Model theory: an introduction

We introduce a notion of the space of types in positive model theory based on Stone duality for distributive lattices. We show that this space closely mirrors the Stone space of types in the full first-order model theory with negation (Tarskian model theory). We use this to generalise some classical results on countable models from the Tarskian setting to positive model theory.

The notions of recursively saturated and resplendent models grew out of the study of admissible sets with urelements and admissible fragments of Lω1ω, but, when applied to ordinary first order model theory, give us new tools for research and exposition. We will discuss their history in §3.The notion of saturated model has proven to be important in model theory. Its most important property for applications is that if , are saturated and of the same cardinality then = iff ≅ . See, e.g., Chang-Keisler [3]. The main drawback is that saturated models exist only under unusual assumptions of set theory. For example, if 2κ = κ+ then every theory T of L has a saturated model of power κ+. (Similarly, if κ is strongly inaccessible, then every T has a saturated model of power κ.) On the other hand, a theory T like Peano arithmetic, with types, cannot have a saturated model in any power κ with ω ≤ κ ≤ .One method for circumventing these problems of existence (or rather non-existence) is the use of “special” models (cf. [3]). If κ = Σλ<κ2λ, κ < ω, then every theory T of L has a special model of power κ. Such cardinals are large and, themselves, rather special. There are definite aesthetic objections to the use of these large, singular models to prove results about first order logic.

A tame theory is a decidable first-order theory with only countably many countable models, and all complete types recursive.It is shown here that the recursive complexity of countable homogeneous models of tame theories is unbounded in the hyperarithmetic hierarchy.Recursive model theory considers model-theoretic objects and constructions from the point of view of recursion-theoretic complexity.For instance, must isomorphism types of countable models that are structurally simple have models of low recursion-theoretic complexity?This paper considers one such question involving countable homogeneous models.If is a countable homogeneous model of a complete theory T, then is necessarily decidable in the degree of Tl Since there are complete theories with 2m complete types, the answer is obviously "no".On the other hand, by results in [1], if T has only finitely many countable models and all of the complete types of the theory are recursive, then the answer is "yes".This paper considers decidable theories T that have at most countably many countable models and only recursive complete types.Call such a theory tame.The notational conventions in the paper are standard.Fact.Every countable homogeneous model of a tame theory T is hyperarithmetic.Proof.Let {p^i < co} be an effective enumeration of all the partial recursive functions : N - TV, and let {B(\i < co} be an effective enumeration of all formulas in L(T).Since all of the types of T are recursive, we can fix a hyperarithmetic set 7 C TV such that (i) Vi I[{0j\i(j) = 17 < <y} is a complete type of T]; and(ii) vr complete type of T miel Vj[8j G T iff .(J) = I].For a countable model il I-T, say that W CI is a witness set for il iff there is some enumeration {jS.\i < co} of |il|<cu such that (1) Ve W [n is total];

Book reviewed in this article:“Brief Course in Chemistry” by Lyman C. Newell, Professor of Chemistry, Boston University, Boston, Mass. Author of Experimental Chemistry, Descriptive Chemistry, etc. In two parts—Part I, Minimum Essentials, Part II, Supplementary Topics. First edition. Pp.General Science—For Reviews, by W. Dean Pulvermacher, Chairman, General Science Department, Jamaica High School, New York, and Charles H. Vosburgh, Principal, Jamaica High School, New York. First edition.Our Wonderful Universe, by Clarence Augustus Chant, Professor of Astrophysics, University of Toronto. Published by the World Book Company.Six‐Place Tables by Edward S. Allen, Associate Professor of Mathematics, Iowa State College. Third Edition. Cloth.College Algebra, by H. L. Rietz, Professor of Mathematics, University of Iowa, and A. R. Chrathorne, Associate Professor of Mathematical Statistics, University of Illinois. Third Edition.Origin Through Evolution by Nathan Fasten, Professor of Zoology in Oregon State Agricultural College. Cloth.Mathematics Preparatory to Statistics and Finance, by George N. Bauer, University of New Hampshire. Pp. vii+337. 13.5times19.5 cm.A Short Table of Integrals by B. O. Peirce, Hollis Professor of Mathematics and Natural Philosophy in Harvard University. Third Revised Edition. Cloth. 156 pages.Standard Service Geometry Work‐Book, H. D. Welte, Principal of the State Normal School, New Britain, Connecticut, and F. B. Knight, College of Education, University of Iowa. 74 pages. 20times28 cm. Paper.Plane Geometry by Joseph A. Nyberg, M. S., Instructor in Mathematics, Hyde Park High School, Chicago. 302 pages. 14times19 cm.General Science, by Richard W. Sharpe, George Washington High School, New York. First edition. 1929.A First Course in Physics, Herbert Brownell, Professor of the Technique of Instruction in Science, Teachers' College, University of Nebraska. The John C. Winston Company, Philadelphia.Magnetism and Electricity by Morris Meister, New York Training School for Teachers, College of the City of New York. Cloth.Solutions of Electrolytes, with particular application to Qualitative Analysis, by Louis P. Hammett, Ph.D., Associate Professor of Chemistry in Columbia University. First Edition. ppix+211. 15times21times2.7 cm.

