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Applications of topology to L ω 1 ω
... For first-order theories this correspondence is essentially known (see e.g. Morley [1974]). It is suggested in Macintyre [2003] that type spaces are more fundamental than models themselves. ...
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 first-order model theory. We use this to generalise some classical results on countable models from first-order model theory to positive model theory.
... Theorem 1 below can easily be derived from Theorem 6.1 of [1]. For completeness, and as the object of later comment, we present a direct proof. ...
Let L be a countable first-order language and T a fixed complete theory in L . If is a model of T , is an n -sequence of variables, and ā=〈a 1 ,…, a n 〉 is an n -sequence of elements of M , the universe of , we let where ranges over formulas of L containing freely at most the variables υ 1 ,…υ n . ā is said to realize in We let be where is the sequence of the first n variables of L .
We identify and highlight certain landmark results in Samson Abramsky’s work which we believe are fundamental to current developments and future trends. In particular, we focus on the use of
topological duality methods to solve problems in logic and computer science;
category theory and, more particularly, free (and co-free) constructions;
these tools to unify the ‘power’ and ‘structure’ strands in computer science.
KeywordsDuality theoryTopological methods in logicVietoris spaceQuantifiers and measuresStructural limitsLindenbaum-Tarski algebrasFree constructions
By utilizing the topological concept of pseudocompactness, we simplify and improve a proof of Caicedo, Dueñez, and Iovino concerning Terence Tao's metastability. We also pinpoint the exact relationship between the Omitting Types Theorem and the Baire Category Theorem by developing a machine that turns topological spaces into abstract logics.
In [3], Flum asked whether, given an L .... formula, if it and its negation are both co-compact, it is equivalent to a finitary formula. A related result indicating a possible affirmative answer was shown by Lindstrom [7] : if L is a generalized first order logic, satisfying the downward LowenheimSkolem theorem and the compactness theorem for countable sets of sentences, L is equivalent to elementary logic. Thus, if one takes the language L formed by building formulas from eeL .... and atomic formulas by -n, finite V, and 3 quantification, L is such a generalized language, and satisfies the downward Lowenheim-Skolem theorem. Further, if both ¢ and ~ ¢ are w-compact, each by itself satisfies the compactness theorem for countable sets of sentences not involving the other. However, L doesn't quite satisfy the hypothesis of the Lindstrom theorem, for ~b and -1 q5 can both appear in sentences of L and in that case it's not clear that compactness holds; in fact, in general we shall show it doesn't. In this paper we answer Flum's question in the negative. We also show that the answer is affirmative when co-compact is replaced by compact. We also discuss various other properties of compact and w-compact formulas. A general background for the subject will be found in [6], and the definition of and information about admissible sets, used in Section 3, will be found in Eli. This paper is a modified version of the author's Ph.D. thesis [4]. It is my pleasure here to thank Professor M. Morley for his many suggestions and encouragement throughout. Also I would like to thank Professor J. Burgess for several helpful observations, and Professors C.Jockush and A.Nerode for helpful conversations.
We describe an infinitary logic for metric structures which is analogous to
. We show that this logic is capable of expressing
several concepts from analysis that cannot be expressed in finitary continuous
logic. Using topological methods, we prove an omitting types theorem for
countable fragments of our infinitary logic. We use omitting types to prove a
two-cardinal theorem, which yields a strengthening of a result of Ben Yaacov
and Iovino concerning separable quotients of Banach spaces.
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