The problem mentioned in the title has already been investigated by J. Baumgartner, J. Stern, A. Miller and many others (see [2] and [5]). We prove here some generalizations of theorems of Miller and Stern from [2] and [5]. We use standard set-theoretical notation. Let

One can check that in the above definition we can replace “compact subset of ωω” by “closed nowhere dense subset of ω2” or “Fσ and meager subset of ω2” (as any Fσ subset of ω2 can be presented as a disjoint countable union of compact sets).

For functions f, g ϵ ωω we define f ≼ g if for all but finitely many n ϵ ω we have f(n) ≤ g(n). Let denote the least cardinality of a family A ⊆ ωω such that for any f ϵ ωω there is g ϵ A for which f ≼ g. It is easy to see that ≤ κω ≤ κ1. If f ϵ ωω then let ≼(f) = {h ϵωω: h ≼ f}.

We find an axiom which implies = ω1 → κ1 = ω1, and which can be preserved by any ccc notion of forcing of “small cardinality”. We construct also in a generic model many partitions of ωω into compact sets preserved not only by any random real extension, but also by Sacks' notion of forcing. This shows that from some point of view Miller's modification of Sacks' forcing (from [2]) is the “minimal” one able to destroy a partition of ωω into compact sets.

We prove that MM (Martin maximum) is equivalent (in ZFC) to the older axiom SPFA (semiproper forcing axiom). We also prove that SPFA does not imply SPFA+ or even PFA+ (using the consistency of a large cardinal).

In this paper we give a closer analysis of the elementary properties of the Banach spaces C(K), where K is a totally disconnected, compact Hausdorff space, in terms of the Boolean algebra B(K) of clopen subsets of K. In particular we sharpen a result in [4] by showing that if B(K1) and B(K2) satisfy the same sentences with ≤ n alternations of quantifiers, then the same is true of C(K1) and C(K2). As a consequence we show that for each n there exist C(K) spaces which are elementarily equivalent for sentences with ≤ n quantifier alternations, but which are not elementary equivalent in the full sense. Thus the elementary properties of Banach spaces cannot be determined by looking at sentences with a bounded number of quantifier alternations.

The notion of elementary equivalence for Banach spaces which is studied here was introduced by the second author [4] and is expressed using the language of positive bounded formulas in a first-order language for Banach spaces. As was shown in [4], two Banach spaces are elementarily equivalent in this sense if and only if they have isometrically isomorphic Banach space ultrapowers (or, equivalently, isometrically isomorphic nonstandard hulls.)

We consider Banach spaces over the field of real numbers. If X is such a space, Bx will denote the closed unit ball of X, Bx = {x ϵ X∣ ∣∣x∣∣ ≤ 1}. Given a compact Hausdorff space K, we let C(K) denote the Banach space of all continuous real-valued functions on K, under the supremum norm. We will especially be concerned with such spaces when K is a totally disconnected compact Hausdorff space. In that case B(K) will denote the Boolean algebra of all clopen subsets of K. We adopt the standard notation from model theory and Banach space theory.

We shall be concerned here with weak axiomatic systems of set theory with a universal set. The language in which they are expressed is that of set theory—two primitive predicates, = and ϵ, and no function symbols (though some function symbols will be introduced by definitional abbreviation). All the theories will have stratified axioms only, and they will all have Ext (extensionality: (∀x)(∀y)(x = y· ↔ ·(∀z)(z ϵ x ↔ z ϵ y))). In fact, in addition to extensionality, they have only axioms saying that the universe is closed under certain set-theoretic operations, viz. all of the form

and these will always include singleton, i.e., ι′x exists if x does (the iota notation for singleton, due to Russell and Whitehead, is used here to avoid confusion with {x: Φ}, set abstraction), and also x ∪ y, x ∩ y and − x (the complement of x). The system with these axioms is called NF2 in the literature (see [F]). The other axioms we consider will be those giving ⋃x, ⋂x, {y: y ⊆x} and {y: x ⊆ y}. We will frequently have occasion to bear in mind that 〈 V, ⊆ 〉 is a Boolean algebra in any theory extending NF2. There is no use of the axiom of choice at any point in this paper. Since the systems with which we will be concerned exhibit this feature of having, in addition to extensionality, only axioms stating that V is closed under certain operations, we will be very interested in terms of the theories in question. A T-term, for T such a theory, is a thing (with no free variables) built up from V or ∧ by means of the T-operations, which are of course the operations that the axioms of T say the universe is closed under.

