What is the theory ZFC without power set?

Journal article

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 are models of ZFC in which ω 1 is singular, in which every set of reals is countable, yet ω 1 exists, in which there are sets of reals of every size ℵ n , but none of size ℵ ω , and therefore, in which the collection axiom fails; there are models of ZFC for which the Łoś 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 → N of ZFC models that is Σ 1 -elementary and cofinal, but not elementary; there are elementary embeddings j : M → N of ZFC- models whose cofinal restriction j : M → υ j”M is not elementary. Moreover, the collection of formulas that are provably equivalent in ZFC to a Σ 1 -formula or a Π 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.

Authors: Gitman, V; Hamkins, J; Johnstone, T

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Wiley
• Journal: Mathematical Logic Quarterly
• Volume: 62
• Issue: 4-5
• Pages: 391-406
• Publication date: 2016-07-25
• Acceptance date: 2015-08-22
• DOI: 10.1002/malq.201500019
• EISSN: 1521-3870
• ISSN: 0942-5616
• Language: English
• Keywords: FFR