AbstractA model $${\mathcal {M}}$$
M
of ZF is said to be condensable if $$ {\mathcal {M}}\cong {\mathcal {M}}(\alpha )\prec _{\mathbb {L}_{{\mathcal {M}}}} {\mathcal {M}}$$
M
≅
M
(
α
)
≺
L
M
M
for some “ordinal” $$\alpha \in \mathrm {Ord}^{{\mathcal {M}}}$$
α
∈
Ord
M
, where $$\mathcal {M}(\alpha ):=(\mathrm {V}(\alpha ),\in )^{{\mathcal {M}}}$$
M
(
α
)
:
=
(
V
(
α
)
,
∈
)
M
and $$\mathbb {L}_{{\mathcal {M}}}$$
L
M
is the set of formulae of the infinitary logic $$\mathbb {L}_{\infty ,\omega }$$
L
∞
,
ω
that appear in the well-founded part of $${\mathcal {M}}$$
M
. The work of Barwise and Schlipf in the 1970s revealed the fact that every countable recursively saturated model of ZF is cofinally condensable (i.e., $${\mathcal {M}}\cong {\mathcal {M}}(\alpha ) \prec _{\mathbb {L}_{{\mathcal {M}}}}{\mathcal {M}}$$
M
≅
M
(
α
)
≺
L
M
M
for an unbounded collection of $$\alpha \in \mathrm {Ord}^{{\mathcal {M}}}$$
α
∈
Ord
M
). Moreover, it can be readily shown that any $$\omega $$
ω
-nonstandard condensable model of $$\mathrm {ZF}$$
ZF
is recursively saturated. These considerations provide the context for the following result that answers a question posed to the author by Paul Kindvall Gorbow.Theorem A.Assuming a modest set-theoretic hypothesis, there is a countable model $${\mathcal {M}}$$
M
of ZFC that is bothdefinably well-founded (i.e., every first order definable element of $${\mathcal {M}}$$
M
is in the well-founded part of $$\mathcal {M)}$$
M
)
andcofinally condensable. We also provide various equivalents of the notion of condensability, including the result below.Theorem B.The following are equivalent for a countable model$${\mathcal {M}}$$
M
of $$\mathrm {ZF}$$
ZF
: (a) $${\mathcal {M}}$$
M
is condensable. (b) $${\mathcal {M}}$$
M
is cofinally condensable. (c) $${\mathcal {M}}$$
M
is nonstandard and $$\mathcal {M}(\alpha )\prec _{\mathbb {L}_{{\mathcal {M}}}}{\mathcal {M}}$$
M
(
α
)
≺
L
M
M
for an unbounded collection of $$ \alpha \in \mathrm {Ord}^{{\mathcal {M}}}$$
α
∈
Ord
M
.
Condensable models of set theory
