Saturated models constitute one of the powerful methods of conventional model theory, with many applications. Here we develop a categorical abstract model theoretic approach to saturated models within the theory of institutions. The most important consequence is that the method of saturated models becomes thus available to a multitude of logical systems from logic or from computing science. In this paper we define the concept of saturated model at an abstract institution-independent level and (...) develop the fundamental existence and uniqueness theorems. As an application we prove a general institution-independent version of the Keisler–Shelah isomorphism theorem “any two elementarily equivalent models have isomorphic ultrapowers” (assuming Generalized Continuum Hypothesis). (shrink)

We put in print a classical result that states that for most purposes, there is no harm in assuming the existence of saturated models in model theory. The presentation is aimed for model theorists with only basic knowledge of axiomatic set theory.

This book contains the material for a first course in pure model theory with applications to differentially closed fields.

This paper is devoted to the problem of existence of saturated models for first-order many-valued logics. We consider a general notion of type as pairs of sets of formulas in one free variable that express properties that an element of a model should, respectively, satisfy and falsify. By means of an elementary chains construction, we prove that each model can be elementarily extended to a $\kappa $-saturated model, i.e. a model where as many types as possible are (...) realized. In order to prove this theorem we obtain, as by-products, some results on tableaux (understood as pairs of sets of formulas) and their consistency and satisfiability and a generalization of the Tarski–Vaught theorem on unions of elementary chains. Finally, we provide a structural characterization of $\kappa $-saturation in terms of the completion of a diagram representing a certain configuration of models and mappings. (shrink)

We study reducts of Peano arithmetic for which conditions of saturation imply the corresponding conditions for the whole model. It is shown that very weak reducts (like pure order) have such a property for κ-saturation in every κ ≥ ω 1 . In contrast, other reducts do the job for ω and not for $\kappa > \omega_1$ . This solves negatively a conjecture of Chang.

A notion called Herbrand saturation is shown to provide the model-theoretic analogue of a proof-theoretic method, Herbrand analysis, yielding uniform model-theoretic proofs of a number of important conservation theorems. A constructive, algebraic variation of the method is described, providing yet a third approach, which is finitary but retains the semantic flavor of the model-theoretic version.

We use the language of categorical logic to construct generic saturated models of intuitionistic theories. Our main technique is the thorough study of the filter construction on categories with finite limits, which is the completion of subobject lattices under filtered meets. When restricted to coherent or Heyting categories, classifying categories of intuitionistic first-order theories, the resulting categories are filtered meet coherent categories, coherent categories with complete subobject lattices such that both finite disjunctions and existential quantification distribute over filtered (...) meets. Such categories naturally embed into Grothendieck toposes which then contain saturated models of the theory we started with. (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.

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 ($${\cal N}$$).

The paper presents four open problems concerning recursively saturated models of Peano Arithmetic. One problems concerns a possible converse to Tarski's undefinability of truth theorem. The other concern elementary cuts in countable recursively saturated models, extending automorphisms of countable recursively saturated models, and Jonsson models of PA. Some partial answers are given.

In this paper we study the automorphism groups of countable arithmetically saturated models of Peano Arithmetic. The automorphism groups of such structures form a rich class of permutation groups. When studying the automorphism group of a model, one is interested to what extent a model is recoverable from its automorphism group. Kossak-Schmerl [12] show that ifMis a countable, arithmetically saturated model of Peano Arithmetic, then Aut(M) codes SSy(M). Using that result they prove:Let M1. M2be countable arithmetically (...) class='Hi'>saturated models of Peano Arithmetic such that Aut(M1)≅ Aut(M2).ThenSSy(M1) = SSy(M2).We show that ifMis a countable arithmetically saturated of Peano Arithmetic, then Aut(M) can recognize if some maximal open subgroup is a stabilizer of a nonstandard element, which is smaller than any nonstandard definable element. That fact is used to show the main theorem:Let M1,M2be countable arithmetically saturated models of Peano Arithmetic such thatAut(M1) ≅ Aut(M2).Then for every n<ωHere RT2nis Infinite Ramsey's Theorem stating that every 2-coloring of [ω]nhas an infinite homogeneous set. Theorem 0.2 shows that for models of a false arithmetic the converse of Kossak-Schmerl Theorem 0.1 is not true. Using the results of Reverse Mathematics we obtain the following corollary:There exist four countable arithmetically saturated models of Peano Arithmetic such that they have the same standard system but their automorphism groups are pairwise non-isomorphic. (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)

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)

In this paper we investigate the properties of automorphism groups of countable short recursively saturated models of arithmetic. In particular, we show that Kaye's Theorem concerning the closed normal subgroups of automorphism groups of countable recursively saturated models of arithmetic applies to automorphism groups of countable short recursively saturated models as well. That is, the closed normal subgroups of the automorphism group of a countable short recursively saturated model of PA are exactly the (...) stabilizers of the invariant cuts of the model which are closed under exponentiation. This Galois correspondence is used to show that there are countable short recursively saturated models of arithmetic whose automorphism groups are not isomorphic as topological groups. Moreover, we show that the automorphism groups of countable short arithmetically saturated models of PA are not topologically isomorphic to the automorphism groups of countable short recursively saturated models of PA which are not short arithmetically saturated. (shrink)

We give some information about the action of Aut on M, where M is a countable arithmetically saturated model of Peano Arithmetic. We concentrate on analogues of moving gaps and covering gaps inside M.

We give a survey of automorphisms of countable recursively saturated models of Peano Arithmetic.

