scholarly journals Understanding preservation theorems: chapter VI of Proper and Improper Forcing, I

2013 ◽  
Vol 53 (1-2) ◽  
pp. 171-202
Author(s):  
Chaz Schlindwein
Keyword(s):  
Preservation Theorems

scholarly journals On generalized γµ-closed sets and related continuity

2021 ◽  
Vol 13 (2) ◽  
pp. 483-493
Author(s):  
Ritu Sen
Keyword(s):  
Topological Space ◽  
Main Interest ◽  
Closed Sets ◽  
Separation Axioms ◽  
Preservation Theorems ◽  
Generalized Topological Space ◽  
New Type ◽  
Weak Separation

Abstract In this paper our main interest is to introduce a new type of generalized open sets defined in terms of an operation on a generalized topological space. We have studied some properties of this newly defined sets. As an application, we have introduced some weak separation axioms and discussed some of their properties. Finally, we have studied some preservation theorems in terms of some irresolute functions.

Preservation theorems for MTL-chains

2010 ◽  
Vol 19 (3) ◽  
pp. 490-511 ◽  
Author(s):  
C. J. Van Alten
Keyword(s):  
Preservation Theorems

Preservation Theorems

2012 ◽  
pp. 227-252
Author(s):  
Jaroslav Nešetřil ◽  
Patrice Ossona de Mendez
Keyword(s):  
Preservation Theorems

Syntactic characterisations of amalgamation, convexity and related properties

1974 ◽  
Vol 39 (3) ◽  
pp. 433-451 ◽  
Author(s):  
Paul D. Bacsich ◽  
Dafydd Rowlands Hughes
Keyword(s):  
General Idea ◽  
Semantic Property ◽  
Cylindric Algebras ◽  
Real Numbers ◽  
Elementary Class ◽  
Preservation Theorems ◽  
Semantic Properties ◽  
Extra Structure

We prove that certain syntactic conditions similar to separation principles on a theory are equivalent to semantic properties such as amalgamation and strong amalgamation, by showing that appropriate classes of structures are definable by Lω1ω-sentences. Then we characterise the elements of core models and thus give a natural proof of Rabin's characterisation of convex theories.The notion of a syntactic characterisation of a semantic property of a theory is by now fairly well known. The earliest such were the classical preservation theorems: For example, a theorem of Lyndon characterised the theories whose models were closed under homomorphic images as those with a set of positive axioms.Presumably the notion of syntactic characterisation can be made precise, but it is probably better at this stage to leave it vague. The general idea is that theories are “algebras” (cylindric algebras, or logical categories, with suitable extra structure) and that a semantic property P of theories is syntactically characterisable if the class of theories with P is an “elementary” class of “algebras.”When one codes countable theories as real numbers, a syntactically characterisable property will be arithmetical. The converse does not seem reasonable, especially as it is often fairly easy to prove a property arithmetical (using extra predicates, usually), when we may not be able to find a syntactic characterisation.

Some useful preservation theorems

1983 ◽  
Vol 48 (2) ◽  
pp. 427-440 ◽  
Author(s):  
Kevin J. Compton
Keyword(s):  
Current Model ◽  
Classical Model ◽  
Analytical Techniques ◽  
Order Logic ◽  
First Order Logic ◽  
Central Position ◽  
Combinatorial Structures ◽  
Asymptotic Growth ◽  
First Order ◽  
Preservation Theorems

The study of preservation theorems for first order logic was the focus of much research by model theorists in the 1960's. These theorems, which came to form the foundation for classical model theory, characterize first order sentences and theories that are preserved under operations such as the taking of unions or submodels (see Chang and Keisler [5] for a discussion of preservation theorems for first order logic). In current model theoretic research, logics richer than first order logic and applications of logic to other parts of mathematics have assumed the central position. In the former area, preservation theorems are not so important; in the latter, especially in applications to algebra, many of the techniques developed for proving these theorems have been useful.In this paper I prove several preservation theorems for first order logic which I discovered while investigating the asymptotic growth of classes of finite combinatorial structures. The significance of these theorems lies in their applications to problems in finite combinatorics. Since the applications require combinatorial and analytical techniques that are not pertinent to logical questions discussed here, I shall present them in another paper [7].

Approximation theorems and model theoretic forcing

1976 ◽  
Vol 41 (1) ◽  
pp. 59-72 ◽  
Author(s):  
Victor Harnik
Keyword(s):  
Approximation Theorem ◽  
Invariant Set ◽  
Borel Set ◽  
Approximation Theorems ◽  
First Approximation ◽  
Preservation Theorems ◽  
Preservation Theorem

We suggest the name “approximation theorems” for a new kind of theorems which are strong versions of preservation theorems. A typical preservation theorem has the form:A sentence φ is preserved by the relation R (i.e. and imply ) iff there exists φ* ϵ Φ such that ⊧φ↔ φ*where R is a relation between structures and Φ is a class of sentences (depending, of course, on R). Usually, Φ is described in syntactical terms and it is easy to see that every element of it is, indeed, preserved by R. A typical approximation theorem has the form:For every sentence Φ there is a sentence Φ* ϵ Φ (the “approximation” of Φ) such that, for all sentences δ which are preserved under R,(a) if ⊧δ → φ then ⊧δ → φ* and(b) if ⊧φ → δ then ⊧φ* → δ.An approximation theorem obviously implies the corresponding preservation theorem.The first approximation theorem was proved by Vaught in [14] (see Corollary 2.3 below). That paper inspired the present one. Vaught's result is stated in topological terms. It says that for each Borel set B there is an invariant Borel set B* explicitly defined by an Lω1ω sentence, such that B− ⊆ B* ⊆ B+ where B− (B+) is the largest (smallest) invariant set included in (containing) B.

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