Topic Archive: axioms
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 are models of ZFC- in which a countable union of countable sets is not countable. There are models of ZFC- for which the Los ultrapower theorem fails, even for wellfounded ultrapowers on a measurable cardinal. Moreover, the theory ZFC- is not sufficient to establish that the union of Σn and Πn sets is closed under bounded quantification. Lastly, there are models of ZFC- for which the Gaifman theorem fails, in that there exists cofinal embeddings j:M–>N between ZFC- models that are Σ1-elementary, but not fully elementary.
Nevertheless, these deficits of ZFC- are completely repaired by strengthening it to the theory obtained by using collection rather than replacement in the axiomatization above. This is joint work with Joel David Hamkins and Victoria Gitman, and it extends prior work of Andrzej Zarach.