1: Structures. 1.1. Structures. 1.2. Sentences. 1.3. Embeddings. 1.4. The Compactness Theorem. 1.5. Elementary classes and theories. 1.6. Complete theories. 1.7. Definable sets. 1.8. References. 2: Quantifier Elimination. 2.1. Elimination sets. 2.2. Discrete linear orders. 2.3. Dense linear orders. 2.4. Algebraically closed fields (and Tarski). 2.5. Tarski again: Real closed fields. 2.6. pp-elimination of quantifiers and modules. 2.7. Strongly minimal theories. 2.8. o-minimal theories. 2.9. Computational aspects of q. e. 2.10. References. 3: Model Completeness. 3.1. An introduction. 3.2. Abraham Robinson's test. 3.3. Model completeness and algebra. 3.4. p-adic fields and Artin's conjecture. 3.5. Existentially closed fields. 3.6. DCF0. 3.7. SCFp and DCFp. 3.8. ACFA. 3.9. References. 4: Elimination of Imaginaries. 4.1. Interpretability. 4.2. Imaginary elements. 4.3. Algebraically closed fields. 4.4. Real closed fields. 4.5. The elimination of imaginaries sometimes fails. 4.6. References. 5: Morley Rank. 5.1. A tale of two chapters. 5.2. Definable sets. 5.3. Types. 5.4. Saturated models. 5.5. A parenthesis: pure injective models. 5.6. Omitting types. 5.7. The Morley rank, at last. 5.8. Strongly minimal sets. 5.9. Algebraic closure and definable closure. 5.10. References. 6: Omega-stability. 6.1. Totally transcendental theories. 6.2. omega-stable groups. 6.3. omega-stable fields. 6.4. Prime models. 6.5. DCF0 revisited. 6.6. Ryll-Nardzewski's Theorem and other things. 6.7. References. 7: Classifying. 7.1. Shelah's Classification Theory. 7.2. Simple theories. 7.3. Stable theories. 7.4. Superstable theories. 7.5. w-stable theories. 7.6. Classifiable theories. 7.7. Shelah's Uniqueness Theorem. 7.8. Morley's Theorem. 7.9. Biinterpretability and Zilber Conjecture. 7.10. Two algebraic examples. 7.11. References. 8: Model Theory and Algebraic Geometry. 8.1. Introduction. 8.2. Algebraic varieties, ideals, types. 8.3. Dimension and Morley rank. 8.4. Morphisms and definable functions. 8.5. Manifolds. 8.6. Algebraic groups. 8.7. The Mordell-Lang Conjecture. 8.8. References. 9: O-Minimality. 9.1. Introduction. 9.2. The Monotonicity Theorem. 9.3. Cells. 9.4. Cell decomposition and other theorems. 9.5. Their proofs. 9.6. Definable groups in o-minimal structures. 9.7. O-minimality and Real Analysis. 9.8. Variants on the o-minimal theme. 9.9. No rose without thorns. 9.10. References. Bibliography. Index.

