Martin's axiom in the model theory of LA

1980 ◽  
Vol 45 (1) ◽  
pp. 172-176
Author(s):  
W. Richard Stark
Keyword(s):  
Model Theory ◽  
Equivalence Theorem ◽  
Compactness Theorem ◽  
Compactness Result ◽  
First Order ◽  
Martin's Axiom ◽  
New Class ◽  
Martin’S Axiom ◽  
Order Language ◽  
Omitting Types

Working in ZFC + Martin's Axiom we develop a generalization of the Barwise Compactness Theorem which holds in languages of cardinality less than . Next, using this compactness theorem, an omitting types theorem for fewer than types is proved. Finally, in ZFC, we prove that this compactness result implies Martin's Axiom (the Equivalence Theorem). Our compactness theorem applies to a new class of theories—ccΣ-theories—which generalize the countable Σ-theories of Barwise's theorem. The Omitting Types Theorem and the Equivalence Theorem serve as examples illustrating the use of ccΣ-theories.Assume = (A, ε) or = (A, ε R1,…,Rm) where is admissible. L() is the first-order language with constants for elements of A and relation symbols for relations in . LA is A ⋂ L∞ω where the L of L∞ω is any language in A. A theory T in LA is consistent if there is no derivation in A of a contradiction from T. is LA with new constants ca for each a and A. The basic terms of consist of the constants of and the terms f(ca1,…,cam) built directly from constants using functions f of . The symbol t is used for basic terms. A theory T in LA is Σ if it is defined by a formula of L(). The formula φ⌝ is a logical equivalent of ¬φ defined by: (1) φ⌝ = ¬φ if φ is atomic; (2) (¬φ)⌝ = φ (3) (⋁φ∈Φ φ)⌝ = ⋀φ∈Φ φ⌝; (4) (⋀φ∈Φ φ) ⋁φ∈Φ φ⌝; (5) (∃χφ(x))⌝ ∀χφ⌝(x); ∀χφ(x))⌝ = ∃χφ⌝(x).

Weakly atomic-compact relational structures

1971 ◽  
Vol 36 (1) ◽  
pp. 129-140 ◽  
Author(s):  
G. Fuhrken ◽  
W. Taylor
Keyword(s):  
Abelian Group ◽  
Model Theory ◽  
Compact Abelian Group ◽  
Finite Subset ◽  
Relational Structure ◽  
Similarity Type ◽  
First Order ◽  
Relational Structures ◽  
Order Language ◽  
Weakly Atomic

A relational structure is called weakly atomic-compact if and only if every set Σ of atomic formulas (taken from the first-order language of the similarity type of augmented by a possibly uncountable set of additional variables as “unknowns”) is satisfiable in whenever every finite subset of Σ is so satisfiable. This notion (as well as some related ones which will be mentioned in §4) was introduced by J. Mycielski as a generalization to model theory of I. Kaplansky's notion of an algebraically compact Abelian group (cf. [5], [7], [1], [8]).

Theories without countable models

1972 ◽  
Vol 37 (3) ◽  
pp. 562-568
Author(s):  
Andreas Blass
Keyword(s):  
Standard Model ◽  
Countable Model ◽  
Compactness Theorem ◽  
Complete Theory ◽  
Natural Numbers ◽  
Axiom Schema ◽  
First Order ◽  
Order Language ◽  
The Standard Model ◽  
Skolem Theorem

Consider the Löwenheim-Skolem theorem in the form: If a theory in a countable first-order language has a model, then it has a countable model. As is well known, this theorem becomes false if one omits the hypothesis that the language be countable, for one then has the following trivial counterexample.Example 1. Let the language have uncountably many constants, and let the theory say that they are unequal.To motivate some of our future definitions and to introduce some notation, we present another, less trivial, counterexample.Example 2. Let L0 be the language whose n-place predicate (resp. function) symbols are all the n-place predicates (resp. functions) on the set ω of natural numbers. Let be the standard model for L0; we use the usual notation Th() for its complete theory. Add to L0 a new constant e, and add to Th() an axiom schema saying that e is infinite. By the compactness theorem, the resulting theory T has models. However, none of its models are countable. Although this fact is well known, we sketch a proof in order to refer to it later.By [5, p. 81], there is a family {Aα ∣ < α < c} of infinite subsets of ω, the intersection of any two of which is finite.

A new omitting types theorem for L(Q)

1979 ◽  
Vol 44 (4) ◽  
pp. 507-521
Author(s):  
Matt Kaufmann
Keyword(s):  
General Theorem ◽  
First Order ◽  
Order Language ◽  
Omitting Types ◽  
Compactness Theorems ◽  
Countable Theory ◽  
Analogous Theorem

For L a countable first-order language, let L(Q) be logic with the quantifier Qx which means “there exist uncountably many x”. We assume a little familiarity with Keisler's paper [8]. One finds there completeness and compactness theorems for L(Q), as well as an omitting types theorem: a syntactic condition is given for a consistent countable theory to have a model satisfying ∀x⋁Σ(x), where Σ is a countable set of formulas of L(Q). (See also Chang and Keisler [3] for the first-order omitting types theorem, due to Henkin and Orey.) An analogous theorem is proved in Barwise, Kaufmann, and Makkai [1] and in Kaufmann [6] for stationary logic. However, a more general theorem than just an anlaogue to Keisler's is proved there. Conditions are given which are sufficient for a theory T to have models satisfying sentences such as aas1aas2 … aasn⋁Σ(s1, … sn), ∀xaas ∨ Σ(x, s), and so forth. Bruce [2] had asked whether such a theorem can be proved for L(Q). with “aa” replaced by “Q*”, where Q* is ¬Q¬ (“for all but countably many”).