We introduce a well-founded relation < between filters on the space of descending sequences of ordinals. For each regular uncountable cardinal κ, the length of the relation is an ordinal o(κ) ≤ (2κ)+.

In [1, p. 51] A. V. Arhangel'skiĭ, in connection with the problems of L-spaces and S-spaces, examined further the notions of hereditary separability and hereditary Lindelöfness. In particular he considered the following property P: “Every regular topological space has a countable net weight provided its countable product is hereditarily Lindelöf and hereditarily separable.” He noticed that the continuum hypothesis implies negation of the property P and posed a question: “Do Martin's Axiom and the negation of the continuum hypothesis imply P?” The purpose of this paper is to give a negative answer to this question.

The set-theoretical and topological notation that we use is standard and can be found in [6] and [5] respectively.

Throughout the paper we will use the notation H(X, Y) to denote the set of all finite functions from a set X to Y.

Theorem. Con(ZFC) → Con(ZFC + MA + ¬CH + there exists a 0-dimensional Hausdorff space X such that nw(X) = с and nw(Y) = ω for any Y ϵ [X]<с).

Proof. Let M be a model of ZFC satisfying CH and let F be an M-generic filter over the Cohen forcing {H(ω2 × ω2, 2), ⊃). Then f = ⋃F is a function and f: ω2 × ω2 → 2.

Dans Poizat [1981], le second auteur a montré qu'un sous-groupe infiniment définissable d'un groupe stable était intersection de sous-groupes définissables; il a posé la question de savoir si une relation d'équivalence E, infiniment définissable dans un modèle M d'une théorie stable T, était conjonction de relations d'équivalence définissables. Nous allons voir ici que c'est presque exact: c'est vrai si T est totalement transcendante, et, dans le cas général de stabilité E a toujours un raffinement E1 (plus précisément, E1 est la conjonction de E et de la relation “x et y ont même type”) qui a cette propriété; cela montre que cette relation E n'introduit pas d'imaginaires d'une nature vraiment différente de celle des imaginaires de Shelah: dans une théorie stable, un imaginaire infinitaire n'est rien d'autre qu'un ensemble d'imaginaires finis.

La démonstration du théorème principal de cette note s'appuie lourdement sur la construction Meq de Shelah, la machinerie de la déviation, les paramètres imaginaires canoniques pour la définition d'un type stable, etc…. Pour tout cela, les références adéquates sont Shelah [1978], Pillay [1983], et Poizat [1985, Chapitre 16].

Nouscommençons par préciser ce que nous entendons par “relation d'équivalence infiniment définissable”: une collection de formules e(, ȳ), et ȳ étant de longueur n, telle que, pour tout modèle M de T, les couples (, ȳ) qui les satisfont toutes forment une rélation d'équivalence E.

By analyzing how one obtains the Stone space of the reduced product of an indexed collection of Boolean algebras from the Stone spaces of those algebras, we derive a topological construction, the “reduced coproduct”, which makes sense for indexed collections of arbitrary Tichonov spaces. When the filter in question is an ultrafilter, we show how the “ultracoproduct” can be obtained from the usual topological ultraproduct via a compactification process in the style of Wallman and Frink. We prove theorems dealing with the topological structure of reduced coproducts (especially ultracoproducts) and show in addition how one may use this construction to gain information about the category of compact Hausdorff spaces.

In [2], Hiller and myself have introduced what we have called self-referential systems. Under this name we categorize models of a theory whose predicates refer to objects which are themselves predicates of the language—this particular interpretation being considered as the intended one.

is formulated in a language whose variables are typed: as types become “larger”, the corresponding objects become “smaller”; for this reason, we have suggestively chosen types to be negative integers: 0, −1, −2, and so on.

