We show that the theory ZFC-, consisting of the usual axioms of ZFC but with
the power set axiom removed-specifically axiomatized by extensionality,
foundation, pairing, union, infinity, separation, replacement and the assertion
that every set can be well-ordered-is weaker than commonly supposed and is
inadequate to establish several basic facts often desired in its context. For
example, there
... [Show full abstract] are models of ZFC- in which $\omega_1$ is singular, in which
every set of reals is countable, yet $\omega_1$ exists, in which there are sets
of reals of every size $\aleph_n$, but none of size $\aleph_\omega$, and
therefore, in which the collection axiom sceme fails; there are models of ZFC-
for which the Los theorem fails, even when the ultrapower is well-founded and
the measure exists inside the model; there are models of ZFC- for which the
Gaifman theorem fails, in that there is an embedding $j:M\to N$ of ZFC- models
that is $\Sigma_1$-elementary and cofinal, but not elementary; there are
elementary embeddings $j:M\to N$ of ZFC- models whose cofinal restriction
$j:M\to \bigcup j``M$ is not elementary. Moreover, the collection of formulas
that are provably equivalent in ZFC- to a $\Sigma_1$-formula or a
$\Pi_1$-formula is not closed under bounded quantification. Nevertheless, these
deficits of ZFC- are completely repaired by strengthening it to the theory
$ZFC^-$, obtained by using collection rather than replacement in the
axiomatization above. These results extend prior work of Zarach.
