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INITIAL SELF-EMBEDDINGS OF MODELS OF SET THEORY
Published online by Cambridge University Press: 13 August 2021
Abstract
By a classical theorem of Harvey Friedman (1973), every countable nonstandard model 
$\mathcal {M}$
of a sufficiently strong fragment of ZF has a proper rank-initial self-embedding j, i.e., j is a self-embedding of 
$\mathcal {M}$
such that 
$j[\mathcal {M}]\subsetneq \mathcal {M}$
, and the ordinal rank of each member of 
$j[\mathcal {M}]$
is less than the ordinal rank of each element of 
$\mathcal {M}\setminus j[\mathcal {M}]$
. Here, we investigate the larger family of proper initial-embeddings j of models 
$\mathcal {M}$
of fragments of set theory, where the image of j is a transitive submodel of 
$\mathcal {M}$
. Our results include the following three theorems. In what follows, 
$\mathrm {ZF}^-$
is 
$\mathrm {ZF}$
without the power set axiom; 
$\mathrm {WO}$
is the axiom stating that every set can be well-ordered; 
$\mathrm {WF}(\mathcal {M})$
is the well-founded part of 
$\mathcal {M}$
; and 
$\Pi ^1_\infty \text{-}\mathrm {DC}_\alpha $
is the full scheme of dependent choice of length 
$\alpha $
.
Theorem A.
There is an 
$\omega $
-standard countable nonstandard model 
$\mathcal {M}$
of
$\mathrm {ZF}^-+\mathrm {WO}$
that carries no initial self-embedding 
$j:\mathcal {M} \longrightarrow \mathcal {M}$
other than the identity embedding.
Theorem B.
Every countable 
$\omega $
-nonstandard model 
$\mathcal {M}$
of 
$\ \mathrm {ZF}$
is isomorphic to a transitive submodel of the hereditarily countable sets of its own constructible universe 
$L^{\mathcal {M}}$
.
Theorem C.
The following three conditions are equivalent for a countable nonstandard model 
$\mathcal {M}$
of 
$\mathrm {ZF}^{-}+\mathrm {WO}+\forall \alpha \ \Pi ^1_\infty \text{-}\mathrm {DC}_\alpha $
.
(I) There is a cardinal in
$\mathcal {M}$
that is a strict upper bound for the cardinality of each member of 
$\mathrm {WF}(\mathcal {M})$
.(II)
$\mathrm {WF}(\mathcal {M})$
satisfies the powerset axiom.(III) For all
$n \in \omega $
and for all 
$b \in M$
, there exists a proper initial self-embedding 
$j: \mathcal {M} \longrightarrow \mathcal {M}$
such that 
$b \in \mathrm {rng}(j)$
and 
$j[\mathcal {M}] \prec _n \mathcal {M}$
.
MSC classification
- Type
- Article
- Information
- Copyright
- © Association for Symbolic Logic 2021
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