On countable fractions from an elementary class

1994 ◽  
Vol 59 (4) ◽  
pp. 1410-1413
Author(s):  
C. J. Ash
Keyword(s):  
Model Theory ◽  
Relation Symbol ◽  
First Order ◽  
Elementary Class ◽  
Free Variables ◽  
Order Language ◽  
Skolem Theorem ◽  
Elementary Result ◽  
Countable Models

The following fairly elementary result seems to raise possibilities for the study of countable models of a theory in a countable language. For the terminology of model theory we refer to [CK].Let L be a countable first-order language. Let L′ be the relational language having, for each formula φ of L and each sequence υ1,…,υn of variables including the free variables of φ, an n-ary relation symbol Pφ. For any L-structure and any formula Ψ(υ) of L, we define the Ψ-fraction of to be the L′-structure Ψ whose universe consists of those elements of satisfying Ψ(υ) and whose relations {Rφ}φϵL are defined by letting a1,…,an satisfy Rφ in Ψ if, and only if, a1,…, an satisfy φ in .An L-elementary class means the class of all L-structures satisfying each of some set of sentences of L. The countable part of an L-elementary class K means the class of all countable L-structures from K.Theorem. Let K be an L-elementary class and let Ψ(υ) be a formula of L. Then the class of countable Ψ-fractions of structures in K is the countable part of some L′-elementary class.Comment. By the downward Löwenheim-Skolem theorem, the countable Ψ-fractions of structures in K are the same as the Ψ-fractions of countable structures in K.Proof. We give a set Σ′ of L′-sentences whose countable models are exactly the countable Ψ-fractions of structures in K.

Truth Definitions, Skolem Functions and Axiomatic Set Theory

1998 ◽  
Vol 4 (3) ◽  
pp. 303-337 ◽  
Author(s):  
Jaakko Hintikka
Keyword(s):  
Set Theory ◽  
Model Theory ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Negative Results ◽  
Skolem Functions ◽  
Order Language ◽  
Definition Of ◽  
Axiomatic Set Theory

§1. The mission of axiomatic set theory. What is set theory needed for in the foundations of mathematics? Why cannot we transact whatever foundational business we have to transact in terms of our ordinary logic without resorting to set theory? There are many possible answers, but most of them are likely to be variations of the same theme. The core area of ordinary logic is by a fairly common consent the received first-order logic. Why cannot it take care of itself? What is it that it cannot do? A large part of every answer is probably that first-order logic cannot handle its own model theory and other metatheory. For instance, a first-order language does not allow the codification of the most important semantical concept, viz. the notion of truth, for that language in that language itself, as shown already in Tarski (1935). In view of such negative results it is generally thought that one of the most important missions of set theory is to provide the wherewithal for a model theory of logic. For instance Gregory H. Moore (1994, p. 635) asserts in his encyclopedia article “Logic and set theory” thatSet theory influenced logic, both through its semantics, by expanding the possible models of various theories and by the formal definition of a model; and through its syntax, by allowing for logical languages in which formulas can be infinite in length or in which the number of symbols is uncountable.

Boolean-valued structures

2018 ◽  
Author(s):  
Tim Button ◽  
Sean Walsh
Keyword(s):  
Model Theory ◽  
Order Logic ◽  
First Order Logic ◽  
Inference Rules ◽  
Mathematical Concepts ◽  
Semantic Framework ◽  
First Order ◽  
Standard Models ◽  
Boolean Valued Model ◽  
Semantic Values

Chapters 6-12 are driven by questions about the ability to pin down mathematical entities and to articulate mathematical concepts. This chapter is driven by similar questions about the ability to pin down the semantic frameworks of language. It transpires that there are not just non-standard models, but non-standard ways of doing model theory itself. In more detail: whilst we normally outline a two-valued semantics which makes sentences True or False in a model, the inference rules for first-order logic are compatible with a four-valued semantics; or a semantics with countably many values; or what-have-you. The appropriate level of generality here is that of a Boolean-valued model, which we introduce. And the plurality of possible semantic values gives rise to perhaps the ‘deepest’ level of indeterminacy questions: How can humans pin down the semantic framework for their languages? We consider three different ways for inferentialists to respond to this question.

scholarly journals Simpson type inequalities and applications

2021 ◽  
Author(s):  
Muhammad Uzair Awan ◽  
Muhammad Zakria Javed ◽  
Michael Th. Rassias ◽  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor
Keyword(s):  
Quadrature Formula ◽  
Convex Functions ◽  
Integral Identity ◽  
Higher Order ◽  
Auxiliary Result ◽  
First Order ◽  
New Class ◽  
Differentiable Functions

AbstractA new generalized integral identity involving first order differentiable functions is obtained. Using this identity as an auxiliary result, we then obtain some new refinements of Simpson type inequalities using a new class called as strongly (s, m)-convex functions of higher order of $$\sigma >0$$ σ > 0 . We also discuss some interesting applications of the obtained results in the theory of means. In last we present applications of the obtained results in obtaining Simpson-like quadrature formula.

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