Initial self-embeddings of models of set theory

Abstract

By a classical theorem of Harvey Friedman (1973), every countable nonstandard model MM of a sufficiently strong fragment of ZF has a proper rank-initial self-embedding j, i.e., j is a self-embedding of MM such that j[M]⊊Mj[M]⊊M , and the ordinal rank of each member of j[M]j[M] is less than the ordinal rank of each element of M∖j[M]M∖j[M] . Here, we investigate the larger family of proper initial-embeddings j of models MM of fragments of set theory, where the image of j is a transitive submodel of MM . Our results include the following three theorems. In what follows, ZF−ZF− is ZFZF without the power set axiom; WOWO is the axiom stating that every set can be well-ordered; WF(M)WF(M) is the well-founded part of MM ; and Π1∞-DCαΠ∞1-DCα is the full scheme of dependent choice of length αα . Theorem A. There is an ωω -standard countable nonstandard model MM of ZF−+WOZF−+WO that carries no initial self-embedding j:M⟶Mj:M⟶M other than the identity embedding. Theorem B. Every countable ωω -nonstandard model MM of ZF ZF is isomorphic to a transitive submodel of the hereditarily countable sets of its own constructible universe LMLM . Theorem C. The following three conditions are equivalent for a countable nonstandard model MM of ZF−+WO+∀α Π1∞-DCαZF−+WO+∀α Π∞1-DCα . 1. (I) There is a cardinal in MM that is a strict upper bound for the cardinality of each member of WF(M)WF(M) . 2. (II) WF(M)WF(M) satisfies the powerset axiom. 3. (III) For all n∈ωn∈ω and for all b∈Mb∈M , there exists a proper initial self-embedding j:M⟶Mj:M⟶M such that b∈rng(j)b∈rng(j) and j[M]≺nMj[M]≺nM .