This paper concerns ultrafilters on a cardinal γ extending the filter of λ-closed, unbounded sets, λ < γ. The history of these ultrafilters is closely connected with that of the axiom of determinacy (AD). Solovay noticed first that, under AD, there was such an ultrafilter for γ = ℵ1; λ = ω. Later, Kleinberg found that the existence of such ultrafilters followed from the partition relation γ → (γ)λ+λ . Specific instances of this and more powerful relations on cardinals were then proved from AD by Martin, Kunen, Paris, and others. The axiom of determinacy was recently shown consistent with ZF relative to something less than a supercompact cardinal by Martin and Steel. Solovay's and Kleinberg's results were actually stronger, and we discuss this at the end of the paper. Good references for these results include [K2] and [KM].

We are interested here in the case where γ is the ultrapower of a strong partition cardinal κ (a cardinal satisfying for all α < κ). Such cardinals exist in great abundance assuming AD, and in fact, if sufficiently many cardinals are strong, then AD holds in L[R] [KKMW].

Frege's attempts to formulate a theory of properties to serve as a foundation for logic, mathematics and semantics all dissolved under the weight of the logicial paradoxes. The language of Frege's theory permitted the representation of the property which holds of everything which does not hold of itself. Minimal logic, plus Frege's principle of abstraction, leads immediately to a contradiction. The subsequent history of foundational studies was dominated by attempts to formulate theories of properties and sets which would not succumb to the Russell argument. Among such are Russell's simple theory of types and the development of various iterative conceptions of set. All of these theories ban, in one way or another, the self-reference responsible for the paradoxes; in this sense they are all “typed” theories. The semantical paradoxes, involving the concept of truth, induced similar nightmares among philosophers and logicians involved in semantic theory. The early work of Tarski demonstrated that no language that contained enough formal machinery to respresent the various versions of the Liar could contain a truth-predicate satisfying all the Tarski biconditionals. However, recent work in both disciplines has led to a re-evaluation of the limitations imposed by the paradoxes.

In the foundations of set theory, the work of Gilmore [1974], Feferman [1975], [1979], [1984], and Aczel [1980] has clearly demonstrated that elegant and useful type-free theories of classes are feasible. Work on the semantic paradoxes was given new life by Kripke's contribution (Kripke [1975]). This inspired the recent work of Gupta [1982] and Herzberger [1982]. These papers demonstrate that much room is available for the development of theories of truth which meet almost all of Tarski's desiderata.

We review briefly the attempts to define random sequences (§0). These attempts suggest two theorems: one concerning the number of subsequence selection procedures that transform a random sequence into a random sequence (§§1–3 and 5); the other concerning the relationship between definitions of randomness based on subsequence selection and those based on statistical tests (§4).

Subtle cardinals were first introduced in a paper by Jensen and Kunen [JK]. They show that if κ is subtle then ◇κ holds. Subtle cardinals also play an important role in [B1], where Baumgartner proposed that certain large cardinal properties should be considered as properties of their associated normal ideals. He shows that in the case of ineffables, the ideals are particularly useful, as can be seen by the following theorem, κ is ineffable if and only if κ is subtle and Π½-indescribable and the subtle and Π½-indescribable ideals cohere, i.e. they generate a proper, normal ideal (which in fact turns out to be the ineffable ideal).

In this paper we examine properties of subtle cardinals and consider methods of forcing that destroy the property of subtlety while maintaining other properties. The following is a list of results.

1) We relativize the following two facts about subtle cardinals:

i) if κ is n-subtle then {α < κ: α is not n-subtle} is n-subtle, and

ii) if κ is (n + 1)-subtle then {α < κ: α is n-subtle} is in the (n + 1)-subtle filter to subsets of κ:

i′) if A is an n-subtle subset of κ then {α ϵ A: A ∩ α is not n-subtle} is n-subtle, and

ii′) if A is an (n + 1)-subtle subset of κ then {α ϵ A: A ∩ α is n-subtle} is (n + 1)-subtle.

2) We show that although a stationary limit of subtles is subtle, a subtle limit of subtles is not necessarily 2-subtle.

3) In §3 we use the technique of forcing to turn a subtle cardinal into a κ-Mahlo cardinal that is no longer subtle.

4) In §4 we extend the results of §3 by showing how to turn an (n + 1)-subtle cardinal into an n-subtle cardinal that is no longer (n + 1)-subtle.

We investigate some relations between omitting types of a countable theory and some notions defined in terms of the real line, such as for example the ideal of meager subsets of R. We also try to express connections between the logical structure of a theory and the existence of its countable models omitting certain families of types.

It is well known that assuming MA we can omit < nonisolated types. But MA is rather a strong axiom. We prove that in order to be able to omit < nonisolated types it is sufficient to assume that the real line cannot be covered by less than meager sets; and this is in fact the weakest possible condition. It is worth pointing out that by means of forcing we can easily obtain the model of ZFC in which R cannot be covered by < meager sets. It suffices to add to the ground model Cohen generic reals.

