Set theory with a proper class of indiscernibles

TL;DR

This paper explores an extension of ZFC with a proper class of indiscernibles, showing its consequences align with ZFC augmented by schemes asserting the existence of certain Mahlo cardinals with specific elementary submodel properties.

Contribution

It introduces a set-theoretic extension with a proper class of indiscernibles and proves its consequences match those of ZFC plus schemes for Mahlo cardinals.

Findings

Consequences of the extension match ZFC plus Mahlo cardinal schemes.

Establishes equivalence between indiscernibles and Mahlo cardinal schemes.

Provides insights into the structure of set-theoretic extensions.

Abstract

We investigate an extension of ZFC set theory (in an extended language) that stipulates the existence of a proper class of indiscernibles over the universe. One of the main results of the paper shows that the purely set-theoretical consequences of this extension of ZFC coincide with the theorems of the system of set theory obtained by augmenting ZFC with the (Levy) scheme whose instances assert, for each natural number $n$ in the metatheory, that there is an $n$-Mahlo cardinal $\kappa$ with the property that the initial segment of the universe determined by $\kappa$ is a $\Sigma_n$-elementary submodel of the universe.