Set theory with a proper class of indiscernibles

The talk concerns the metamathematics of an extension ZFCI of ZFC set theory (in an extended language) that stipulates the existence of a proper class of indiscernibles. We will identify the purely set-theoretical consequences of ZFCI, and will also compare and contrast the metamathemetics of ZFCI with that of PAI (Peano arithmetic with a class of indiscernibles).