Let $M$ be a saturated model of a superstable theory and let $G = \operatorname{Aut}(M)$. We study subgroups $H$ of $G$ which contain $G_{(A)}, A$ the algebraic closure of a finite set, generalizing results of Lascar [L] as well as giving an alternative characterization of the simple superstable theories of [P]. We also make some observations about good, locally modular regular types $p$ in the context of $p$-simple types.

In this paper we discuss automorphism groups of saturated models and boundedly saturated models of $\mathsf{PA}$. We show that there are saturated models of $\mathsf{PA}$ of the same cardinality with nonisomorphic automorphism groups. We then show that every saturated model of $\mathsf{PA}$ has short saturated elementary cuts with nonisomorphic automorphism groups.

A model of Peano Arithmetic is short recursively saturated if it realizes all its bounded finitely realized recursive types. Short recursively saturated models of $\PA$ are exactly the elementary initial segments of recursively saturated models of $\PA$. In this paper, we survey and prove results on short recursively saturated models of $\PA$ and their automorphisms. In particular, we investigate a certain subgroup of the automorphism group of such models. This subgroup, denoted $G|_{M(a)}$, (...) contains all the automorphisms of a countable short recursively saturated model of which can be extended to an automorphism of the countable recursively saturated elementary end extension of the model. (shrink)

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.

Some highly saturated models of Peano Arithmetic are constructed in this paper, which consists of two independent sections. In § 1 we answer a question raised in [10] by constructing some highly saturated, rather classless models of PA. A question raised in [7], [3], ]4] is answered in §2, where highly saturated, nonstandard universes having no bad cuts are constructed.Highly saturated, rather classless models of Peano Arithmetic were constructed in [10]. The main result (...) proved there is the following theorem. If λ is a regular cardinal andis aλ-saturated model of PA such that ∣M∣ >λ, thenhas an elementary extensionof the same cardinality which is alsoλ-saturated and which, in addition, is rather classless. The construction in [10] produced a modelfor which cf() =λ+. We asked in Question 5.1 of [10] what other cofinalities could such a modelhave. This question is answered here in Theorem 1.1 of §1 by showing that any cofinality not immediately excluded is possible. Its proof does not depend on the theorem from [10]; in fact, the proof presented here gives a proof of that theorem which is much simpler and shorter than the one in [10].Recursively saturated, rather classless κ-like models of PA were constructed in [9]. In the case of singular κ such models were constructed whenever cf(κ) > ℵ0; no additional set-theoretic hypothesis was needed. (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)

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.

We discuss automorphisms of saturated models of PA and boundedly saturated models of PA. We show that Smoryński's Lemma and Kaye's Theorem are not only true for countable recursively saturated models of PA but also true for all boundedly saturated models of PA with slight modifications.

We give a characterization for those stable theories whose $\omega_{1}$-saturated models have a "Shelah-style" structure theorem. We use this characterization to prove that if a theory is countable, stable, and 1-based without dop or didip, then its $\omega_{1}$-saturated models have a structure theorem. Prior to us, this is proved in a paper of Hart, Pillay, and Starchenko . Some other remarks are also included.

We show that is a quasi-minimal torsion-free divisible abelian group. After discussing the axiomatization of the theory of this structure, we present its ω-saturated quasi-minimal model.

The aims of this paper are: (1) to present tense-logical versions of such classical notions as saturated and special models; (2) to establish several fundamental existence theorems about these notions; (3) to apply these powerful techniques to tense complexity.In this paper we are concerned exclusively with quantifiedK 1 (for linear time) with constant domain. Our present research owes much to Bowen [2], Fine [5] and Gabbay [6].

In this paper we shall discuss properties of saturation in monotonic neighbourhood models and study some applications, like a characterization of compact and modally saturated monotonic models and a characterization of the maximal Hennessy-Milner classes. We shall also show that our notion of modal saturation for monotonic models naturally extends the notion of modal saturation for Kripke models.

In the early eighties, answering a question of A. Macintyre, J. H. Schmerl ([13]) proved that every countable recursively saturated structure, equipped with a function β encoding the finite functions, is the β-closure of an infinite indiscernible sequence. This result implies that every countably saturated structure, in a countable but not necessarily recursive language, is an Ehrenfeucht-Mostowski model, by which we mean that the structure expands, in a countable language, to the Skolem hull of an infinite indiscernible sequence (...) (in the new language). More recently, D. Lascar ([5]) showed that the saturated model of cardinality ℵ 1 of an ω-stable theory is also an Ehrenfeucht-Mostowski model. These results naturally raise the following problem: which (countable) complete theories have an uncountably saturated Ehrenfeucht-Mostowski model. We study a generalization of this question. Namely, we call ACI-model a structure which can be expanded, in a countable language L', to the algebraic closure (in L') of an infinite indiscernible sequence (in L'). And we try to characterize the λ-saturated structures which are ACI-models. The main results are the following. First it is enough to restrict ourselves to ℵ 1 -saturated structures: if T has an ℵ 1 -saturated ACI-model then, for every infinite λ, T has a λ-saturated ACI-model. We obtain a complete answer in the case of stable theories: if T is stable then the three following properties are equivalent: (a) T is ω-stable, (b) T has a ℵ 1 saturated ACI-model, (c) every saturated model of T is an Ehrenfeucht-Mostowski model. The unstable case is more complicated, however we show that if T has an ℵ 1 -saturated ACI-model then T doesn't have the independence property. (shrink)