A fundamental notion in a large part of mathematics is the notion of equicardinality. The language with Härtig 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 (§1), in §§2 and 3 we give the fundamental results about LI. In §4 the known model-theoretic properties are discussed. The next section is devoted to properties of mathematical theories in LI. In §6 the spectra of sentences of LI are discussed, and §7 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 Härtig quantifier.Contents. §1. Introduction. §2. Preliminaries. §3. Basic results. §4. Model-theoretic properties of LI. §5. Decidability of theories with I. §6. Spectra of LI-sentences. §7. Independence results. §8. What is not yet known about LI. Bibliography.
In [5] Henkin defined a quantifier, which we shall denote by QH: linking four variables in one formula. This quantifier is related to the notion of formulas in which the usual universal and existential quantifiers occur but are not linearly ordered. The original definition of QH wasHere (QHx1x2y1y2)φ is true if for every x1 there exists y1 such that for every x2 there exists y2, whose choice depends only on x2 not on x1 and y1 such that φ(x14, x2, y1, y2). Another way of writing this isIn [5] it was observed that the logic L(QH) obtained by adjoining QH defined as in (1) is more powerful than first-order logic. In particular, it turned out that the quantifier “there exist infinitely many” denoted Q0 was definable from QH because
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