We also formulate similar results for omitting pairwise contradictory types. It turns out that from some point of view it is much more difficult to find the family of pairwise contradictory types which cannot be omitted by a model of T, than to find such a family of possibly noncontradictory types. Moreover, for any two countable theories T1, T2 without prime models, the existence of a family of κ types which cannot be omitted by a model of T1 is equivalent to the existence of such a family for T2. This means that from the point of view of omitting types all theories without prime models are identical. Similar results hold for omitting pairwise contradictory types.

The language of propositional modal logic is extended by the introduction of sequents. Validity of a modal sequent on a frame is defined, and modal sequent-axiomatic classes of frames are introduced. Through the use of modal algebras and general frames, a study of the properties of such classes is begun.

This paper presents a unified treatment of the propositional and first-order many-valued logics through the method of tableaux. It is shown that several important results on the proof theory and model theory of those logics can be obtained in a general way.

We obtain, in this direction, abstract versions of the completeness theorem, model existence theorem (using a generalization of the classical analytic consistency properties), compactness theorem and Löwenheim-Skolem theorem.

The paper is completely self-contained and includes examples of application to particular many-valued formal systems.

We study the sets definable in an algebraic number field by first order formulas of various simple types, showing in particular that N and Z do not have very simple definitions.

In this paper we introduce the notion of a first order topological structure, and consider various possible conditions on the complexity of the definable sets in such a structure, drawing several consequences thereof.

Our aim is to develop, for a restricted class of unstable theories, results analogous to those for stable theories. The “material basis” for such an endeavor is the analogy between the field of real numbers and the field of complex numbers, the former being a “nicely behaved” unstable structure and the latter the archetypal stable structure. In this sense we try here to situate our work on o-minimal structures [PS] in a general topological context. Note, however, that the p-adic numbers, and structures definable therein, will also fit into our analysis.

In the remainder of this section we discuss several ways of studying topological structures model-theoretically. Eventually we fix on the notion of a structure in which the topology is “explicitly definable” in the sense of Flum and Ziegler [FZ]. In §2 we introduce the hypothesis that every definable set is a Boolean combination of definable open sets. In §3 we introduce a “dimension rank” on (closed) definable sets. In §4 we consider structures on which this rank is defined, and for which also every definable set has a finite number of definably connected definable components. We show that prime models over sets exist under such conditions.

For a countable complete o-minimal theory T, we introduce the notion of a sequentially complete model of T. We show that a model of T is sequentially complete if and only if ≺ for some Dedekind complete model . We also prove that if T has a Dedekind complete model of power greater than , then T has Dedekind complete models of arbitrarily large powers. Lastly, we show that a dyadic theory—namely, a theory relative to which every formula is equivalent to a Boolean combination of formulas in two variables—that has some Dedekind complete model has Dedekind complete models in arbitrarily large powers.

Une conséquence de la classification des théories complètes d'algèbres de Boole par Tarski [5] est que la théorie élémentaire d'une algèbre de Boole A est déterminée par le type d'isomorphisme du treillis de ses idéaux définissables et, pour chacun de ces idéaux, par le nombre d'atomes du quotient de A par cet idéal lorsque ce nombre est fini. Une remarque analogue peut être faite à propos des cas particuliers d'algèbres de Boole munies d'un idéal distingué étudiés par Ershov [1] et par Jurie et Touraille [3]; dans to us ces cas, c'est la simplicité des treillis possibles qui permet la classification des théories complètes. Le résultat principal de cet article est que, dans le cas général d'une algèbre de Boole munie d'une famille quelconque d'idéaux distingués, la théorie d'un modèle peut encore être caractérisée grâce à une structure algébrique sur l'ensemble de ses idéaux définissables. Il s'agit d'une structure d'algèbre de Heyting munie d'une opération unaire sa définie par sa(K) = {a: a/K est sans atome}, et cette structure s'avère être engendrée par les idéaux distingués du modèle. La méthode utilisée est l'élimination directe des quantificateurs, par réductions successives des formules. Elle nécessite des propriétés algébriques et topologiques qui sont données aux §§1 et 2: on introduit au §1 la notion d'algèbre de Heyting étoilée, c'est-à-dire d'algèbre de Heyting munie d'une opération unaire * vérifiant des égalités qui permettent de rendre compte, d'une certaine façon, de la dérivation de Cantor-Bendixon; le §2 est consacré à des propriétés topologiques qui, dans le cas de l'espace de Stone d'une algèbre de Boole A, permettent d'éclaircir les relations possibles entre les atomes des quotients de A par des idéaux différents.

