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.

Let D be a strongly minimal set in the language L, and D′ ⊃ D an elementary extension with infinite dimension over D. Add to L a unary predicate symbol D and let T′ be the theory of the structure (D′, D), where D interprets the predicate D. It is known that T′ is ω-stable. We prove

Theorem A. If D is not locally modular, then T′ has Morley rank ω.

We say that a strongly minimal set D is pseudoprojective if it is nontrivial and there is a k < ω such that, for all a, b ∈ D and closed X ⊂ D, a ∈ cl(Xb) ⇒ there is a Y ⊂ X with a ∈ cl(Yb) and ∣Y∣ ≤ k. Using Theorem A, we prove

Theorem B. If a strongly minimal set D is pseudoprojective, then D is locally projective.

The following result of Hrushovski's (proved in §4) plays a part in the proof of Theorem B.

Theorem C. Suppose that D is strongly minimal, and there is some proper elementary extension D1 of D such that the theory of the pair (D1, D) is ω1-categorical. Then D is locally modular.

We consider the following properties of uncountable-dimensional quadratic spaces (E, Φ):

(*) For all subspaces U ⊆ E of infinite dimension: dim U˔ < dim E.

(**) For all subspaces U ⊆ E of infinite dimension: dim U˔ < ℵ0.

Spaces of countable dimension are the orthogonal sum of straight lines and planes, so they cannot have (*), but (**) is trivially satisfied.

These properties have been considered first in [G/O] in the process of investigating the orthogonal group of quadratic spaces. It has been shown there (in ZFC) that over arbitrary uncountable fields (**)-spaces of uncountable dimension exist.

In [B/G], (**)-spaces of dimension ℵ1 (so (*) = (**)) have been constructed over arbitrary finite or countable fields. But this could be done only under the assumption that the continuum hypothesis (CH) holds in the underlying set theory.

For a complete theory of Boolean algebras T, let MT denote the class of countable models of T. For B1, B2 ∈ MT, let B1 ≤ B2 mean that B2 is elementarily embeddable in B2. Theorem 1. For every complete theory of Boolean algebras T, if T ≠ Tω, then ‹MT, ≤› is well-quasi-ordered. ∎ We define Tω. For a Boolean algebra B, let I(B) be the ideal of all elements of the form a + s such that B ↾ a is an atomic Boolean algebra and B ↾ s is an atomless Boolean algebra. The Tarski derivative of B is defined as follows: B(0) = B and B(n + 1) = B(n)/I(B(n)). Define Tω to be the theory of all Boolean algebras such that for every n ∈ ω, B(n) ≠ {0}. By Tarski [1949], Tω is complete. Recall that ‹A, < › is partial well-quasi-ordering, it is a partial quasi-ordering and for every {ai, ⃒ i ∈ ω} ⊆ A, there are i < j < ω such that ai ≤ aj. Theorem 2. contains a subset M such that the partial orderings ‹M, ≤ ↾ M› and are isomorphic. ∎ Let M′0 denote the class of all countable Boolean algebras. For B1, B2 ∈ M′0, let B1 ≤′ B2 mean that B1 is embeddable in B2. Remark. ‹M′0, ≤′› is well-quasi-ordered. ∎ This follows from Laver's theorem [1971] that the class of countable linear orderings with the embeddability relation is well-quasi-ordered and the fact that every countable Boolean algebra is isomorphic to a Boolean algebra of a linear ordering.

The axiom of infinity can be expressed by stating the existence of sets satisfying a formula which involves restricted universal quantifiers only, even if the axiom of foundation is not assumed.

The theory of immediate pairs of Henselian valued fields, with a given residual theory (of characteristic zero) and a given theory of valuation group (nonzero), is undecidable and has completions.

Elementary axioms describe a category of categories. Theorems of category theory follow, including some on adjunctions and triples. A new result is that associativity of composition in categories follows from cartesian closedness of the category of categories. The axioms plus an axiom of infinity are consistent iff the axioms for a well-pointed topos with separation axiom and natural numbers are. The theory is not finitely axiomatizable. Each axiom is independent of the others. Further independence and definability results are proved. Relations between categories and sets, the latter defined as discrete categories, are described, and applications to foundations are discussed.

In this paper we prove undecidability of first-order definability of propositional formulas. The main result is proved for intuitionistic formulas, but it remains valid for other kinds of propositional formulas by analogous arguments or with the help of various translations.

For general background on correspondence theory the reader is referred to van Benthem [1], [2] (see [3] for a survey of recent results).

The method for the proofs of undecidability in this paper will be to simulate calculations of a Minsky machine by intuitionistic formulas. §3 concerns this simulation. Effective procedures for the construction of simulating modal formulas can be found in the literature (cf. [4]).

The principal results of the paper are in §4. §5 gives some further undecidability results, that will be proved in another paper by modification of the method of this paper.

I am indebted to the referee for drawing my attention to some errors in an earlier version of this paper.

We discuss in the paper the following problem: Given a function in a given Baire class, into “how many” (in terms of cardinal numbers) functions of lower classes can it be decomposed? The decomposition is understood here in the sense of the set-theoretical union.

