One more aspect of forcing and omitting types

1976 ◽  
Vol 41 (1) ◽  
pp. 73-80
Author(s):  
Zofia Adamowicz
Keyword(s):  
Boolean Algebra ◽  
Set Theory ◽  
Canonical Models ◽  
Defining Relations ◽  
Original Sense ◽  
Tarski Algebra ◽  
Cohen Forcing ◽  
Omitting Types ◽  
True Relation ◽  
Dense Set

As has been shown by the investigations of some mathematicians, there are numerous analogies between forcing and omitting types. Evidence of that can also be found in the application of forcing to model theory. Both methods use the Rasiowa-Sikorski lemma on the existence in a Boolean algebra of an ultrafilter intersecting every dense set from a given denumerable family.In this paper we will not use the terms Cohen forcing and omitting types in their original sense. We shall deal mainly with the Scott-Boolean kind of forcing as a transformation of Cohen's idea and with methods used in logic that consist in finding a suitable ultrafilter in the Lindenbaum-Tarski algebra for a given theory and defining canonical models—under the name of omitting types.The two methods will be confronted to show that Cohen's forcing stems from logical methods. Attention will also be drawn to some differences between forcing in set theory and the general methods of logic.In logic we construct a model by defining relations in a set of constants. In particular when defining a model for set theory we define a certain relation E in a set of constants. It is usually immaterial what E is, in particular whether it is the true relation of membership up to isomorphism.

scholarly journals The Application of Boolean Algebra in Modelling of Leakage Condition of a Car Hydraulic Braking System

2013 ◽  
Vol 18 (2) ◽  
pp. 353-363
Author(s):  
A. Idzikowski ◽  
S. Salamon
Keyword(s):  
Boolean Algebra ◽  
Mathematical Models ◽  
Set Theory ◽  
Boolean Functions ◽  
General Model ◽  
Graphical Model ◽  
Braking System

A general characteristics of a car hydraulic braking system (CHBS) is presented in this publication. A graphical model of properties-component objects is developed for the above-mentioned system. Moreover, four mathematical models in terms of logic, the set theory and the Boolean algebra of Boolean functions are developed. The examination is ended with a general model of the CHBS for n - Boolean variables and the construction and mathematical-technical interpretation of this model is presented.

scholarly journals Aleph–zero categorical Stone algebras

1978 ◽  
Vol 26 (3) ◽  
pp. 337-347 ◽  
Author(s):  
Philip Olin
Keyword(s):  
Boolean Algebra ◽  
Characterization Problem ◽  
A Chain ◽  
Dense Set

AbstractThis paper is a contribution to the problem of characterizing the ℵ0-categorical Stone algebras. If the dense set is either finite or a chain, the problem is solved by reducing it to the ℵ0-categoricity of the skeleton and the dense set, solutions for these being known. If the dense set is a Boolean algebra, we show that this type of reduction works for certain subclasses but not for all such algebras. For generalized Post algebras the characterization problem is solved completely.

Teaching Modeling and Axiomatization with Boolean Algebra

1987 ◽  
Vol 80 (7) ◽  
pp. 528-532
Author(s):  
Michael D. De Villiers
Keyword(s):  
Boolean Algebra ◽  
Set Theory ◽  
Computer Programming ◽  
Programming Courses

Traditionally, Boolean algebra is largely taught in connection with computer programming courses, logic, or set theory. Since Boolean algebra arose from George Boole's application of algebraic principles to the study of logic in 1854, this approach would seem natural.

Omitting types in set theory and arithmetic

1976 ◽  
Vol 41 (1) ◽  
pp. 25-32 ◽  
Author(s):  
Julia F. Knight
Keyword(s):  
Set Theory ◽  
Conservative Extension ◽  
Alternate Proof ◽  
Hanf Number ◽  
Omitting Types ◽  
Infinite Set ◽  
Models Of Set Theory

In [7] it is shown that if Σ is a type omitted in the structure = ω, +, ·, < and complete with respect to Th() then Σ is omitted in models of Th() of all infinite powers. The proof given there extends readily to other models of P. In this paper the result is extended to models of ZFC. For pre-tidy models of ZFC, the proof is a straightforward combination of the methods in [7] and in Keisler and Morley ([9], [6]). For other models, the proof involves forcing. In particular, it uses Solovay and Cohen's original forcing proof that GB is a conservative extension of ZFC (see [2, p. 105] and [5, p. 77]).The method of proof used for pre-tidy models of set theory can be used to obtain an alternate proof of the result for This new proof yields more information. First of all, a condition is obtained which resembles the hypothesis of the “Omitting Types” theorem, and which is sufficient for a theory T to have a model omitting a type Σ and containing an infinite set of indiscernibles. The proof that this condition is sufficient is essentially contained in Morley's proof [9] that the Hanf number for omitting types is so the condition will be called Morley's condition.If T is a pre-tidy theory, Morley's condition guarantees that T will have models omitting Σ in all infinite powers.

