Types and Stone spaces

2018 ◽  
Author(s):  
Tim Button ◽  
Sean Walsh
Keyword(s):  
Topological Space ◽  
Model Theory ◽  
Algebraic Structure ◽  
Possible Worlds ◽  
Elementary Extension ◽  
Philosophical Significance ◽  
Related Algebra ◽  
Contemporary Study ◽  
Stone Spaces

Types are one of the cornerstones of contemporary model theory. Simply put, a type is the collection of formulas satisfied by an element of some elementary extension. The types can be organised in an algebraic structure known as a Lindenbaum algebra. But the contemporary study of types also treats them as the points of a certain kind of topological space. These spaces, called ‘Stone spaces’, illustrate the richness of moving back-and-forth between algebraic and topological perspectives. Further, one of the most central notions of contemporary model theory—namely stability—is simply a constraint on the cardinality of these spaces. We close the chapter by discussing a related algebra-topology ‘duality’ from metaphysics, concerning whether to treat propositions as sets of possible worlds or vice-versa. We show that suitable regimentations of these two rival metaphysical approaches are biinterpretable (in the sense of chapter 5), and discuss the philosophical significance of this rapprochement.

Many-valued logics of extended Gentzen style II

1970 ◽  
Vol 35 (4) ◽  
pp. 493-528 ◽  
Author(s):  
Moto-o Takahashi
Keyword(s):  
Topological Space ◽  
Model Theory ◽  
Natural Extension ◽  
Preceding Paper ◽  
Considerable Extent ◽  
Continuous Logic ◽  
First Order ◽  
Natural Extensions ◽  
Fundamental Theorems ◽  
Order Continuous

In the monograph [1] of Chang and Keisler, a considerable extent of model theory of the first order continuous logic (that is, roughly speaking, many-valued logic with truth values from a topological space) is ingeniously developed without using any notion of provability.In this paper we shall define the notion of provability in continuous logic as well as the notion of matrix, which is a natural extension of one in finite-valued logic in [2], and develop the syntax and semantics of it mostly along the line in the preceding paper [2]. Fundamental theorems of model theory in continuous logic, which have been proved with purely model-theoretic proofs (i.e. those proofs which do not use any proof-theoretic notions) in [1], will be proved with proofs which are natural extensions of those in two-valued logic.

Subsemigroups of Stone-Čech compactifications

1994 ◽  
Vol 116 (1) ◽  
pp. 99-118 ◽  
Author(s):  
Talin Budak ◽  
Nilgün Işik ◽  
John Pym
Keyword(s):  
Positive Integer ◽  
Topological Space ◽  
Algebraic Structure ◽  
Discrete Group ◽  
Continuous Homomorphism ◽  
Discrete Space ◽  
Publication Date ◽  
Finite Subgroups ◽  
Discrete Subspace

The Stone–Čech compactification βℕ of the discrete space ℕ of positive integer is a very large topological space; for example, any countable discrete subspace of the growth ℕ* = βℕ/ℕ has a closure which is homeomorphic to βℕ itself ([23], §3·5] Now ℕ, while hardly inspiring as a discrete topological space, has a rich algebrai structure. That βℕ also has a semigroup structure which extends that of (ℕ, +) and in which multiplication is continuous in one variable has been apparent for about 30 years. (Civin and Yood [3] showed that βG was a semigroup for each discrete group G, and any mathematician could then have spotted that βℕ was a subsemigroup of βℕ.) The question which now appears natural was explicitly raised by van Douwen[6] in 1978 (in spite of the recent publication date of his paper), namely, does ℕ* contain subspaces simultaneously algebraically isomorphic and homeomorphic to βℕ? Progress on this question was slight until Strauss [22] solved it in a spectacular fashion: the image of any continuous homomorphism from βℕ into ℕ* must be finite, and so the homomorphism cannot be injective. This dramatic advance is not the end of the story. It is still not known whether that image can contain more than one point. Indeed, what appears to be one of the most difficult questions about the algebraic structure of βℕ is whether it contains any non-trivial finite subgroups

Kripke, Saul Aaron (1940–)

2018 ◽  
Author(s):  
Michael Jubien
Keyword(s):  
Philosophy Of Mind ◽  
Philosophy Of Language ◽  
Possible Worlds ◽  
A Priori ◽  
Philosophical Significance ◽  
Essential Properties ◽  
Proper Name ◽  
Late Twentieth ◽  
Concept Of Truth ◽  
And Mathematics

Saul Kripke is one of the most important and influential philosophers of the late twentieth century. He is also one of the leading mathematical logicians, having done seminal work in areas including modal logic, intuitionistic logic and set theory. Although much of his work in logic has philosophical significance, it will not be discussed here. Kripke’s main contributions fall in the areas of metaphysics, philosophy of language, epistemology, philosophy of mind and philosophy of logic and mathematics. He is particularly well known for his views on and discussions of the following topics: the concepts of necessity, identity and ‘possible worlds’; ‘essentialism’ – the idea that things have significant essential properties; the question of what determines the referent of an ordinary proper name and the related question of whether such names have meanings; the relations among the concepts of necessity, analyticity, and the a priori; the concept of belief and its problems; the concept of truth and its problems; and scepticism, the idea of following a rule, and Ludwig Wittgenstein’s ‘private language argument’. This entry will be confined to the topics of identity, proper names, necessity and essentialism.

