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Set theory with a proper class of indiscernibles

Tom 259 / 2022

Fundamenta Mathematicae 259 (2022), 33-76 MSC: Primary 03E55; Secondary 03F25, 03C62, 03E02. DOI: 10.4064/fm999-2-2022 Opublikowany online: 28 June 2022

Streszczenie

We investigate an extension of $\mathrm {ZFC}$ set theory, denoted $\mathrm {ZFI}_{ \lt }$, which is equipped with a well-ordering $\lt$ of the universe V of set theory, and a proper class $I$ of indiscernibles over $(\mathrm {V},\in , \lt )$. Our main results are Theorems A, B, and C below. Note that the equivalence of conditions (ii) and (iii) in Theorem A was established in an earlier (2004) published work of the author. In what follows, $\mathrm{GBC}$ is the Gödel–Bernays theory of classes with global choice. In Theorem C the symbol $\rightarrow$ is the usual Erdős-arrow notation for partition calculus.

Theorem A. The following are equivalent for a sentence $\varphi$ in the language $\{=,\in \}$ of set theory:

(i) $\mathrm {ZFI}_{\mathrm { \lt }}\vdash \varphi .$

(ii) $\mathrm {ZFC}+{\mit\Lambda} \vdash \varphi ,$ where ${\mit\Lambda} =\{\lambda _{n}:n\in \omega \}$, and $\lambda _{n}$ is the sentence asserting the existence of an $n$-Mahlo cardinal $\kappa$ such that $\mathrm {V}(\kappa )$ is a ${\mit\Sigma} _{n}$-elementary submodel of the universe $\mathrm {V}$.

(iii) $\mathrm{GBC}$ $+$ “$\textrm{Ord}$ is weakly compact” $\vdash \varphi$.

Theorem B. Every $\omega$-model of $\mathrm {ZFI}_{\mathrm { \lt }}$satisfies $\mathrm {V}\neq \mathrm {L}$.

Theorem C. The sentence expressing $\forall m,n\in \omega$ $( \mathrm {Ord}\rightarrow ( \mathrm {Ord}) _{m}^{n})$ is not provable in the theory $T= \textrm{GBC}$ $+$ “$\textrm {Ord}$ is weakly compact” , assuming $T$ is consistent.

The paper also includes results about the interpretability relationship between the theories $\mathrm {ZFC}+{\mit\Lambda}$, $\mathrm {ZFI}_{\mathrm { \lt }}$, and $\mathrm{GBC}$ $+$ “$\textrm{Ord}$ is weakly compact”.

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