By Solovay's theorem [16], the modal logic of provability GL gives a complete description of the propositional schemata involving the provability predicate PrPA(x) for Peano arithmetic PA, which are provable in PA. However, many important aspects of provability cannot be fully expressed in terms of PrPA(x). For this reason, many authors have introduced extensions of GL which take account either of Rosser constructions or of other important metamathematical formulas (see, for example, [5], [6], [14], [16], and [19]). In this paper, we concentrate on the modal logic of the provability predicate for finitely axiomatizable subtheories of PA; the interest of this modal logic is based on the following facts. First of all, it provides a modal translation of a very important property of PA, namely the essential reflexiveness. Secondly, in view of Orey's theorem [10] it constitutes a possible approach to the study of interpretability of finite extensions of PA. Indeed, by Orey's theorem PA + θ is interpretable in PA + θ′ iff for every n, and, therefore, relative interpretability of finitely axiomatizable extensions of PA can be expressed by means of the provability predicate for finitely axiomatizable subtheories of PA.

In §1, we introduce three modal logics extending GL and discuss their arithmetical interpretations; §2 deals with Kripke semantics for two of these logics. In §3, a theorem on arithmetical completeness is shown, which characterizes the logic of the provability predicate for finitely axiomatizable subtheories of PA; a uniform version of this theorem is proved in §4.

The formalism of P(redicate) P(rovability) L(ogic) is the result of adjoining the unary operator □ to first-order logic without identity, constants, or function symbols. The term “provability” indicates that □ is to be “read” as “it is provable in P(eano) A(rithmetic) that…” and that the formulae of predicate provability logic are to be interpreted via formulae of PA as follows.

Pr(x), alias Bew(x), is the standard provability predicate of PA. For any formula F of PA, Pr[F] is the formula of PA that expresses the PA-provability of F “of” the values of the variables free in F, i.e., it is the formula of PA with the same free variables as F that expresses the PA-provability of the result of substituting for each variable free in F the numeral for the value of that variable. For the details of the construction of Pr[F], the reader may consult [B2, p. 42]. If F is a sentence of PA, then Pr[F] = Pr(‘F’), the sentence that expresses the PA-provability of F.

Let υ 1, υ 2,… be an enumeration of the variables of PA. An interpretation * of a formula ϕ of PPL is a function which assigns to each predicate symbol P of ϕ a formula P* of the language of arithmetic whose free variables are the first n variables of PA, where n is the degree of P.

This paper continues the investigation of inconsistent arithmetical structures. In §2 the basic notion of a model with identity is defined, and results needed from elsewhere are cited. In §3 several nonisomorphic inconsistent models with identity which extend the (=, <) theory of the usual classical denumerable nonstandard model of arithmetic are exhibited. In §4 inconsistent nonstandard models of the classical theory of finite rings and fields modulo m, i.e. Zm, are briefly considered. In §5 two models modulo an infinite nonstandard number are considered. In the first, it is shown how to model inconsistently the arithmetic of the rationals with all names included, a strengthening of earlier results. In the second, all inconsistency is confined to the nonstandard integers, and the effects on Fermat's Last Theorem are considered. It is concluded that the prospects for a good inconsistent theory of fields may be limited.

We characterize the model companions of universal Horn classes generated by a two-element algebra (or ordered two-element algebra). We begin by proving that given two mutually model consistent classes M and N of (respectively ) structures, with , , provided that an -definability condition for the function and relation symbols of holds. We use this, together with Post's characterization of ISP(A), where A is a two-element algebra, to show that the model companions of these classes essentially lie in the classes of posets and semilattices, or characteristic two groups and relatively complemented distributive lattices.

The interpretation of the ZF set theory reported in this paper is, actually, part of a wider effort, namely, a new approach to the foundation of mathematics, which is referred to as The Cybernetic Foundation. A detailed exposition of the Cybernetic Foundation will be published elsewhere. Our approach leads to a full acceptance of the formalism of the classical set theory, but interprets it using only the idea of potential, but not actual (completed) infinity, and dealing only with finite objects that can actually be constructed. Thus we have a finitist proof of the consistency of ZF. This becomes possible because we set forth a metatheory of mathematics which goes beyond the classical logic and set theory and, of course, cannot be formalized in ZF, yet yields proofs which are as convincing—at least, from the author's viewpoint—as any mathematical proof can be.

Our metatheory is based on the following two ideas. Firstly, we define the semantics of the mathematical language using the cybernetical concept of knowledge. According to this concept, to say that a cybernetic system (a human being, in particular) has some knowledge is to say that it has some models of reality. In the Cybernetic Foundation we consider mathematics as the art of constructing linguistic models of reality.

This paper is concerned with the combinator representation of numeral systems with logarithmic space complexity of symbols. The principle used is based on the lexicographic ordering of words over a finite alphabet.

Let K be a finite set of finite structures. We give a syntactic characterization of the property: every element of K is injective in ISP(K). We use this result to establish that is injective in ISP( ) for every two-element algebra .