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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2601.23165 |
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| _version_ | 1866917235485835264 |
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| author | Gitman, Victoria Hamkins, Joel David Johnstone, Thomas A. |
| author_facet | Gitman, Victoria Hamkins, Joel David Johnstone, Thomas A. |
| contents | 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 $φ(x,X)$, then there is a class $Z\subseteq V\times V$ for which $φ(x,Z_x)$ on every section. This scheme can fail with KM even in low-complexity first-order instances $φ$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_23165 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Class choice and the surprising weakness of Kelley-Morse set theory Gitman, Victoria Hamkins, Joel David Johnstone, Thomas A. Logic 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 $φ(x,X)$, then there is a class $Z\subseteq V\times V$ for which $φ(x,Z_x)$ on every section. This scheme can fail with KM even in low-complexity first-order instances $φ$ 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. |
| title | Class choice and the surprising weakness of Kelley-Morse set theory |
| topic | Logic |
| url | https://arxiv.org/abs/2601.23165 |