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. (shrink)

Connections between partially ordered connectives and Henkin quantifiers are considered. It is proved that the logic with all partially ordered connectives and the logic with all Henkin quantifiers coincide. This implies that the hierarchy of partially ordered connectives is strongly hierarchical and gives several nondefinability results between some of them. It is also deduced that each Henkin quantifier can be defined by a quantifier of the form equation imagewhat is a strengthening of the Walkoe result. MSC: 03C80.

We say that a semantical function is correlated with a syntactical function F iff for any structure A and any sentence we have A F A .It is proved that for a syntactical function F there is a semantical function correlated with F iff F preserves propositional connectives up to logical equivalence. For a semantical function there is a syntactical function F correlated with iff for any finitely axiomatizable class X the class –1X is also finitely axiomatizable (i.e. iff is (...) continuous in model class topology). (shrink)

We investigate theories of initial segments of the standard models for arithmetics. It is easy to see that if the ordering relation is definable in the standard model then the decidability results can be transferred from the infinite model into the finite models. On the contrary we show that the Σ₂—theory of multiplication is undecidable in finite models. We show that this result is optimal by proving that the Σ₁—theory of multiplication and order is decidable in finite models as well (...) as in the standard model. We show also that the exponentiation function is definable in finite models by a formula of arithmetic with multiplication and that one can define in finite models the arithmetic of addition and multiplication with the concatenation operation. We consider also the spectrum problem. We show that the spectrum of arithmetic with multiplication and arithmetic with exponentiation is strictly contained in the spectrum of arithmetic with addition and multiplication. (shrink)

Simple formula should contain only few quantifiers. In the paper the methods to estimate quantity and quality of quantifiers needed to express a sentence equivalent to given one.

Krynicki, M. and M. Mostowski, Decidability problems in languages with Henkin quantifiers, Annals of Pure and Applied Logic 58 149–172.We consider the language L with all Henkin quantifiers Hn defined as follows: Hnx1…xny1…yn φ iff f1…fnx1. ..xn φ, ...,fn). We show that the theory of equality in L is undecidable. The proof of this result goes by interpretation of the word problem for semigroups.Henkin quantifiers are strictly related to the function quantifiers Fn defined as follows: Fnx1…xny1…yn φ iff fx1…xn φ,...,f). (...) In contrast with the first result we show that the theory of equality with all quantifiers Fn is decidable.We also consider decidability problems for other theories in languages L and L. (shrink)

The theorem to the effect that the languageL introduced in [2] is mutually interpretable with the first order language is proved. This yields several model-theoretical results concerningL.