We consider extensions of Peano arithmetic suitable for doing some of nonstandard analysis, in which there is a predicate N(x) for an elementary initial segment, along with axiom schemes approximating ω 1 -saturation. We prove that such systems have the same proof-theoretic strength as their natural analogues in second order arithmetic. We close by presenting an even stronger extension of Peano arithmetic, which is equivalent to ZF for arithmetic statements.

In this paper we consider some axiomatic systems of set theory related to the system NF (New Foundations) of Quine. In particular we discuss the possible relations of cardinality between a finite set x and its subset class SC(x) = {y | y ∩ x} and also between x and its unit set class USC(x) = {{y} | y ε x}. Specker [5] has shown that in NF the cardinal of a finite set x can never be the same as (...) the cardinal of SC(x). (shrink)