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 modifcation of Quine's `New Foundations' set
theory that allows non-sets into the domain of discourse. The axioms
AxCount, AxCount_≤ and 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 set theory we separate the consistency
strengths of these three extensions of NFU. We show that
NFU + AxCount proves the consistency of NFU +
AxCount_≤, and NFU + AxCount_≤ proves the
consistency of NFU + AxCount_≥.
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