Letκbe the cardinality of some saturated model of Peano Arithmetic. There is a set of${2^{{\aleph _0}}}$saturated models of PA, each having cardinalityκ, such that wheneverMandNare two distinct models from this set, then Aut(${\cal M}$) ≇ Aut ($${\cal N}$$).

In this paper we will show that for every cutIof any countable nonstandard model$\mathcal {M}$of$\mathrm {I}\Sigma _{1}$, eachI-small$\Sigma _{1}$-elementary submodel of$\mathcal {M}$is of the form of the set of fixed points of some proper initial self-embedding of$\mathcal {M}$iffIis a strong cut of$\mathcal {M}$. Especially, this feature will provide us with some equivalent conditions with the strongness of the standard cut in a given countable model$\mathcal {M}$of$ \mathrm {I}\Sigma _{1} $. In addition, we will find some criteria for extendability of initial (...) self-embeddings of countable nonstandard models of$ \mathrm {I}\Sigma _{1} $to larger models. (shrink)

We prove a number of results concerning the variety of first-order theories and isomorphism types of pairs of the form $(N,M)$ , where $N$ is a countable recursively saturated model of Peano Arithmetic and $M$ is its cofinal submodel. We identify two new isomorphism invariants for such pairs. In the strongest result we obtain continuum many theories of such pairs with the fixed greatest common initial segment of $N$ and $M$ and fixed lattice of interstructures $K$ , such that $M\prec (...) K\prec N$. (shrink)

In a countable, recursively saturated model of Peano Arithmetic, an interstice is a maximal convex set which does not contain any definable elements. The interstices are partitioned into intersticial gaps in a way that generalizes the partition of the unbounded interstice into gaps. Continuing work of Bamber and Kotlarski [1], we investigate extensions of Kotlarski's Moving Gaps Lemma to the moving of intersticial gaps.

If${\cal M},{\cal N}$are countable, arithmetically saturated models of Peano Arithmetic and${\rm{Aut}}\left( {\cal M} \right) \cong {\rm{Aut}}\left( {\cal N} \right)$, then the Turing-jumps of${\rm{Th}}\left( {\cal M} \right)$and${\rm{Th}}\left( {\cal N} \right)$are recursively equivalent.

The main result of this paper partially answers a question raised in about the existence of countable just recursively saturated models of Peano Arithmetic with non‐isomorphic automorphism groups. We show the existence of infinitely many countable just recursively saturated models of Peano Arithmetic such that their automorphism groups are not topologically isomorphic. We also discuss maximal open subgroups of the automorphism group of a countable arithmetically saturated model of in a very good interstice.

We develop the method of iterated ultrapower representation to provide a unified and perspicuous approach for building automorphisms of countable recursively saturated models of Peano arithmetic . In particular, we use this method to prove Theorem A below, which confirms a long-standing conjecture of James Schmerl.Theorem AIf is a countable recursively saturated model of in which is a strong cut, then for any there is an automorphism j of such that the fixed point set of j is isomorphic to .We (...) also fine-tune a number of classical results. One of our typical results in this direction is Theorem B below, which generalizes a theorem of Kaye–Kossak–Kotlarski .Theorem BSuppose is a countable recursively saturated model of in which is a strong cut. There is a group embedding from into such that for each that is fixed point free, moves every undefinable element of. (shrink)