Logic of reduced power structures

1983 ◽  
Vol 48 (1) ◽  
pp. 53-59
Author(s):  
G.C. Nelson
Keyword(s):  
Boolean Algebra ◽  
Complete Theory ◽  
Power Structures ◽  
Index Set ◽  
First Order ◽  
Order Language ◽  
Power Set ◽  
Definition Of ◽  
Proper Filter ◽  
Horn Sentence

We start with the framework upon which this paper is based. The most useful reference for these notions is [2]. For any nonempty index set I and any proper filter D on S(I) (the power set of I), we denote by I/D the reduced power of modulo D as defined in [2, pp. 167–169]. The first-order language associated with I/D will always be the same language as associated with . We denote the 2-element Boolean algebra 〈{0, 1}, ⋂, ⋃, c, 0, 1〉 by 2 and 2I/D denotes the reduced power of it modulo D. We point out the intimate connection between the structures I/D and 2I/D given in [2, pp. 341–345]. Moreover, we assume as known the definition of Horn formula and Horn sentence as given in [2, p. 328] along with the fundamental theorem that φ is a reduced product sentence iff φ is provably equivalent to a Horn sentence [2, Theorem 6.2.5/ (iff φ is a 2-direct product sentence and a reduced power sentence [2, Proposition 6.2.6(ii)]). For a theory T(any set of sentences), ⊨ T denotes that is a model of T.In addition to the above we assume as known the elementary characteristics (due to Tarski) associated with a complete theory of a Boolean algebra, and we adopt the notation 〈n, p, q〉 of [3], [10], or [6] to denote such an elementary characteristic or the corresponding complete theory. We frequently will use Ershov's theorem which asserts that for each 〈n, p, q〉 there exist an index set I and filter D such that 2I/D ⊨ 〈n, p, q〉 [3] or [2, Lemma 6.3.21].

Recursive categoricity and persistence

1986 ◽  
Vol 51 (2) ◽  
pp. 430-434 ◽  
Author(s):  
Terrence Millar
Keyword(s):  
First Order ◽  
Existential Sentence ◽  
Order Language ◽  
Recursive Structures ◽  
Definition Of

This paper is concerned with recursive structures and the persistance of an effective notion of categoricity. The terminology and notational conventions are standard. We will devote the rest of this paragraph to a cursory review of some of the assumed conventions. If θ is a formula, then θk denotes θ if k is zero, and ¬θ if k is one. If A is a sequence with domain a subset of ω, then A∣n denotes the sequence obtained by restricting the domain of A to n. For an effective first order language L, let {ci∣i<ω} be distinct new constants, and let {θi∣i<ω} be an effective enumeration of all sentences in the language L ∪ {ci∣j<ω}. An infinite L-structure is recursive iff it has universe ω and the set is recursive, where cn is interpreted by n. In general we say that a set of formulas is recursive if the set of its indices with respect to an enumeration such as above is recursive. The ∃-diagram of a structure is recursive if the structure is recursive and the set and θi is an existential sentence} is also recursive. The definition of “the ∀∃-diagram of is recursive” is similar.

Theories with finitely many models

1986 ◽  
Vol 51 (2) ◽  
pp. 374-376 ◽  
Author(s):  
Simon Thomas
Keyword(s):  
Natural Number ◽  
Complete Theory ◽  
Elementary Equivalence ◽  
First Order ◽  
Finite Models ◽  
Simple Observation ◽  
Finite Structure ◽  
Order Language ◽  
Complete Theories

If L is a first order language and n is a natural number, then Ln is the set of formulas which only make use of the variables x1,…,xn. While every finite structure is determined up to isomorphism by its theory in L, the same is no longer true in Ln. This simple observation is the source of a number of intriguing questions. For example, Poizat [2] has asked whether a complete theory in Ln which has at least two nonisomorphic finite models must necessarily also have an infinite one. The purpose of this paper is to present some counterexamples to this conjecture.Theorem. For each n ≤ 3 there are complete theories in L2n−2andL2n−1having exactly n + 1 models.In our notation and definitions, we follow Poizat [2]. To test structures for elementary equivalence in Ln, we shall use the modified Ehrenfeucht-Fraïssé games of Immerman [1]. For convenience, we repeat his definition here.Suppose that L is a purely relational language, each of the relations having arity at most n. Let and ℬ be two structures for L. Define the Ln game on and ℬ as follows. There are two players, I and II, and there are n pairs of counters a1, b1, …, an, bn. On each move, player I picks up any of the counters and places it on an element of the appropriate structure.

