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)

The paper presents an outline of the general theory of countable arithmetically saturated models of PA and some of its applications. We consider questions concerning the automorphism group of a countable recursively saturated model of PA. We prove new results concerning fixed point sets, open subgroups, and the cofinality of the automorphism group. We also prove that the standard system of a countable arithmetically saturated model of PA is determined by the lattice of its elementary substructures.

In a nonstandard universe, the κ-saturation property states that any family of fewer than κ internal sets with the finite intersection property has a nonempty intersection. An ordered field F is said to have the λ-Bolzano-Weierstrass property iff F has cofinality λ and every bounded λ-sequence in F has a convergent λ-subsequence. We show that if $\kappa < \lambda$ are uncountable regular cardinals and $\beta^\alpha < \lambda$ whenever $\alpha < \kappa$ and $\beta < \lambda$, then there is a κ-saturated nonstandard (...) universe in which the hyperreal numbers have the λ-Bolzano-Weierstrass property. The result also applies to certain fragments of set theory and second order arithmetic. (shrink)

In a nonstandard universe, the $\kappa$-saturation property states that any family of fewer than $\kappa$ internal sets with the finite intersection property has a nonempty intersection. An ordered field $F$ is said to have the $\lambda$-Bolzano-Weierstrass property iff $F$ has cofinality $\lambda$ and every bounded $\lambda$-sequence in $F$ has a convergent $\lambda$-subsequence. We show that if $\kappa < \lambda$ are uncountable regular cardinals and $\beta^\alpha < \lambda$ whenever $\alpha < \kappa$ and $\beta < \lambda$, then there is a $\kappa$-saturated nonstandard (...) universe in which the hyperreal numbers have the $\lambda$-Bolzano-Weierstrass property. The result also applies to certain fragments of set theory and second order arithmetic. (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.

Bounded lattices (that is lattices that are both lower bounded and upper bounded) form a large class of lattices that include all distributive lattices, many nondistributive finite lattices such as the pentagon lattice N₅, and all lattices in any variety generated by a finite bounded lattice. Extending a theorem of Paris for distributive lattices, we prove that if L is an ℵ₀-algebraic bounded lattice, then every countable nonstandard model ������ of Peano Arithmetic has a cofinal elementary extension ������ such that (...) the interstructure lattice Lt(������ / ������) is isomorphic to L. (shrink)

Let ℳ be a countable, recursively saturated model of Peano Arithmetic, and let Aut be its automorphism group considered as a topological group with the pointwise stabilizers of finite sets being the basic open subgroups. Kaye proved that the closed normal subgroups are precisely the obvious ones, namely the stabilizers of invariant cuts. A proof of Kaye's theorem is given here which, although based on his proof, is different enough to yield consequences not obtainable from Kaye's proof.

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)

A model $\mathscr{M} = (M,+,\times, 0,1,<)$ of Peano Arithmetic $({\sf PA})$ is boundedly saturated if and only if it has a saturated elementary end extension $\mathscr{N}$. The ordertypes of boundedly saturated models of $({\sf PA})$ are characterized and the number of models having these ordertypes is determined. Pairs $(\mathscr{N},M)$, where $\mathscr{M} \prec_{\sf end} \mathscr{N} \models({\sf PA})$ for saturated $\mathscr{N}$, and their theories are investigated. One result is: If $\mathscr{N}$ is a $\kappa$-saturated model of $({\sf PA})$ and $\mathscr{M}_0, \mathscr{M}_1 \prec_{\sf end} (...) \mathscr{N}$ are such that $\aleph_1 \leq \mathrm{min}(\mathrm{cf}(M_0),\mathrm{dcf}(M_0)) \leq \mathrm{min}(\mathrm{cf}(M_1), \mathrm{dcf}(M_1)) < \kappa$, then $(\mathscr{N},M_0) \equiv (\mathscr{N},M_1)$. (shrink)

We divide the class of infinite computable trees into three types. For the first and second types, 0' computes a nontrivial self-embedding while for the third type 0'' computes a nontrivial self-embedding. These results are optimal and we obtain partial results concerning the complexity of nontrivial self-embeddings of infinite computable trees considered up to isomorphism. We show that every infinite computable tree must have either an infinite computable chain or an infinite Π01 antichain. This result is optimal and has connections (...) to the program of reverse mathematics. (shrink)

For a model M of Peano Arithmetic, let Lt(M) be the lattice of its elementary substructures, and let Lt⁺(M) be the equivalenced lattice (Lt(M), ≅M), where ≅M is the equivalence relation of isomorphism on Lt(M). It is known that Lt⁺(M) is always a reasonable equivalenced lattice. Theorem. Let L be a finite distributive lattice and let (L,E) be reasonable. If M₀ is a nonstandard prime model of PA, then M₀ has a confinal extension M such that Lt⁺(M) ≅ (L,E). A (...) general method for proving such theorems is developed which, hopefully, will be able to be applied to some nondistributive lattices. (shrink)

