Journal of Symbolic Logic

The Hartig Quantifier: A Survey

Heinrich Herre, Michal Krynicki, Alexandr Pinus, and Jouko Vaananen

Full-text is available via JSTOR, for JSTOR subscribers. Go to this article in JSTOR.


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.

Article information

J. Symbolic Logic, Volume 56, Issue 4 (1991), 1153-1183.

First available in Project Euclid: 6 July 2007

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier



Herre, Heinrich; Krynicki, Michal; Pinus, Alexandr; Vaananen, Jouko. The Hartig Quantifier: A Survey. J. Symbolic Logic 56 (1991), no. 4, 1153--1183.

Export citation