The härtig quantifier: A survey

Journal of Symbolic Logic 56 (4):1153-1183 (1991)
A fundamental notion in a large part of mathematics is the notion of equicardinality. The language with Hartig quantifier is, roughly speaking, a first-order language in which the notion of equicardinality is expressible. Thus this language, denoted by LI, is in some sense very natural and has in consequence special interest. Properties of LI are studied in many papers. In [BF, Chapter VI] there is a short survey of some known results about LI. We feel that a more extensive exposition of these results is needed. The aim of this paper is to give an overview of the present knowledge about the language LI and list a selection of open problems concerning it. After the Introduction $(\S1)$ , in $\S\S2$ and 3 we give the fundamental results about LI. In $\S4$ the known model-theoretic properties are discussed. The next section is devoted to properties of mathematical theories in LI. In $\S6$ the spectra of sentences of LI are discussed, and $\S7$ is devoted to properties of LI which depend on set-theoretic assumptions. The paper finishes with a list of open problem and an extensive bibliography. The bibliography contains not only papers we refer to but also all papers known to us containing results about the language with Hartig quantifier. Contents. $\S1$ . Introduction. $\S2$ . Preliminaries. $\S3$ . Basic results. $\S4$ . Model-theoretic properties of $LI. \S5$ . Decidability of theories with $I. \S6$ . Spectra of LI- sentences. $\S7$ . Independence results. $\S8$ . What is not yet known about LI. Bibliography
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DOI 10.2307/2275466
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References found in this work BETA
Some Applications of Model Theory in Set Theory.Jack H. Silver - 1971 - Annals of Mathematical Logic 3 (1):45-110.
Δ-Logics and Generalized Quantifiers.J. A. Makowsky, Saharon Shelah & Jonathan Stavi - 1976 - Annals of Mathematical Logic 10 (2):155-192.
Axioms for Abstract Model Theory.K. Jon Barwise - 1974 - Annals of Mathematical Logic 7 (2-3):221-265.
A Remark On The Härtig Quantifier.Gebhard Fuhrken - 1972 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 18 (13-15):227-228.
On Some Applications of Games for Härtig Quantifier.Michal Krynicki - 1987 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 33 (4):359-370.

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Citations of this work BETA
Decidability Problems in Languages with Henkin Quantifiers.Michał Krynicki & Marcin Mostowski - 1992 - Annals of Pure and Applied Logic 58 (2):149-172.

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