An Ehrenfeucht theory is a complete first order theory with exactly n countable models up to isomorphism, 1 &lt; n &lt; ω. Numerous results have emerged regarding these theories ([1]–[15]). A general question in model theory is whether or not the number of countable models of a complete theory can be different than the number of countable models of a complete consistent extension of the theory by finitely many constant symbols. Examples are known of Ehrenfeucht theories that have complete extensions by finitely many constant symbols such that the extensions fail to be Ehrenfeucht ([4], [8], [13]). These examples are easily modified to allow finite increases in the number of countable models.This paper contains examples in the other direction—complete theories that have consistent extensions by finitely many constant symbols such that the extensions have fewer countable models. This answers affirmatively a question raised by, among others, Peretyat'kin [8]. The first example will be an Ehrenfeucht theory with exactly four countable models with an extension by a constant symbol that has only three countable models. The second example will be a complete theory that is not Ehrenfeucht, but which has an extension by a constant symbol that is Ehrenfeucht. The notational conventions for this paper are standard.Peretyat'kin introduced the theory of a dense binary branching tree with a meet operator [7]. Dense ω-branching trees have also proven useful [5], [11]. Both of the Theories that will be constructed make use of dense ω-branching trees.

Nerode asked if there could be a complete decidable theory with only finitely many countable models up to isomorphism, such that not all of the countable models were decidable. Morley, Lachlan, and Peretyatkin produced examples of such theories. However, all the countable models of those theories were decidable in 0 ′ 0’ . The question then arose whether all countable models of such theories had to be, for example, arithmetic. In this paper we provide a negative answer to that question by showing that there are such examples with countable models of arbitrarily high hyperarithmetic degree. It is not difficult to show that any countable model of a hyperarithmetic theory which has only finitely many countable models must be decidable in some hyperarithmetic degree.

In the 2012–13 academic year, the Mathematical Sciences Research Institute, Berkeley, hosted programs in Commutative Algebra (Fall 2012 and Spring 2013) and Noncommutative Algebraic Geometry and Representation Theory (Spring 2013). There have been many significant developments in these fields in recent years; what is more, the boundary between them has become increasingly blurred. This was apparent during the MSRI program, where there were a number of joint seminars on subjects of common interest: birational geometry, D-modules, invariant theory, matrix factorizations, noncommutative resolutions, singularity categories, support varieties, and tilting theory, to name a few. These volumes reflect the lively interaction between the subjects witnessed at MSRI. The Introductory Workshops and Connections for Women Workshops for the two programs included lecture series by experts in the field. The volumes include a number of survey articles based on these lectures, along with expository articles and research papers by participants of the programs. Volume 2 focuses on the most recent research.

Alex Lipton is Managing Director, Quantitative Solutions Executive at Bank of America and a Visiting Professor of Quantitative Finance at University of Oxford. Prior to his current role, he was Managing Director, Co-Head of the Global Quantitative Group at Bank of America Merrill Lynch, and a Visiting Professor of Mathematics at Imperial College London. Previously, he was Managing Director and Head of Capital Structure Quantitative Research at Citadel Investment Group in Chicago; he has also worked at Credit Suisse, Deutsche Bank and Bankers Trust. Before switching to finance, Alex was a Full Professor of Mathematics at the University of Illinois at Chicago and a Consultant at Los Alamos National Laboratory. He received his undergraduate and graduate degrees from Lomonosov Moscow State University. His current interests include industrial strength derivative pricing including capital calculations and valuation adjustments, as well as technical trading strategies. In 2000 Alex was awarded the first Quant of the Year Award by Risk Magazine. Alex is the author of two books (Magnetohydrodynamics and Spectral Theory and Mathematical Methods for Foreign Exchange) and the editor of four more, including, most recently, The Oxford Handbook of Credit Derivatives (jointly with Andrew Rennie). He has published numerous research papers on hydrodynamics, magnetohydrodynamics, astrophysics, and financial engineering. Alex is a founding patron of The 14-10 Club at the Royal Institution (jointly with David Harding).

SAMUEL WESLEY STRATTON was born at Litchfield, Illinois, in 1861. He graduated in 1884 from the University of Illinois, where he remained for the next eight years, at first as an instructor, then as professor of mathematics, and finally as professor of physics and electrical engineering. In 1892 he became assistant professor of physics and later professor in the University of Chicago. Nine years afterwards he was invited by the Assistant Secretary of the Treasury to formulate a project for the development of work on weights and measures, and in 1901 was mainly responsible for the Act which established the Bureau of Standards at Washington. Of this he became the first director, retaining the position until 1923, when he was appointed president of the Massachusetts Institute of Technology. He had a lifelong connexion with the American Navy, and in 1898 served during the Spanish American War. This brief summary of his activities shows that the most important part of his work was connected with the Bureau of Standards, which is indebted to him for its existence and for many of its most striking developments.

