove the class choice scheme, asserting that whenever every set $x$ admits a class $X$ with $\varphi(x,X)$, then there is a class $Z\subseteq V\times V$ for which $\varphi(x,Z_x)$ on every section. This scheme can fail with KM even in low-complexity first-order instances $\varphi$ and even when only a set of indices $x$ are relevant. For closely related reasons, (ii) the theory KM does not prove the Łoś theorem scheme for internal second-order ultrapowers, even for large cardinal ultrapowers, such as the ultrapower by a normal measure on a measurable cardinal. Indeed, the theory KM itself is not generally preserved by internal ultrapowers. Finally, (iii) KM does not prove that the $Σ^1_n$ logical complexity is invariant under first-order quantifiers, even bounded first-order quantifiers. For example, $\forall α{<}δ ψ(α,X)$ is not always provably equivalent to a $Σ^1_1$ assertion when $ψ$ is. Nevertheless, these various weaknesses in KM are addressed by augmenting it with the class choice scheme, thereby forming the theory KM+, which we propose as a robust KM alternative for the foundations of second-order set theory.