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.

scholarly journals Semantics for Hyperclassical Logic and the Problem of Negation in the Formal Language

2018 ◽  
Vol 16 (3) ◽  
pp. 5-15
Author(s):  
V. V. Tselishchev
Keyword(s):  
Theoretical Model ◽  
Formal Language ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Order Language ◽  
Winning Strategies ◽  
Game Theoretic ◽  
Definition Of ◽  
Game Theoretic Semantics

The application of game-theoretic semantics for first-order logic is based on a certain kind of semantic assumptions, directly related to the asymmetry of the definition of truth and lies as the winning strategies of the Verifier (Abelard) and the Counterfeiter (Eloise). This asymmetry becomes apparent when applying GTS to IFL. The legitimacy of applying GTS when it is transferred to IFL is based on the adequacy of GTS for FOL. But this circumstance is not a reason to believe that one can hope for the same adequacy in the case of IFL. Then the question arises if GTS is a natural semantics for IFL. Apparently, the intuitive understanding of negation in natural language can be explicated in formal languages in various ways, and the result of an incomplete grasp of the concept in these languages can be considered a certain kind of anomalies, in view of the apparent simplicity of the explicated concept. Comparison of the theoretical-model and game theoretic semantics in application to two kinds of language – the first-order language and friendly-independent logic – allows to discover the causes of the anomaly and outline ways to overcome it.

Products of two-sorted structures

1972 ◽  
Vol 37 (1) ◽  
pp. 75-80 ◽  
Author(s):  
Philip Olin
Keyword(s):  
Finite Number ◽  
Direct Product ◽  
Direct Products ◽  
Direct Sums ◽  
First Order ◽  
Order Language ◽  
Universal Equivalence ◽  
Relational Systems ◽  
Multiple Operation ◽  
Definition Of

First order properties of direct products and direct sums (weak direct products) of relational systems have been studied extensively. For example, in Feferman and Vaught [3] an effective procedure is given for reducing such properties of the product to properties of the factors, and thus in particular elementary equivalence is preserved. We consider here two-sorted relational systems, with the direct product and sum operations keeping one of the sorts stationary. (See Feferman [4] for a similar definition of extensions.)These considerations are motivated by the example of direct products and sums of modules [8], [9]. In [9] examples are given to show that the direct product of two modules (even having only a finite number of module elements) does not preserve two-sorted (even universal) equivalence for any finite or infinitary language Lκ, λ. So we restrict attention here to direct powers and multiples (many copies of one structure). Also in [9] it is shown (for modules, but the proofs generalize immediately to two-sorted structures with a finite number of relations) that the direct multiple operation preserves first order ∀E-equivalence and the direct power operation preserves first order ∀-equivalence. We show here that these results for general two-sorted structures in a finite first order language are, in a sense, best-possible. Examples are given to show that does not imply , and that does not imply .

A Note on the Realization of Types

1980 ◽  
Vol 23 (1) ◽  
pp. 95-98
Author(s):  
Alan Adamson
Keyword(s):  
Complete Theory ◽  
First Order ◽  
Order Language ◽  
The Universe

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=〈a1,…, an〉 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 υ1,…υn. ā is said to realize in We let be where is the sequence of the first n variables of L.

Forcing and reducibilities. III. Forcing in fragments of set theory

1983 ◽  
Vol 48 (4) ◽  
pp. 1013-1034
Author(s):  
Piergiorgio Odifreddi
Keyword(s):  
Set Theory ◽  
Recursion Theory ◽  
Second Order ◽  
Analytic Hierarchy ◽  
Second Order Arithmetic ◽  
First Order ◽  
Order Language ◽  
Order Arithmetic ◽  
Definition Of ◽  
Generic Set

