We prove that in IST, Nelson's internal set theory, the Uniqueness and Collection principles, hold for all (including external) formulas. A corollary of the Collection theorem shows that in IST there are no definable mappings of a set X onto a set Y of greater (not equal) cardinality unless both sets are finite and #(Y) ≤ n #(X) for some standard n. Proofs are based on a rather general technique which may be applied to other nonstandard structures. In particular we (...) prove that in a nonstandard model of PA, Peano arithmetic, every hyperinteger uniquely definable by a formula of the PA language extended by the predicate of standardness, can be defined also by a pure PA formula. (shrink)

We give an examination of the automorphism group Aut of a countable recursively saturated model M of PA. The main result is a characterisation of strong elementary initial segments of M as the initial segments consisting of fixed points of automorphisms of M. As a corollary we prove that, for any consistent completion T of PA, there are recursively saturated countable models M1, M2 of T, such that Aut[ncong]Aut, as topological groups with a natural topology. Other results include a classification (...) of the normal subgroups of Aut of the form [lcub]g: g [uharr] A = idA[rcub], for sets A M, and a highly homogeneous representation of Aut as a subgroup of Aut. (shrink)

We show that the stabilizer of an element a of a countable recursively saturated model of arithmetic M is a maximal subgroup of Aut iff the type of a is selective. This is a point of departure for a more detailed study of the relationship between pointwise and setwise stabilizers of certain subsets of M and the types of elements in those subsets. We also show that a complete type of PA is 2-indiscernible iff it is minimal in the sense (...) of Gaifman. (shrink)