Set theory with a proper class of indiscernibles

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.

35 pages. This version corrects some minor issues detected in the previous draft, and includes refinements of certain results (see Theorem 4.1 and Remark 6.6). Note that a section dealing with models of Peano arithmetic has been excised from this draft (a much expanded version of it will appear as a separate paper)

References in corpus (2)

• Inner-model reflection principles
• Set theoretical analogues of the Barwise-Schlipf theorem