In this paper we study the notion of a D-saturated model, which occupies an intermediate position between the notions of a homogeneous model and a saturated model. Both the homogeneous models and the saturated models play a very important role in model theory. For example, the saturated models are the universal domains of the corresponding theories in the sense that is used in algebraic geometry (one can also notice that the algebraically closed fields of infinite transcendence degree, which are the universal domains in algebraic geometry, are the saturated models of the theory of algebraically closed fields); this means that the saturated models realize all what is necessary for the effective study of the corresponding theory.The D-saturated models proved to be useful in various situations. Naturally the question arises on the conditions under which a model is D-saturated. In this paper we indicate some conditions of such sort for weakly o-minimal models and models of stable theories. Namely, for such models we prove that homogeneity and a certain approximation of Dsaturation imply D-saturation.

We shall characterize the complete first order theories of torsion free abelian groups having algebraically prime models, elementarily prime models, minimal models, etc., using the well known invariants from [9], [3]. This study was suggested by the questions raised in [2]. Group here will always mean abelian group. For reference on groups see Fuchs [5], [6], for model theory Chang-Keisler [1]. All theories considered are theories in the first order language of group theory. A group G is minimal (strictly minimal) if H < G implies H = G (H-= G), and elementarily prime (algebraically prime) if G is elementarily (algebraically) embeddable in every H G. p, q always denote prime numbers, P the set of all primes. The p-rank of a group G, p-rk(G), is the dimension of G/pG as a vector space over Z/pZ. We define tf(p, G): = min {p-rk(G), ~o}. By the results of [9], [3], two nontrivial torsion free abelian groups G and H are elementarily equivalent iff for all primes p tf(p, G) = tf(p, H). This gives a natural 1-1 correspondence between the complete theories of nontrivial torsion free abelian groups and the functions /3 : P --~ ~o + 1. /3 will always stand for such a function, and the corresponding complete theory will be denoted by T 0. S o is the set of primes p for which/3(p) ~ 0. For more definitions see section 3. Let 13 be a function from P to o) + 1. We shall study the following questions: Does T o have an elementarily (algebraically) prime model? Does T o have a minimal (strictly minimal) model, and if this is the case, how many such models does T o have? Do the minimal and strictly minimal models of T o coincide? These questions have been studied for the special case of the theory of the

We give effectiveness conditions on a strongly minimal theory T guaranteeing that all countable models have computable copies. In particular, we show that if T is strongly minimal and for all n\geq 1 , T\cap\exists_{n+2} is \Delta^0_n , uniformly in n , then every countable model has a computable copy. A longstanding question of computable model theory asked whether for a strongly minimal theory with one computable model, every countable model has an arithmetical copy. Relativizing our main result, we get the fact that if there is one computable model, then every countable model has a \Delta^0_4 copy.

We employ an infinite-signature Hrushovski amalgamation construction to yield two results in Recursive Model Theory. The first result, that there exists a strongly minimal theory whose only recursively presentable models are the prime and saturated models, adds a new spectrum to the list of known possible spectra. The second result, that there exists a strongly minimal theory in a finite language whose only recursively presentable model is saturated, gives the second non-trivial example of a spectrum produced in a finite language.

One of the most significant by-products of the study of admissible sets with urelements is the emphasis it has given to recursively saturated models. As suggested in [Schlipf, 1977], countable recursively saturated models (for finite languages) possess many of the desirable properties of saturated and special models. The notion of resplendency was introduced to isolate some of these desirable properties. In §§1 and 2 of this paper we study these parallels, showing how they can be exploited to give new proofs of some traditional model theoretic theorems. This yields both pedagogical and philosophical advantages: pedagogical since countable recursively saturated models are easier to build and manipulate than saturated and special models; philosophical since it shows that uncountable models — which the downward Lowenheim–Skolem theorem tells us are in some sense not basic in the study of countable theories — are not needed in model theoretic proofs of these theorems. In §3 we apply our local results to get results about resplendent models of ZF set theory and PA (Peano arithmetic). In §4 we shall examine certain analogous results for admissible languages, most similar to, and seemingly generally slightly weaker than, already known results. (The Chang–Makkai sort of result, however, is new.)Although this paper is an outgrowth of work with admissible sets with urelements, I have tried to keep it as accessible as possible to those with a background only in finitary model theory. Thus §§1,2, and 3 should not involve any work with admissible sets. §4, however, is concerned with some admissible analogues to results in §2 and necessarily uses certain technical results of §11 5 of [Schlipf, 1977].