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 ≤ 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 subsystems of set theory we separate the consistency strengths of these three extensions of NFU. More specifically, we show that NFU + AxCount proves the consistency of NFU +AxCount ≤ , and NFU+AxCount ≤ proves the consistency of NFU + AxCount ≥ .

the longest initial segment of Ord M that is pointwise fixed byǰ for every nontrivial j ∈ Aut(Q). In Theorem C, Aut(X ) is the group of automorphisms of the structure X , and Q is the ordered set of rationals.

No abstract

Let M be the basic set theory that consists of the axioms of extensionality, emptyset, pair, union, powerset, infinity, transitive containment, 0 -separation and set foundation. This paper studies the relative strength of set theories obtained by adding fragments of the set-theoretic collection scheme to M. We focus on two common parameterisations of the collection: n -collection, which is the usual collection scheme restricted to n -formulae, and strong n -collection, which is equivalent to n -collection plus n+1 -separation. The main result of this paper shows that for all n ≥ 1,(1) M + n+1 -collection + n+2 -induction on ω proves that there exists a transitive model of Zermelo Set Theory plus n -collection, (2) the theory M + n+1 -collection is n+3 -conservative over the theory M + strong n -collection.It is also shown that (2) holds for n = 0 when the Axiom of Choice is included in the base theory. The final section indicates how the proofs of (1) and (2) can be modified to obtain analogues of these results for theories obtained by adding fragments of collection to a base theory (Kripke-Platek Set Theory with Infinity plus V=L) that does not include the powerset axiom.Both Collection and Strong Collection yield ZF when added to M. In § 2, we note that, over M, the restriction of the Strong Collection scheme to n -formulae (strong n -collection) is equivalent to the restriction of the Collection scheme to n -formulae ( n -collection) plus separation for all n+1 -formulae. This means that M plus n+1 -collection proves all instances of strong n -collection. One of the many achievements of [7] is showing that if M is consistent, then so is M plus the Axiom of Choice and strong 0 -collection. In § 3, we investigate the strength of adding ℘ 0 -collection to four of the weak set theories studied in [7]. We show that if T is one of the theories M, Mac, M + H or MOST, then T plus *

We identify complete fragments of the Simple Theory of Types with Infinity (TSTI) and Quine's NF set theory. We show that TSTI decides every sentence φ in the language of type theory that is in one of the following forms: