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Omitting classes of elements
Abstract
This chapter focuses on omitting classes of elements. By a class of elements we mean a class defined by a set of formulas. It is fairly easy to show that there must be some cardinal, x, such that for any theory T and any class Σ the existence of a model of T of power K which omits Σ implies the existence of such models in each infinite power. The principal result of this paper is the determination of this cardinal. The proof depends upon a partition theorem of Erdös and Rado. The letter T will always denote a theory in a countable first-order language L, and Σ will denote a set of formulas in L having a common single free variable. In particular, Ehrenfeucht used it to show that for any theory T (having an infinite model) there will be some countable set of types of elements such that T has arbitrarily large models containing only elements of those types.
... The result that L ∞ω is weak was first proven by Lopez-Escobar (Lopez-Escobar [1966a]). First, by generalizing results by Morley (Morley [1965]) and Helling (Helling [1964]) on an upper bound for h κ (L ω1ω ), Lopez-Escobar showed that h(L κω ) < (2 κ ) + . Then, using methods introduced by Hanf (Hanf [1962]), he could show that if L κω is strong, then L κω pins down the cardinal λ , where λ = 2 2 2 κ , yielding a lower bound for h(L κω ). ...
... This result can be found in (Chang [1968]), building on Morleys proof of m ω = ω1 in (Morley [1965]). The proof makes use of a construction with indiscernibles and the Erdős-Rado theorem. ...
In this paper we will study the expressive power, measured by the ability to define certain classes, of some extensions of first order logic. The central concepts will be definability of classes of ordinals and the well-ordering number w of a logic. First we discuss the partial orders ≤, ≤P C and ≤RP C on logics and how these relate to each other and to our definability concept. Then we study the division between bounded and unbounded logics. An interrest-ing result in this direction is the theorem due to Lopez-Escobar stating that L∞ω is weak in the sense that it does not define the entire class of well-orderings, even though it has no well-ordering number, whereas Lω 1 ω 1 is strong in the same sense.
... The presentation theorem asserts that an AEC with arbitrarily large models can be defined as the reducts of models of a first order theory which omit a family of types. But Morley [44] had calculated the Hanf number for such syntactically defined classes. Thus, using the syntactical presentation as a tool, Shelah obtains a purely semantic theorem: if an AEC with Löwenheim number ℵ 0 has a model of cardinality ω1 it has arbitrarily large models. ...
... To be precise, we take the formalization in[49] since it has the same vocabulary: points, lines, and incidence for two and multidimensional geometry. The notion of plane is introduced by explicit definition as in Definition 4.3.13.44 Note that the union of the theories PP and PG is inconsistent. ...
We emphasize the role of the choice of vocabulary in formalization of a mathematical area and remark that this is a particular preoccupation of logicians. We use this framework to discuss Kennedy’s notion of ‘formalism freeness’ in the context of various schools in model theory. Then we clarify some of the mathematical issues in recent discussions of purity in the proof of the Desargues proposition. We note that the conclusion of ‘spatial content’ from the Desargues proposition involves arguments which are algebraic and even metamathematical. Hilbert showed that the Desargues proposition implies the coordinatizing ring is associative, which in turn implies the existence of a three-dimensional geometry in which the given plane can be embedded. With W. Howard we give a new proof, removing Hilbert’s ‘detour’ through algebra, of the ‘geometric’ embedding theorem.
Finally, our investigation of purity leads to the conclusion that even the introduction of explicit definitions in a proof can violate purity. We argue that although both involve explicit definition, our proof of the embedding theorem is pure while Hilbert’s is not. Thus the determination of whether an argument is pure turns on the content of the particular proof. Moreover, formalizing the situation does not provide a tool for characterizing purity.
... The method of model construction employed in this chapter was first introduced by Ehrenfeucht and Mostowski [ 2] in order to produce models with many automorphisms. The technique was later exploited by Morley, both in his dissertation [22] and in a paper [23] for the model theory symposium, for a variety of purposes. The author was led to the applications here in attempting to find another proof of results of Gaifman [9] concerning the constructible universe. ...
... Since well-ordering is equivalent to the absence of descending chains of type co, we might well inquire whether there is a corresponding theorem for structures having no descending chains of type co 1. The answer is already evident from work of Morley [23], who has shown that if T is a tidy countable theory having a model 9J of cardinality ~1~o (~1o~ is the smallest uncountable cardinal ja such that, if X < ,u, then 2 x < ta), then there is a set of formulas ~; including T,such that: (a) ifH is a finite subset of ~, then there is an ordered structure ~ of power arbitrarily close to ~1,o whose universe is a subset of 19~1 and such that any increasing sequence from ~f satisfies the formulas of H; (b) if 9C is an ordered structure, then there exists a structure ~ for which K(~, 9C, ~ ). Now suppose that this structure 9J of cardinality ~1~o is also a partly ordered structure having no descending chain of type to 1 (i.e., having no subset of order type co ~' under <~ ). ...
... (2 2 LS(K) ) + (see Section 4 of Chapter VII of [Shc]). Morley's proof [Mo2] gives a better upper bound in certain situations: for a class K that is P C ℵ 0 , the Hanf number of K is ≤ ω 1 . ...
The results in this paper are in a context of abstract elementary classes identified by Shelah and Villaveces in which the amalgamation property is not assumed. The long-term goal is to solve Shelah's Categoricity Conjecture in this context. Here we tackle a problem of Shelah and Villaveces by proving that in their context, the uniqueness of limit models follows from categoricity under the assumption that the subclass of amalgamation bases is closed under unions of bounded, increasing chains.
... This problem is related to a wider model theoretical program. Recall that Morley's Theorem states that for any countable theory T and a type p, there are models of T that omit p of arbitrarily large cardinality, if and only if the possible cardinalities of such models are unbounded below ω 1 , [10]. Shelah has generalized this statement and showed that there is a connection between the cardinalities of models that satisfy a theory T and omits a type p and the supremum of definable orders which can be guaranteed to be well founded in ω-models of a related theory 1 . ...
In this paper we study the spectrum of heights of transitive models of theories extending , under various definitions. In particular, we investigate the consistency strength of making those spectra as simple as possible.
... Every infinite model of a first order theory has a proper elementary extension and so each theory has arbitrarily large models. Morley [Mor65] showed that every sentence of L ω1,ω that has models up to ω1 has arbitrarily large models and provided counterexamples showing that cardinal was minimal. Thus he showed the Hanf number for existence of L ω1,ω -sentences in a countable vocabulary is ω1 . ...
Theorem: There is a {\em complete sentence} of such that has maximal models in a set of cardinals that is cofinal in the first measurable while has no maximal models in any .
... Morley [26] showed for an arbitrary sentence of L ω 1 ,ω (τ ) the Hanf number for existence is ω 1 when τ is countable (More generally, it is (2 |τ | ) + . [28]); the situation for complete sentences is much more complicated. ...
This paper contains portions of Baldwin’s talk at the Set Theory and Model Theory Conference (Institute for Research in Fundamental Sciences, Tehran, October 2015) and a detailed proof that in a suitable extension of ZFC, there is a complete sentence of Lω1,ω that has maximal models in cardinals cofinal in the first measurable cardinal and, of course, never again.
... With the extensive preliminaries out of the way, we are now able to obtain our characterization of sentences of L ω 1 ,ω that have Borel complete expansions. We begin with a classical result of Morley [14] that characterizes which sentences of L ω 1 ,ω have Ehrenfeucht-Mostowski models. The novelty will be that we apply this result to a class of flat structures. ...
We develop general machinery to cast the class of potential canonical Scott sentences of an infinitary sentence as a class of structures in a related language. From this, we show that has a Borel complete expansion if and only if divides Aut(M) for some countable model . Using this, we prove that for theories asserting that is a countable family of cross cutting equivalence relations with h(n) classes, if h(n) is uniformly bounded then is not Borel complete, providing a converse to Theorem~2.1 of \cite{LU}.
... This means that for every large finitely accessible category K there is a faithful functor Lin → K preserving directed colimits. [8] does not contain a proof of this result -Makkai and Paré just say that it essentially follows from the work of Morley [9]. Since there are many applications of this result (see, e.g., [5]), it might be useful to give an explicit proof of it. ...
We give a purely category-theoretic proof of the result of Makkai and Par\'e saying that the category of linearly ordered sets and order preserving injective mappings is a minimal finitely accessible category. We also discuss the existence of a minimal -accessible category.
... Any AEC in a countable vocabulary with countable Löwenheim-Skolem number with models up to ω1 has arbitrarily large models. And Morley [Mor65] gave easy examples showing this bound was tight for arbitrary sentences of L ω1,ω . But it was almost 40 years later that Hjorth [Hjo02,Hjo07] showed this bound is also tight for complete-sentences of L ω1,ω . ...
We prove that a strongly compact cardinal is an upper bound for a Hanf number
for amalgamation, etc. in AECs using both semantic and syntactic methods. To
syntactically prove non-disjoint amalgamation, a different presentation theorem
than Shelah's is needed. This relational presentation theorem has the added
advantage of being {\it functorial}, which allows the transfer of amalgamation.
... Since Shelah proved that any AEC K is a P CΓ(κ, 2 κ ) class (i.e., K = P C L (T, Γ) with T = κ and Γ = 2 κ ) with LS(K) = κ, as a consequence of the M. Morley's omitting types theorem -see [Mo65]-we have that there exists a cardinality H 1 ∶= ℶ (2 µ ) + such that if K is an AEC of Lstructures with LS(K) = κ such that if there exists M ∈ K of cardinality > H 1 then there exists a model in K in any cardinality > H 1 . But its existence strongly depends on the existence of Hanf numbers in (discrete) omiting type classes (which depends on infinitary logics, see [Mo65]). Despite there are some recent works about some intends of providing suitable notions of metric infinitary logics (see [Ea14]), it is still open if this can yields a suitable analysis of Hanf numbers which implies that metric PC classes have a Hanf number, and so MAECs does as well. ...
In my PhD thesis a version of Shelah's Presentation Theorem in the setting of
Metric Abstract Elementary Classes was proved, where we claimed that the new
function symbols are not necessarily uniformly continuous. In this paper we
provide a proof they are in fact uniformly continuous.
... Elimination of quantifiers can arise in two radically different ways. Morley [71] introduced the idea of making a definitional extension of a first order theory by adding a predicate symbol for each definable relation . 8 At the time this appeared only a technical convenience. ...
We propose a criterion to regard a property of a theory (in first or second order logic) as virtuous: the property must have significant mathematical consequences for the theory (or its models). We then rehearse results of Ajtai, Marek, Magidor, H. Friedman and Solovay to argue that for second order logic, ‘categoricity’ has little virtue. For first order logic, categoricity is trivial; but ‘categoricity in power’ has enormous structural consequences for any of the theories satisfying it. The stability hierarchy extends this virtue to other complete theories. The interaction of model theory and traditional mathematics is examined by considering the views of such as Bourbaki, Hrushovski, Kazhdan, and Shelah to flesh out the argument that the main impact of formal methods on mathematics is using formal definability to obtain results in ‘mainstream’ mathematics. Moreover, these methods (e.g., the stability hierarchy) provide an organization for much mathematics which gives specific content to dreams of Bourbaki about the architecture of mathematics.
... We describe the next example, based on [11], in detail since the particular formulation is important. ...
We introduce the notion of a ‘pure’ Abstract Elementary Class to block trivial counterexamples. We study classes of models of bipartite graphs and show: Main Theorem (cf. Theorem 3.34 and Corollary 3.38): If is a strictly increasing sequence of characterizable cardinals (Definition 2.1) whose models satisfy JEP, there is an -sentence whose models form a pure AEC and(1)
The models of satisfy JEP, while JEP fails for all larger cardinals and AP fails in all infinite cardinals.
(2)
There exist non-isomorphic maximal models of in , for all , but no maximal models in any other cardinality; and
(3)
has arbitrarily large models.
In particular this shows the Hanf number for JEP and the Hanf number for maximality for pure AEC with Löwenheim number are at least . We show that although for each implies the full amalgamation property, for each does not imply the full joint embedding property. We prove the main combinatorial device of this paper cannot be used to extend the main theorem to a complete sentence.
... A key distinction between first order and infinitary logic is the upwards Löwenheim-Skolem theorem. While a first order sentence that has an infinite model has models of every infinite cardinal, there are sentences [15,9,11,13,21] φ κ of L ω1,ω with maximal models in many cardinalities κ below ω1 ; φ κ is said to characterize κ. But most (all?) of these examples have many models in each cardinal where they have a model. ...
We place the following new result in a context with other small cardinal axioms. The following statement is relatively consistent with ZFC. Suppose κ is a regular cardinal with κ <κ = κ. Then, there is a complete sentence ψ of Lω 1 ,ω such that ψ is categorical in every λ < κ but has 2 κ models of cardinality κ. We say a cardinal κ is small if κ ≤ sup(ℵ ω , 2 ω). By a small cardinal axiom we mean one which describes the properties of only small cardinals. Two examples are Martin's axiom and the very weak generalized continuum hypothesis (VWGCH): 2 ℵn < 2 ℵn+1 for n < ω. Although neither of these axioms says anything about larger cardinals we show that they have contradictory mathematical consequences for structures of arbitrary cardinal-ity. VWGCH implies that for sentences of L ω1,ω categoricity up ℵ ω implies existence and categoricity of models in all powers. Under Martin's axiom there is a sentence categorical in all κ < 2 ℵ0 and with no models beyond the continuum.
... Since Shelah proved that any AEC is a PCΓ (κ, 2 κ ) class with LS(K) = κ, as a consequence of the M. Morley's omitting types theorem -see [Mor65]-we have that there exists a cardinality H 1 := (2 µ ) + such that if K is an AEC of L-structures with LS(K) = κ such that if there exists M ∈ K of cardinality > H 1 then there exists a model in K in any cardinality > H 1 . But its existence strongly depends on the existence of Hanf numbers in (discrete) omiting type classes (which depends on in nitary logics, see [Mor65] ...
... Theorem VII.3.6 of [She78] finds an indiscernible tree in the first order case on ≤ω ω but we want to omit types as well. The basic plan of the proof dates to Morley [Mor65]. We indicate the modifications needed for the more complicated combinatorics to build models to omit types that are over indiscernible trees instead of over linear orders. ...
We prove two results on the stability spectrum for Lω1,ω. Here denotes an appropriate notion (at or mod) of Stone space of m-types over M. (1) Theorem for unstable case: Suppose that for some positive integer m and for every α < δ(T), there is an M ∈ K with . Then for every λ ≥ |T|, there is an M with . (2) Theorem for strictly stable case: Suppose that for every α < δ (T), there is Mα ∈ K such that λα = |Mα| ≥ ℶα and . Then for any μ with μℵ0 > μ, K is not i-stable in μ. These results provide a new kind of sufficient condition for the unstable case and shed some light on the spectrum of strictly stable theories in this context. The methods avoid the use of compactness in the theory under study. In this paper, we expound the construction of tree indiscernibles for sentences of Lω1,ω. Further we provide some context for a number of variants on the Ehrenfeucht–Mostowski construction.
... Our concept of viability is reminiscetu of (and, indeed, was influenced by) -~ a technique employed in model theory by Morley [7], We need a preliminary definition first. ...
... Let L ′ be a language with built-in Skolem functions that extends L and that includes constants for the elements of B. Let T ′ be a Skolem theory in L ′ that extends the theory of A , b b∈B . By Morley's Omitting Types Theorem and the choice of λ, there is an Ehrenfeucht-Mostowski model N ′ of T ′ over a dense ordering U of indiscernibles of cardinality ≥ λ such that N ′ omits the type p (see Morley [1965], Theorem 3.1, or Chang-Keisler [1973], Exercise 7.2.4). Let V be a dense ...
We investigate a notion called uniqueness in power � that is akin to categoricity in power �, but is based on the cardinality of the generating sets of models instead of on the cardinality of their universes. The notion is quite useful for formulating categoricity-like questions regarding powers below the cardinality of a theory. We prove, for (uncountable) universal theories T , that if T is �-unique for one uncountable �, then it is �-unique for every uncountable �; in particular, it is categorical in powers greater than the cardinality of T . It is well known that the notion of categoricity in power exhibits certain irregu-
... In contrast, well-order is not definable in the languages L λ ω (even with extra predicates) [LE66], and methods developed first in [Mor65] give the estimates ...
Let λ be an infinite cardinal number. The ordinal number δ(λ) is the least ordinal γ such that if ϕ is any sentence of L λ + ω , with a unary predicate D and a binary predicate ≺, and ϕ has a model ℳ with 〈D ℳ ,≺ ℳ 〉 a well-ordering of type ≥γ, then ϕ has a model ℳ ' where 〈D ℳ ' ,≺ ℳ ' 〉 is non-well-ordered. One of the interesting properties of this number is that the Hanf number of L λ + ω is exactly ℶ δ(λ) . It was proved by Barwise and Kunen that if ℵ 0 <λ<κ are regular cardinal numbers, then there is a forcing extension, preserving cofinalities, such that in the extension 2 λ =κ and δ(λ)<λ ++ . We improve this result by proving the following: Suppose ℵ 0 <λ<θ≤κ are cardinal numbers such that λ <λ =λ; cf(θ)≥λ + and μ λ <θ whenever μ<θ; κ λ =κ. Then there is a forcing extension preserving all cofinalities, adding no new sets of cardinality <λ, and such that in the extension 2 λ =κ and δ(λ)=θ.