It is shown that Con(ZF) implies Con(ZFC + there exists a Dowker filter on ω2). The problem is also considered on higher cardinals.

It is well known that, in the terminology of Moschovakis, Descriptive set theory (1980), every adequate normed pointclass closed under ∀ω has an effective version of the generalized reduction property (GRP) called the easy uniformization property (EUP). We prove a dual result: every adequate normed pointclass closed under ∃ω has the EUP. Moschovakis was concerned with the descriptive set theory of subsets of Polish topological spaces. We set up a general framework for parts of descriptive set theory and prove results that have as special cases not only the just-mentioned topological results, but also corresponding results concerning the descriptive set theory of classes of classes of structures.

Vaught (1973) asked whether the class of cPCδ classes of countable structures has the GRP. It does. A cPC(A) class is the class of all models of a sentence of the form , where ϕ is a sentence of ℒ∞ω that is in A and is a set of relation symbols that is in A. Vaught also asked whether there is any primitive recursively closed set A such that some effective version of the GRP holds for the class of cPC(A) classes of countable structures. There is: The class of cPC(A) classes of countable structures has the EUP if ω ∈ A and A is countable and primitive recursively closed. Those results and some extensions are obtained by first showing that the relevant classes of classes of structures, which Vaught showed normed, are in a suitable sense adequate and closed under ∃ω, and then applying the dual easy uniformization theorem.

It is shown lhat no infinite group is interpretable in any structure which is homogeneous in a finite relational language. Related questions are discussed for other ω-categorical structures.

For countable structures and , let abbreviate the statement that every sentence true in also holds in . One can define a back and forth game between the structures and that determines whether . We verify that if θ is an Lω,ω sentence that is not equivalent to any Lω,ω sentence, then there are countably infinite models and such that ⊨ θ, ⊨ ¬θ, and . For countable languages ℒ there is a natural way to view ℒ structúres with universe ω as a topological space, Xℒ. Let [] = { ∊ Xℒ∣ ≅ } denote the isomorphism class of . Let and be countably infinite nonisomorphic ℒ structures, and let C ⊆ ωω be any subset. Our main result states that if , then there is a continuous function f: ωω → Xℒ with the property that x ∊ C ⇒ f(x) ∊ [] and x ∉ C ⇒ f(x) ∊ f(x) ∈ []. In fact, for α ≤ 3, the continuous function f can be defined from the relation.

The purpose of this paper is an axiomatic study of the interrelations between certain continuity properties. We show that every mapping is sequentially continuous if and only if it is sequentially nondiscontinuous and strongly extensional, and that “every mapping is strongly extensional”, “every sequentially nondiscontinuous mapping is sequentially continuous”, and a weak version of Markov's principle are equivalent. Also, assuming a consequence of Church's thesis, we prove a version of the Kreisel-Lacombe-Shoenfield-Tseĭtin theorem.

We determine precisely those locally finite varieties of unary algebras of finite type which, when augmented by a ternary discriminator, generate a variety with a decidable theory.

It is shown, assuming the existence of a Woodin cardinal δ, that every tree ordering on some limit ordinal λ < δ with a cofinal branch is the tree ordering of some iteration tree on V.

The paper continues [1]. Let S be a complete theory of ultraflat (e.g. planar) graphs as introduced in [4]. We show a strong form of NOTOP for S: The union of two models M1 and M2, independent over a common elementary submodel M0, is the primary model over M1 ∪ M2 of S. Then by results of [1] Mekler's construction [6] gives for such a theory S of nice ultraflat graphs a superstable 2-step-nilpotent group of exponent p (> 2) with NDOP and NOTOP.

We define a characteristic and definable subgroup F*(G) of any group G of finite Morley rank that behaves very much like the generalized Fitting subgroup of a finite group. We also prove that semisimple subnormal subgroups of G are all definable and that there are finitely many of them.

It is consistent that there exists a set mapping F: [ω2]2 → [ω2]<ω such that F(α,β) ⊆ α for α < β < ω2 and there is no infinite free subset for F. This solves a problem of A. Hajnal and A. Máté.

The linear logic introduced in [3] by J.-Y. Girard keeps one of the so-called structural rules of the sequent calculus: the exchange rule. In a one-sided sequent calculus this rule can be formulated as

The exchange rule allows one to disregard the order of the assumptions and the order of the conclusions of a proof, and this means, when the proof corresponds to a logically correct program, to disregard the order in which the inputs and the outputs occur in a program.

In the linear logic introduced in [3], the exchange rule allows one to prove the commutativity of the multiplicative connectives, conjunction (⊗) and disjunction (⅋). Due to the presence of the exchange rule in linear logic, in the phase semantics for linear logic one starts with a commutative monoid. So, the usual linear logic may be called commutative linear logic.

The aim of the investigations underlying this paper was to see, first, what happens when we remove the exchange rule from the sequent calculus for the linear propositional logic at all, and then, how to recover the strength of the exchange rule by means of exponential connectives (in the same way as by means of the exponential connectives ! and ? we recover the strength of the weakening and contraction rules).