Quinus ab Omni Nævo Vindicatus1

1997 ◽  
Vol 23 ◽  
pp. 25-65 ◽  
Author(s):  
John P. Burgess
Keyword(s):  
Modal Logic ◽  
Model Theory ◽  
Possible Worlds ◽  
Modal Logics ◽  
Singular Terms ◽  
The 1960S

Today there appears to be a widespread impression that W. V. Quine's notorious critique of modal logic, based on certain ideas about reference, has been successfully answered. As one writer put it some years ago: “His objections have been dead for a while, even though they have not yet been completely buried.” What is supposed to have killed off the critique? Some would cite the development of a new ‘possible-worlds’ model theory for modal logics in the 1960s; others, the development of new ‘direct’ theories of reference for names in the 1970s.These developments do suggest that Quine's unfriendliness towards any formal logics but the classical and indifference towards theories of reference for any singular terms but variables were unfortunate.

The metamathematics of model theory: Discovering language in action

1981 ◽  
Vol 46 (3) ◽  
pp. 490-498
Author(s):  
Douglas E. Miller
Keyword(s):  
Topological Space ◽  
Model Theory ◽  
First Order ◽  
Elementary Classes ◽  
Countable Models

AbstractWe discuss the problem of defining the collection of first-order elementary classes in terms of the natural topological space of countable models.

scholarly journals On generalized topological groups

Filomat ◽  
2013 ◽  
Vol 27 (4) ◽  
pp. 567-575 ◽  
Author(s):  
Murad Hussain ◽  
ud Khan ◽  
Cenap Özel
Keyword(s):  
Topological Group ◽  
Topological Space ◽  
Algebraic Structure ◽  
Topological Structure ◽  
Topological Groups ◽  
Generalized Topological Space ◽  
And Inversion

In this paper, we initiate the study of generalized topological groups. A generalized topological group has the algebraic structure of groups and the topological structure of a generalized topological space defined by A. Cs?sz?r [2] and they are joined together by the requirement that multiplication and inversion are G-continuous. Every topological group is a G-topological group whereas converse is not true in general. Quotients of generalized topological groups are defined and studied.

scholarly journals Semigroup structures for families of functions, I. Some Homomorphism Theorems

1967 ◽  
Vol 7 (1) ◽  
pp. 81-94 ◽  
Author(s):  
Kenneth D. Magill
Keyword(s):  
Continuous Function ◽  
Topological Space ◽  
Algebraic Structure ◽  
Topological Structure ◽  
Continuous Functions ◽  
The Family ◽  
Image Position ◽  
Natural Way

This is the first of several papers which grew out of an attempt to provide C (X, Y), the family of all continuous functions mapping a topological space X into a topological space Y, with an algebraic structure. In the event Y has an algebraic structure with which the topological structure is compatible, pointwise operations can be defined on C (X, Y). Indeed, this has been done and has proved extremely fruitful, especially in the case of the ring C (X, R) of all continuous, real-valued functions defined on X [3]. Now, one can provide C(X, Y) with an algebraic structure even in the absence of an algebraic structure on Y. In fact, each continuous function from Y into X determines, in a natural way, a semigroup structure for C(X, Y). To see this, let ƒ be any continuous function from Y into X and for ƒ and g in C(X, Y), define ƒg by each x in X.

scholarly journals LOGICS FOR PROPOSITIONAL CONTINGENTISM

2017 ◽  
Vol 10 (2) ◽  
pp. 203-236 ◽  
Author(s):  
PETER FRITZ
Keyword(s):  
Model Theory ◽  
Possible Worlds ◽  
Second Order ◽  
Order Logic ◽  
Existential Quantifier ◽  
Philosophical Theory ◽  
Propositional Operator ◽  
Second Order Logic

AbstractRobert Stalnaker has recently advocated propositional contingentism, the claim that it is contingent what propositions there are. He has proposed a philosophical theory of contingency in what propositions there are and sketched a possible worlds model theory for it. In this paper, such models are used to interpret two propositional modal languages: one containing an existential propositional quantifier, and one containing an existential propositional operator. It is shown that the resulting logic containing an existential quantifier is not recursively axiomatizable, as it is recursively isomorphic to second-order logic, and a natural candidate axiomatization for the resulting logic containing an existential operator is shown to be incomplete.

Imaginary modules

1992 ◽  
Vol 57 (2) ◽  
pp. 698-723 ◽  
Author(s):  
T. G. Kucera ◽  
M. Prest
Keyword(s):  
Model Theory ◽  
Equivalence Relation ◽  
Algebraic Structure ◽  
Stability Theory ◽  
Equivalence Classes ◽  
General Context ◽  
The Subject ◽  
Fundamental Insight ◽  
Necessary And Sufficient ◽  
Stable Structures

It was a fundamental insight of Shelah that the equivalence classes of a definable equivalence relation on a structure M often behave like (and indeed must be treated like) the elements of the structure itself, and that these so-called imaginary elements are both necessary and sufficient for developing many aspects of stability theory. The expanded structure Meq was introduced to make this insight explicit and manageable. In Meq we have “names” for all things (subsets, relations, functions, etc.) “definable” inside M. Recent results of Bruno Poizat allow a particularly simple and more or less algebraic modification of Meq in the case that M is a module. It is the purpose of this paper to describe this nearly algebraic structure in such a way as to make the usual algebraic tools of the model theory of modules readily available in this more general context. It should be pointed out that some of our discussion has been part of the “folklore” of the subject for some time; it is certainly time to make this “folklore” precise, correct, and readily available.Modules have proved to be good examples of stable structures. Not, we mean, in the sense that they are well-behaved (which, in the main, they are), but in the sense that they are straightforward enough to provide comprehensible illustrations of concepts while, at the same time, they have turned out to be far less atypical than one might have supposed. Indeed, a major feature of recent stability theory has been the ubiquitous appearance of modules or more general “abelian structures” in abstract stable structures.

Sign in / Sign up

Export Citation Format

Share Document