The Härtig quantifier: a survey
Heinrich Herre, Michał Krynicki, A. G. Pinus, Jouko Väänánen · Journal of Symbolic Logic · 1991
Abstract 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 byLI, is in some sense very natural and has in consequence special interest. Properties ofLIare studied in many papers. In [BF, Chapter VI] there is a short survey of some known results aboutLI. 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 languageLIand list a selection of open problems concerning it. After the Introduction (§1), in §§2 and 3 we give the fundamental results aboutLI. In §4 the known model-theoretic properties are discussed. The next section is devoted to properties of mathematical theories inLI. In §6 the spectra of sentences ofLIare discussed, and §7 is devoted to properties ofLIwhich 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 ofLI. §5. Decidability of theories withI. §6. Spectra ofLI-sentences. §7. Independence results. §8. What is not yet known aboutLI. Bibliography.