... Clearly c+ is possible by taking (V(c+), E). The maximum cardinality of models of <p is therefore 3 ., and so the Hanf number of LN is greater than 3 .. By suitably modifying (l)-(9), and I-IX of §2 and imitating the proof of Corollary 1, we obtain the following Chang [2] and Morley [3]. ...
... The class of cardinals k not omitting a type 2 (for some theory T). See Morley [33] [4], and also Exercise 3.2.15, where one will find a finite 2 which admits (k, X) iff X < k < N"(A). ...
For every class of cardinals containing 0 and 1, there exists a finite set T of terms, such that X is precisely the class of cardinals in which T is universal.
We prove the following Löwenheim-Skolem theorems for first-order Gödel logic:
(1)For the Gödel logic G[0,1], a sentence ϕ has models of every infinite cardinality if and only if it has a model of cardinality ℶω(=sup{ℵ0,2ℵ0,…}).
(2)For an arbitrary Gödel logic GT, a sentence ϕ has models of every infinite cardinality if and only if it has a model of cardinality ℶω1.
Moreover, (1) becomes false if ℶω is replaced by a smaller cardinality, and (2) becomes false if ℶω1 is replaced by a smaller cardinality.
We show that ℶ(2LS(K))+ is the lower bound to the Hanf numbers for the length of the order property and for stability in stable abstract elementary classes (AECs). Our examples satisfy the joint embedding property, no maximal model, and (<ℵ0)-tameness but not necessarily the amalgamation property. We also define variations on the order and syntactic order properties insofar as it allows the index set to be linearly ordered rather than well-ordered. Combining with Shelah's stability theorem, we deduce that our examples can have the order property up to any μ<ℶ(2LS(K))+. Boney conjectured that the joint embedding property is needed for two type-counting lemmas. We solved the conjecture by showing it is independent of ZFC.
This paper contains portions of Baldwin's talk at the Set Theory and Model Theory Conference (Institute for Research in Fundamental Sciences, Tehran, October 2015) and a detailed proof that in a suitable extension of ZFC, there is a complete sentence of that has maximal models in cardinals cofinal in the first measurable cardinal and, of course, never again.
Conference: "MODERN ALGEBRA AND ANALYSIS AND THEIR APPLICATIONS" (Dedicated to Professor Mirjana Vuković)
At: Academy of Sciences and Arts of Bosnia and Herzegovina, Bosnia and Herzegovina Sarajevo, September 20-22, 2018
In book: SARAJEVO JOURNAL OF MATHEMATICS
Journal: SARAJEVO JOURNAL OF MATHEMATICS Vol. 14 (27), No. 2
ISSN: 1840 - 0655 (Printed edition) ISSN: 2233 -1964 (Online edition)
This is a book in preparation about large cardinal, infinitary logic and algebra
To buy: https://publicaciones.uacm.edu.mx/gpd-teoria-de-conjuntos-algebra-y-grandes-cardinales.html
Every countable transitive model M of ZF (without choice) has an ordinal preserving extension satisfying ZF, of power '^MC\Orv application to infinitary logic is given.
Let κ and λ be infinite cardinals such that λ ≤ λ (we have new information for the case when κ ≤ λ). Let T be a theory in Lκ +, ω of cardinality at most κ, let . Now define
Our main concept in this paper is is a theory in Lκ +, ω of cardinality κ at most, and φ(x, y) ϵ Lκ +, ω}. This concept is interesting because of
Theorem 1. Let T ⊆ Lκ +, ω of cardinality ≤ κ, and. If
then (∀χ > κ)I(χ, T) = 2χ (where I(χ, T) stands for the number of isomorphism types of models of T of cardinality χ).
Many years ago the second author proved that . Here we continue that work by proving
Theorem 2. .
Theorem 3. For everyκ ≤ λwe have.
For some κ or λ we have better bounds than in Theorem 3, and this is proved via a new two cardinal theorem.
Theorem 4. For every T ⊆ Lκ +, ω, and any set of formulas ⊆ Lκ +, ω such thatT ⊇ Lκ +, ω, if T is (, μ)-unstable for μ satisfyingμμ*(λ,κ) = μ then T is-unstable (i.e. for every χ ≥ λ, T is (, χ)-unstable). Moreover, T is Lκ +, ω-unstable.
In the second part of the paper, we show that always in the applications it is possible to replace the function I(χ, T) by the function IE(χ, T), and we give an application of the theorems to Boolean powers.