We conclude here the treatment of forcing in recursion theory begun in Part I and continued in Part II of [31]. The numbering of sections is the continuation of the numbering of the first two parts. The bibliography is independent.In Part I our language was a first-order language: the only set we considered was the (set constant for the) generic set. In Part II a second-order language was introduced, and we had to interpret the second-order variables in some way. What we did was to consider the ramified analytic hierarchy, defined by induction as:A0 = {X ⊆ ω: X is arithmetic},Aα+1 = {X ⊆ ω: X is definable (in 2nd order arithmetic) over Aα},Aλ = ⋃α<λAα (λ limit),RA = ⋃αAα.We then used (a relativized version of) the fact that (Kleene [27]). The definition of RA is obviously modeled on the definition of the constructible hierarchy introduced by Gödel [14]. For this we no longer work in a language for second-order arithmetic, but in a language for (first-order) set theory with membership as the only nonlogical relation:L0 = ⊘,Lα+1 = {X: X is (first-order) definable over Lα},Lλ = ⋃α<λLα (λ limit),L = ⋃αLα.

Some consequences of the axiom of power-set

1965 ◽  
Vol 30 (3) ◽  
pp. 293-294 ◽  
Author(s):  
Alexander Abian ◽  
Samuel Lamacchia
Keyword(s):  
Predicate Symbol ◽  
Order Theory ◽  
Finite Model ◽  
First Order ◽  
First Order Theory ◽  
Power Set ◽  
Binary Predicate ◽  
Definition Of

In this paper we prove:Theorem 1. Any finite model of the axiom of power-set also satisfies the axioms of extensionality, sum-set and choice.Clearly, it will follow from (2) below that in a finite model the axiom of power-set is satisfied if and only if every set is a power-set. Thus, Theorem 1 follows immediately from Theorem 2 below, where by a theory of sets we mean a first-order theory without identity and with only one binary predicate symbol ∈.Theorem 2. If in a theory of sets every set is a power-set and if the axiom of power-set is valid, then the axioms of extensionality, sum-set and choice are valid.The proof of Theorem 2 will follow from the lemmas which we establish below.We mean by x = y that x and y have the same elements. We denote a power-set of x by P(x) when it exists; similarly, we denote a sum-set of x by Ux.Clearly, in every theory of sets we have:(1) (x ⊂ y) ↔ (P(x) ⊂ P(y)),(2) (x = y) ↔ (P(x) = P(y)),(3) (x = y) → ((x ∈ P(z)) → (y ∈ P(z))),(4) ⋃P(x) = x.In view of (2), (3) and the definition of equality, we have:Lemma 1. If in a theory of sets every set is a power-set, then equal sets are elements of the same sets.We have also, in view of (4):Lemma 2. If in a theory of sets every set is a power-set, then every set has a sum-set.

Modal formulas are either elementary or not ΣΔ-elementary

1976 ◽  
Vol 41 (2) ◽  
pp. 436-438 ◽  
Author(s):  
J. F. A. K. van Benthem
Keyword(s):  
Modal Logic ◽  
Propositional Formula ◽  
First Order ◽  
Propositional Modal Logic ◽  
Order Language ◽  
Power Set ◽  
Binary Predicate ◽  
Truth Definition ◽  
Identity Symbol

In this paper we prove that if L is a set of modal propositional formulas then FR(L) (the class of all frames in which every formula of L holds) is elementary, Δ-elementary or not ΣΔ-elementary. For single modal formulas the second of these cases does not occur.The model theoretic terminology and results used here are from [1]. (The underlying first order language contains only one, binary, predicate letter in addition to the identity symbol.) We presuppose familiarity with the usual notions and notations of propositional modal logic. A structure for our first order language is called a frame. (So a frame is an ordered couple 〈W, R〉 with domain W and R a binary predicate on W, i.e. a subset of W × W.) A valuation V on F is a function from the set of proposition letters to the power set of W. Using the well-known Kripke truth definition V can be extended to a function from the set of all modal propositional formulas to the power set of W. A modal propositional formula φ holds in a frame F (= 〈W, R〉) if, for all V on F, V(φ) = W. Notation: FR(φ) for the class of all frames in which φ holds. For a set L of modal propositional formulas we define FR(L) as ⋂φ∈LFR(φ). Obviously both FR(L) and cFR(L) (the complement of FR(L)) are closed under isomorphisms.

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