Automorphisms of models of set theory and extensions of NFU

In this paper we exploit the structural properties of standard and non-standard models of set theory to produce models of set theory admitting automorphisms that are well-behaved along an initial segment of their ordinals. NFU is Ronald Jensen's modification of Quine's ‘New Foundations’ Set Theory that allows non-sets (urelements) into the domain of discourse. The axioms AxCount, AxCount≤AxCount≤ and AxCount≥AxCount≥ each extend NFU by placing restrictions on the cardinality of a finite set of singletons relative to the cardinality of its union. Using the results about automorphisms of models of subsystems of set theory we separate the consistency strengths of these three extensions of NFU. More specifically, we show that NFU+AxCountNFU+AxCount proves the consistency of NFU+AxCount≤NFU+AxCount≤, and NFU+AxCount≤NFU+AxCount≤ proves the consistency of NFU+AxCount≥NFU+AxCount≥.