Paul Erdős (26 March 1913—20 September 1996) was a mathematician par excellence whose results and initiatives have had a large impact and made a strong imprint on the doing of and thinking about mathematics. A mathematician of alacrity, detail, and collaboration, Erdős in his six decades of work moved and thought quickly, entertained increasingly many parameters, and wrote over 1500 articles, the majority with others. His modus operandi was to drive mathematics through cycles of problem, proof, and conjecture, ceaselessly progressing and ever reaching, and his modus vivendi was to be itinerant in the world, stimulating and interacting about mathematics at every port and capital.
This chapter presents sixteen lectures on the development of mathematical logic and the study of the foundations of mathematics in the years 1930–1964, delivered by the author in the Summer School in Vaasa, Finland in the summer of 1964. The chapter distinguishes three major movements in the philosophy of mathematics: the intuitionism of Brouwer, the logicism of Frege and Russell, and the formalism of Hilbert. The logicism of Frege and Russell tries to reduce mathematics to logic. This seemed to be an excellent program, but when it was put into effect, it turned out that there is simply no logic strong enough to encompass the whole of mathematics. Thus what remained from this program is a reduction of mathematics to set theory. The formalism of Hilbert sets up a program which requires that the whole of mathematics be axiomatized and, that these axiomatic theories be then proved consistent by using very simple combinatorial arguments. The chapter discusses the changes that the three schools underwent during the years 1930–960.
We prove two results on the stability spectrum for Lω1,ω. Here S m i (M) denotes an appropriate notion (at or mod) of Stone space of m-types over M. Theorem A. (unstable case) Suppose that for some positive integer m and for every α < δ(T), there is an M ∈ K with |S m i (M) |> |M | ℶα(|T |). Then for every λ ≥ |T |, there is an M with |S m i (M) |> |M | = λ. Theorem B. (strictly stable case) Suppose that for every α < δ(T), there is Mα ∈ K such that λα = |Mα | ≥ ℶα and |S m i (Mα) |> λα. Then for any µ with µ ℵ0> µ, K is not i-stable in µ. These results provide a new kind of sufficient condition for the unstable case and shed some light on the spectrum of strictly stable theories in this context. The methods avoid the use of compactness in the theory under study. In the Section 4, we expound the construction of tree indiscernibles for sentences of Lω1,ω. Further we provide some context for a number of variants on the Ehrenfeucht-Mostowski construction. 1
This paper explores how the Generalized Continuum Hypothesis (GCH) arose from Cantor's Continuum Hypothesis in the work of Peirce, Jourdain, Hausdorff, Tarski, and how GCH was used up to Gödel's relative consistency result.
This paper is an i exposition of some results that can be obtained by fairly simple methods, namely, the use of cardinality arguments and partition theorems. No results presented here are new but many of the proofs are. These may be of interest even to those already familiar with the results.
We solve a problem of Friedman by showing the existence of a logic stronger than first-order logic even for countable models, but still satisfying the general compactness theorem, assuming e.g. the existence of a weakly compact cardinal. We also discuss several kinds of generalized quantifiers.
In these lectures we survey several fundamental methods of constructing models. We examine which logics admit which constructions,
and illustrate their use with examples. The lectures begin with methods available only in classical first order logic and
proceed to methods available in the infinitary logics Lwi w\mathcal{L}_{\omega _i \omega } and L
∝ ω. We shall focus on three recent developments which have become more prominent in model theory since the publication of the
book Chang and Keisler 7 . They are recursively saturated models (Lectures 2 and 3), model theoretic forcing (Lectures 5 and
6), and soft model theory (Lectures 4 and 8).
In a lecture in Kazan (1977), Goncharov dubbed a number of problems regarding the classification of computable members of various classes of structures. Some of the problems seemed likely to have nice answers, while others did not. At the end of the lecture, Shore asked what would be a convincing negative result. The goal of the present article is to consider some possible answers to Shore's question. We consider structures of some computable language, whose universes are computable sets of constants. In measuring complexity, we identify with its atomic diagram D(), which, via the Gdel numbering, may be treated as a subset of . In particular, is computable if D() is computable. If K is some class, then Kc denotes the set of computable members of K. A computable characterization for K should separate the computable members of K from other structures, that is, those that either are not in K or are not computable. A computable classification (structure theorem) should describe each member of Kc up to isomorphism, or other equivalence, in terms of relatively simple invariants. A computable non-structure theorem would assert that there is no computable structure theorem. We use three approaches. They all give the correct answer for vector spaces over Q, and for linear orderings. Under all of the approaches, both classes have a computable characterization, and there is a computable classification for vector spaces, but not for linear orderings. Finally, we formulate some open problems.
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