In this paper, we study the union axiom of ZFC. After a brief introduction, we sketch a proof of the folklore result that union is independent of the other axioms of ZFC. In the third section, we prove some results in the theory T:= ZFC minus union. Finally, we show that the consistency of T plus the existence of an inaccessible cardinal proves the consistency of ZFC.

Hilbert’s 1st problem: Is there a cardinal strictly between א0 and 2א0? Equivalently, is there A ⊆ R so that R does not inject into A, and A does not inject into N? The continuum hypothesis (CH) asserts that no such A exists. Gödel: CH is consistent with ZFC. In other words, ZFC does not prove the existence of A as above. Cohen: negation of CH also consistent with ZFC. In other words, ZFC does ...

We examine what happens if we replace ZFC with a localistic/relativistic system, LZFC, whose central new axiom, denoted by Loc(ZFC), says that every set belongs to a transitive model of ZFC. LZFC consists of Loc(ZFC) plus some elementary axioms forming Basic Set Theory (BST). Some theoretical reasons for this shift of view are given. All Π2 consequences of ZFC are provable in LZFC. LZFC strongl...

We investigate systems of transitive models ZFC which are elementarily embeddable into each other and the influence definability properties on such systems.