Wilkie proved in 1977 that every countable model ${\mathcal M}$ of Peano Arithmetic has an elementary end extension ${\mathcal N}$ such that the interstructure lattice $\operatorname {\mathrm {Lt}}({\mathcal N} / {\mathcal M})$ is the pentagon lattice ${\mathbf N}_5$. This theorem implies that every countable nonstandard ${\mathcal M}$ has an elementary cofinal extension ${\mathcal N}$ such that $\operatorname {\mathrm {Lt}}({\mathcal N} / {\mathcal M}) \cong {\mathbf N}_5$. It is proved here that whenever ${\mathcal M} \prec {\mathcal N} \models \mathsf {PA}$ and (...) $\operatorname {\mathrm {Lt}}({\mathcal N} / {\mathcal M}) \cong {\mathbf N}_5$, then ${\mathcal N}$ must be either an end or a cofinal extension of ${\mathcal M}$. In contrast, there are ${\mathcal M}^* \prec {\mathcal N}^* \models \mathsf {PA}^*$ such that $\operatorname {\mathrm {Lt}}({\mathcal N}^* / {\mathcal M}^*) \cong {\mathbf N}_5$ and ${\mathcal N}^*$ is neither an end nor a cofinal extension of ${\mathcal M}^*$. (shrink)

This paper is primarily concerned with ℵ 0 -categoricity of theories of partially ordered sets. It contains some general conjectures, a collection of known results and some new theorems on ℵ 0 -categoricity. Among the latter are the following. Corollary 3.3. For every countable ℵ 0 -categorical U there is a linear order of A such that $(\mathfrak{U}, is ℵ 0 -categorical. Corollary 6.7. Every ℵ 0 -categorical theory of a partially ordered set of finite width has a decidable theory. (...) Theorem 7.7. Every ℵ 0 -categorical theory of reticles has a decidable theory. There is a section dealing just with decidability of partially ordered sets, the main result of this section being. Theorem 8.2. If $(P, is a finite partially ordered set and K P is the class of partially ordered sets which do not embed $(P, , then Th(K P ) is decidable iff K P contains only reticles. (shrink)

We study notions of genericity in models of, inspired by lines of inquiry initiated by Chatzidakis and Pillay and continued by Dolich, Miller and Steinhorn in general model‐theoretic contexts. These papers studied the theories obtained by adding a “random” predicate to a class of structures. Chatzidakis and Pillay axiomatized the theories obtained in this way. In this article, we look at the subsets of models of which satisfy the axiomatization given by Chatzidakis and Pillay; we refer to these subsets in (...) models of as CP‐generics. We study a more natural property, called strong CP‐genericity, which implies CP‐genericity. We use an arithmetic version of Cohen forcing to construct (strong) CP‐generics with various properties, including ones in which every element of the model is definable in the expansion, and, on the other extreme, ones in which the definable closure relation is unchanged. (shrink)

The theory PA(aa), which is Peano Arithmetic in the context of stationary logic, is shown to be consistent. Moreover, the first-order theory of the class of finitely determinate models of PA(aa) is characterized.

A structure A = (A; E₀, E₁ , . . . , ${E_{n - 2}}$) is an n-grid if each E i is an equivalence relation on A and whenver X and Y are equivalence classes of, repectively, distinct E i and E j , then X ∩ Y is finite. A coloring χ : A → n is acceptable if whenver X is an equivalence class of E i , then {ϰ Є X: χ(ϰ) = i} is finite. If (...) B is any set, then the n-cube B n = (B n ; E₀ E₁ , . . . , ${E_{n - 2}}$) is considinate axis. Kuratowski [9], generalizing the n = 3 case proved by Sierpiński [17], proved that ℝ n has an acceptable coloring iff ${2^{{N_0}}}$ ≤ ${N_{n - 2}}$. The main result is: of A is a semialgebraic (i.e., first-order definable in the field of reals) n-grid, then the following are equivalent: (1) if A embeds all finite n-cubes, then ${2^{{N_0}}}$ ≤ ${N_{n - 2}}$; if A embeds ℝ n , then ${2^{{N_0}}}$ ≤ ${N_{n - 2}}$; (3) A has an acceptable coloring. (shrink)

For each completion of Peano Arithmetic there is a weakly definable type which is not definable.

We investigate some properties of ordered structures that are related to their having cofinal elementary extensions. Special attention is paid to models of some very weak fragments of Peano Arithmetic.

Every ℵ 0 -categorical partially ordered set of finite width has a finitely axiomatizable theory. Every ℵ 0 -categorical partially ordered set of finite weak width has a decidable theory. This last statement constitutes a major portion of the complete (with three exceptions) characterization of those finite partially ordered sets for which any ℵ 0 -categorical partially ordered set not embedding one of them has a decidable theory.

There is a recursive set of natural numbers which is the difference set of some recursively enumerable set but which is not the difference set of any recursive set.

For each infinite cardinalκ, the set of algebraic hypergraphs having chromatic number no larger thanκis decidable.

A theorem of Enayat's concerning models of ZFC which had been proved using several different additional set-theoretical hypotheses is shown here to be absolute.