Omitting Types in Set Theory and Arithmetic

1976 ◽  
Vol 41 (1) ◽  
pp. 25 ◽  
Author(s):  
Julia F. Knight
Keyword(s):  
Set Theory ◽  
Omitting Types

Omitting types for stable ccc theories

1990 ◽  
Vol 55 (3) ◽  
pp. 1037-1047 ◽  
Author(s):  
Ludomir Newelski
Keyword(s):  
Real Line ◽  
Basic Knowledge ◽  
Complete Theory ◽  
The Real ◽  
Special Cases ◽  
Standard Set ◽  
Tarski Algebra ◽  
Independence Results ◽  
Omitting Types ◽  
Meager Sets

In this paper we investigate omitting types for a certain kind of stable theories which we call stable ccc theories. In Theorem 2.1 we improve Steinhorn's result from [St]. We prove also some independence results concerning omitting types. The main results presented in this paper were part of the author's Ph.D. thesis [N1].Throughout, we use the standard set-theoretic and model-theoretic notation, such as can be found for example in [Sh] or [M]. So in particular T is always a countable complete theory in the language L. We consider all models of T and all sets of parameters subsets of the monster model ℭ, which is very saturated. Ln(A) denotes the Lindenbaum-Tarski algebra of formulas with parameters from A and n free variables. We omit n in Ln(A) when n = 1 or when it is clear from the context what n is. If φ, ψ ∈ L(A) are consistent then we say that φ is below ψ if ψ⊢ψ. For a type p and a set A ⊆ ℭ, p(A) is the set of tuples of elements of A which satisfy p. Formulas are special cases of types. We say that a type p is isolated over A if, for some φ() ∈ L(A), φ() ⊢ p(x), i.e. φ isolates p. For a formula φ, [φ] denotes the class of types which contain φ. We assume that the reader is familiar with some basic knowledge of forking, as presented in [Sh, III] or [M].Throughout, we work in ZFC. and denote (countable) transitive models of ZFC. cov K is the minimal number of meager sets covering the real line R. In this paper we prove theorems showing connections between omitting types and the combinatorics of the real line. More results in this direction are presented in [N2] and [N3].

scholarly journals A class of projective Stone algebras

1981 ◽  
Vol 24 (1) ◽  
pp. 133-147 ◽  
Author(s):  
Ivo Düntsch
Keyword(s):  
Boolean Algebra ◽  
Stone Algebra ◽  
Double Stone Algebra ◽  
Dense Set

We prove that a regular double Stone algebra is protective in the category of Stone algebras if and only if its centre is a projective Boolean algebra and its dense set is countably generated as a filter. It follows that every countable regular double Stone algebra is projective as a Stone algebra.

Boolean algebras in a localic topos

1986 ◽  
Vol 100 (1) ◽  
pp. 43-55 ◽  
Author(s):  
B. Banaschewski ◽  
K. R. Bhutani
Keyword(s):  
Boolean Algebra ◽  
Universal Algebra ◽  
Set Theory ◽  
Abelian Groups ◽  
Boolean Algebras

When a familiar notion is modelled in a certain topos E, the natural problem arises to what extent theorems concerning its models in usual set theory remain valid for its models in E, or how suitable properties of E affect the validity of certain of these theorems. Problems of this type have in particular been studied by Banaschewski[2], Bhutani[5], and Ebrahimi[6, 7], dealing with abelian groups in a localic topos and universal algebra in an arbitrary Grothendieck topos. This paper is concerned with Boolean algebras, specifically with injectivity and related topics for the category of Boolean algebras in the topos of sheaves on a locale and with properties of the initial Boolean algebra in .

Boolean simple groups and boolean simple rings

1988 ◽  
Vol 53 (1) ◽  
pp. 160-173
Author(s):  
Gaisi Takeuti
Keyword(s):  
Boolean Algebra ◽  
Simple Group ◽  
Set Theory ◽  
Finite Simple Group ◽  
Complete Boolean Algebra ◽  
Simple Groups ◽  
Finite Simple Groups

Let be a complete Boolean algebra and G a finite simple group in the Scott-Solovay -valued model V() of set theory. If we observe G outside V(), then we get a new group which is denoted by Ĝ. In general, Ĝ is not finite nor simple. Nevertheless Ĝ satisfies every property satisfied by a finite simple group with some translation. In this way, we can get a class of groups for which we can use a well-developed theory of the finite simple groups. We call Ĝ Boolean simple if G is simple in some V(). In the same way we define Boolean simple rings. The main purpose of this paper is a study of structures of Boolean simple groups and Boolean simple rings. As for Boolean simple rings, K. Eda previously constructed Boolean completion of rings with a certain condition. His construction is useful for our purpose.The present work is a part of a series of systematic applications of Boolean valued method. The reader who is interested in this subject should consult with papers by Eda, Nishimura, Ozawa, and the author in the list of references.

A Boolean ultrapower which is not an ultrapower

1976 ◽  
Vol 41 (1) ◽  
pp. 245-249 ◽  
Author(s):  
Bernd Koppelberg ◽  
Sabine Koppelberg
Keyword(s):  
Boolean Algebra ◽  
Set Theory ◽  
Natural Generalization ◽  
Negative Answer ◽  
Complete Boolean Algebra ◽  
Image Position ◽  
Models Of Set Theory

Several people have independently been studying Boolean ultrapowers recently; see for example [2], [3], [4], [6]. Boolean ultrapowers are a quite natural generalization of the well-known usual ultrapowers, but it seemed to be unknown whether every Boolean ultrapower is isomorphic to an ultrapower. We give a negative answer to that question. We further show that a Boolean ultrapower by an ℵ1-regular ultrafilter need not be ℵ2-universal, i.e. that Theorem 4.3.12 of [1] does not hold for Boolean ultrapowers.Let B be a complete Boolean algebra (we identify the algebra with its underlying set), whose operations are denoted by +, ·, −, 0, 1, Σ, Π Let be a structure for some language ℒ. For those who are familiar with Boolean-valued models of set theory, the B-valued model may be described by its underlying setandif R is an n-place relation in ℒ or equality, its interpretation in , u1 … un Є M(B).

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