Cardinality Quantifiers Research Articles

LOGICS FROM ULTRAFILTERS

Abstract Ultrafilters play a significant role in model theory to characterize logics having various compactness and interpolation properties. They also provide a general method to construct extensions of first-order logic having these properties. A main result of this paper is that every class $\Omega $ of uniform ultrafilters generates a $\Delta $ -closed logic ${\mathcal {L}}_\Omega $ . ${\mathcal {L}}_\Omega $ is $\omega $ -relatively compact iff some $D\in \Omega $ fails to be $\omega _1$ -complete iff ${\mathcal {L}}_\Omega $ does not contain the quantifier “there are uncountably many.” If $\Omega $ is a set, or if it contains a countably incomplete ultrafilter, then ${\mathcal {L}}_\Omega $ is not generated by Mostowski cardinality quantifiers. Assuming $\neg 0^\sharp $ or $\neg L^{\mu }$ , if $D\in \Omega $ is a uniform ultrafilter over a regular cardinal $\nu $ , then every family $\Psi $ of formulas in ${\mathcal {L}}_\Omega $ with $|\Phi |\leq \nu $ satisfies the compactness theorem. In particular, if $\Omega $ is a proper class of uniform ultrafilters over regular cardinals, ${\mathcal {L}}_\Omega $ is compact.

GAMES AND CARDINALITIES IN INQUISITIVE FIRST-ORDER LOGIC

AbstractInquisitive first-order logic, InqBQ, is a system which extends classical first-order logic with formulas expressing questions. From a mathematical point of view, formulas in this logic express properties of sets of relational structures. This paper makes two contributions to the study of this logic. First, we describe an Ehrenfeucht–Fraïssé game for InqBQ and show that it characterizes the distinguishing power of the logic. Second, we use the game to study cardinality quantifiers in the inquisitive setting. That is, we study what statements and questions can be expressed in InqBQ about the number of individuals satisfying a given predicate. As special cases, we show that several variants of the question how many individuals satisfy $\alpha (x)$ are not expressible in InqBQ, both in the general case and in restriction to finite models.

On vectorizations of unary generalized quantifiers

Vectorization of a class of structures is a natural notion in finite model theory. Roughly speaking, vectorizations allow tuples to be treated similarly to elements of structures. The importance of vectorizations is highlighted by the fact that if the complexity class PTIME corresponds to a logic with reasonable syntax, then it corresponds to a logic generated via vectorizations by a single generalized quantifier (Dawar in J Log Comput 5(2):213---226, 1995). It is somewhat surprising, then, that there have been few systematic studies of the expressive power of vectorizations of various quantifiers. In the present paper, we consider the simplest case: the cardinality quantifiers C S . We show that, in general, the expressive power of the vectorized quantifier logic $${{\rm FO}(\{{\mathsf C}_S^{(n)}\, | \, n \in \mathbb{Z}_+\})}$$ is much greater than the expressive power of the non-vectorized logic FO(C S ).

Expressing cardinality quantifiers in monadic second-order logic over chains

AbstractWe investigate the extension of monadic second-order logic of order with cardinality quantifiers “there exists uncountably many sets such that…” and “there exists continuum many sets such that … ”. We prove that over the class of countable linear orders the two quantifiers are equivalent and can be effectively and uniformly eliminated. Weaker or partial elimination results are obtained for certain wider classes of chains. In particular, we show that over the class of ordinals the uncountability quantifier can be effectively and uniformly eliminated. Our argument makes use of Shelah's composition method and Ramsey-like theorem for dense linear orders.

Toward Model-Theoretic Modal Logics

Adding certain cardinality quantifiers into first-order language will give substantially more expressive languages. Thus, many mathematical concepts beyond first-order logic can be handled. Since basic modal logic can be seen as the bisimular invariant fragment of first-order logic on the level of models, it has no ability to handle modally these mathematical concepts beyond first-order logic. By adding modalities regarding the cardinalities of successor states, we can, in principle, investigate modal logics of all cardinalities. Thus ways of exploring model-theoretic logics can be transferred to modal logics.

Expressing Cardinality Quantifiers in Monadic Second-Order Logic over Trees

We study an extension of monadic second-order logic of order with the uncountability quantifier there exist uncountably many sets. We prove that, over the class of finitely branching trees, this extension is equally expressive to plain monadic second-order logic of order. Additionally we find that the continuum hypothesis holds for classes of sets definable in monadic second-order logic over finitely branching trees, which is notable for not all of these classes are analytic. Our approach is based on Shelah's composition method and uses basic results from descriptive set theory. The elimination result is constructive, yielding a decision procedure for the extended logic.

First-order and counting theories ofω-automatic structures

AbstractThe logicextends first-order logic by a generalized form of counting quantifiers (“the number of elements satisfying … belongs to the setC”). This logic is investigated for structures with an injectivelyω-automatic presentation. If first-order logic is extended by an infinity-quantifier, the resulting theory of any such structure is known to be decidable [6]. It is shown that, as in the case of automatic structures [21], also modulo-counting quantifiers as well as infinite cardinality quantifiers (“there aremany elements satisfying …”) lead to decidable theories. For a structure of bounded degree with injectiveω-automatic presentation, the fragment ofthat contains only effective quantifiers is shown to be decidable and an elementary algorithm for this decision is presented. Both assumptions (ω-automaticity and bounded degree) are necessary for this result to hold.

Invariant Version of Cardinality Quantifiers in Superstable Theories

We generalize Shelah's analysis of cardinality quantifiers for a superstable theory from Chapter V of Classification Theory and the Number of Nonisomorphic Models. We start with a set of bounds for the cardinality of each formula in some general invariant family of formulas in a superstable theory (in Classification Theory, a uniform family of formulas is considered) and find a set of derived bounds for all formulas. The set of derived bounds is sharp: up to a technical restriction, every model that satisfies the original bounds has a sufficiently saturated elementary extension that satisfies the original bounds and such that for each formula the set of its realizations in the extension has arbitrarily large cardinality below the corresponding derived bound of the formula.

Recursive logic frames

We define the concept of a logic frame , which extends the concept of an abstract logic by adding the concept of a syntax and an axiom system. In a recursive logic frame the syntax and the set of axioms are recursively coded. A recursive logic frame is called complete (recursively compact , ℵ0-compact ), if every finite (respectively: recursive, countable) consistent theory has a model. We show that for logic frames built from the cardinality quantifiers “there exists at least λ ” completeness always implies .0-compactness. On the other hand we show that a recursively compact logic frame need not be ℵ0-compact. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Axiomatising Tree-Interpretable Structures

We introduce the class of tree-interpretable structures which generalises the notion of a prefix-recognisable graph to arbitrary relational structures. We prove that every tree-interpretable structure is finitely axiomatisable in guarded second-order logic with cardinality quantifiers.

Enhancing fixed point logic with cardinality quantifiers

Enhancing fixed point logic with cardinality quantifiers

Some model theory of modules. III. On infiniteness of sets definable in modules

This is the third and last part of an investigation in several topics of first order model theory of modules which I began in Some model theory of modules. I. On total transcendence of modules (this Journal, vol. 48 (1983), pp. 570–574) and continued in Some model theory of modules. II. On stability and categoricity of flat modules (this Journal, vol. 48 (1983), pp. 970–985). Throughout I refer to these papers as “Part I” and “Part II”.Although these parts are only loosely connected, I will tacitly use the notation and preliminaries already introduced in the preceding ones. Further details are given in §0. Concerning Part II, the reader is assumed to be familiar with most of §1, a very small part of §4, and the criterion for elementary equivalence of modules over regular rings given in §3 (Lemma 20) of that part.Using those tools in this paper I consider completeness of the (elementary) theory of all modules (and of some of its extensions) (§2), and eliminability of cardinality quantifiers in the elementary theories of modules (§3).§2 is almost entirely devoted to a new short proof based on the technique developed in Part II of a theorem of Tukavkin which seems to be the first relevant result concerning the completeness of the theory of all modules (after the simple observation that any theory of infinite vector spaces is complete).

Elimination of Cardinality Quantifiers

Mathematical Logic QuarterlyVolume 28, Issue 4-7 p. 75-81 Article Elimination of Cardinality Quantifiers H. P. Tuschik, H. P. Tuschik Berlin (G.D.R.)Search for more papers by this author H. P. Tuschik, H. P. Tuschik Berlin (G.D.R.)Search for more papers by this author First published: 1982 https://doi.org/10.1002/malq.19820280405Citations: 1AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume28, Issue4-71982Pages 75-81 RelatedInformation

The theorems of beth and Craig in abstract model theory II. Compact logics

Various compact logics such as stationary logic, positive logic, logics with various cardinality quantifiers and cofinality quantifiers are studied. Counterexamples to the theorems of Beth and Craig are given. Back and forth arguments are studied for the first two logics, transfer theorems presented for positive logic and a new compactness proof for the confinality quantifiers is given.

Beth's property fails in L<ω

In this paper we prove that Beth's property does not hold in L<ω. This answers a question posed by Magidor and Malitz in [8]. Beth's property is a natural closure condition on a language which says that everything implicitly definable in the language is also explicitly definable in the language. That the first-order predicate calculus (L0) has the property was first shown by Beth [4]. Lopez-Escobar proved that also has Beth's property [7]. Malitz [9] showed the Beth's property fails in Lκλ where κ ≥ λ ≥ ω1. Friedman and Silver showed that the property fails in Lκλ for κ > ω1. Also Friedman [5] showed that extensions of elementary logic using cardinality quantifiers (L)1κ do not have Beth's property.The undefined notation used here is standard. If further clarification is needed, we refer the reader to [3]. κ and λ denote infinite cardinals. cX and ȣX∣ denote the cardinality of X. cfμ will denote the cofinality of the order μ. All languages discussed are assumed to have no function or constant symbols. All structures are relational. The type of a formula is the set of relation symbols appearing in the formula (excluding equality). The type of a set of formulas is the set of all relation symbols appearing in some formula of the set of formulas. For purpose of this paper we will assume that to each n-ary relation in a structure there is associated an n-ary relation symbol and that in any given structure this association is one-to-one. Using this convention we can define the type of a structure to be the set of those relation symbols to which there is associated a relation in the structure. Sometimes we use the term predicate instead of relation.

Ultraproduct invariant logics

In this paper we give a construction of logics via the property of being preserved from the models to their ultraproduct. Specific examples are given which include some cardinality quantifiers.

Beth’s theorem in cardinality logics

We prove that the Beth definability theorem fails for a comprehensive class of first-order logics with cardinality quantifiers. In particular, we give a counterexample to Beth’s theorem forL(Q), which is finitary first-order logic (with identity) augmented with the quantifier “there exists uncountably many”.