The consistency of has been questioned several times, and it has also been referred to as an “intriguing possibility”. Under such motivation, we are most willing to present a consistency proof for , and we do it relative to the theory ZFC + “there exists a Ramsey cardinal”. The only use of the Ramsey cardinal is to justify the existence of 0#, and the existence of 0# is known to be weaker than the existence of a Ramsey cardinal. For this reason, we note that the minimum we need for our proof is the existence of the set 0#.

To begin with, we prove that the consistency of implies the existence of intended models for : these are models in which every element is definable by a one-free-variable formula of the language. In intended models we also show that the axiom holds, and, besides, these models possess -indescribable cardinals. These two results imply the relative consistency of with respect to and, moreover, that any proof of the consistency of cannot be carried out without the use of some large cardinal axiom: such an axiom which, when added to ZFC, produces a theory equiconsistent to —if it exists—remains unknown. At the end of this article we finally exhibit the consistency proof of , our main result.

We associate with any abstract logic L a family F(L) consisting, intuitively, of the limit ultrapowers which are complete extensions in the sense of L.

For every countably generated [ω, ω]-compact logic L, our main applications are:

(i) Elementary classes of L can be characterized in terms of ≡L only.

(ii) If and are countable models of a countable superstable theory without the finite cover property, then .

(iii) There exists the “largest” logic M such that complete extensions in the sense of M and L are the same; moreover M is still [ω, ω]-compact and satisfies an interpolation property stronger than unrelativized ⊿-closure.

(iv) If L = Lωω(Qx), then cf(ωx) > ω and λω < ωx, for all λ < ωx.

We also prove that no proper extension of Lωω generated by monadic quantifiers is compact. This strengthens a theorem of Makowsky and Shelah. We solve a problem of Makowsky concerning Lκλ-compact cardinals. We partially solve a problem of Makowsky and Shelah concerning the union of compact logics.

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.

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.

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.

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.

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.

Semilattice semantics for relevant logics were discovered independently by Routley and Urquhart over 10 years ago. A semilattice semantics was first published in [10], where the weak theory of implication of [8] and [3] (i.e., R →, the pure implication fragment of the system R of relevant implication) is shown to be consistent and complete with respect to it. That result was extended in [11], But the semantics is explored in greatest detail in [12]. As reported in [4], Fine outfitted the positive semilattice semantics for R+ with a suitable Hilbert-style axiomatisation. (We refer to the system as ◡R+.) In 1980 Charlwood supplied a subscripted system of natural deduction. (See [1] and [2].) A subscripted Gentzen system was devised in [5] and [6].

Obviously, the central idea of the semilattice semantics is to impose relevant-style valuations on a semilattice (with an identity) used as the underlying model structure. However, in [12] the contractionless semantics are obtained (quite reasonably) by dropping the idempotence postulate and thus changing the relatively simple semilattice structure into a commutative monoid. Here we show that the semilattice structure can be regained for positive, contractionless relevant implication. Although we have no proofs as yet, we think that this semantics will pave the way for showing completeness for the corresponding subscripted Gentzen and natural deduction systems, as well as the Hilbert-style axiomatization, ◡RW+.

Throughout this paper, B will always be a Boolean algebra and Γ an ultrafilter on B. We use + and Σ for the Boolean join operation and · and Π for the Boolean meet.

κ is always a regular cardinal. C(κ) is the full structure of κ, the structure with universe κ and whose functions and relations consist of all unitary functions and relations on κ. κB is the collection of all B-valued names for elements of κ. We use symbols f, g, h for members of κB. Formally an element f ∈ κB is a mapping κ → B with the properties that Σα∈κf(α) = 1B and that f(α) · f(β) = 0B whenever α ≠ β. We view f(α) as the Boolean-truth value indicating the extent to which the name f is equal to α, and we will hereafter write ∥f = α∥ for f(α). For every α ∈ κ there is a canonical name fα ∈ κB which has the property that ∥fα = α∥ = 1. Hereafter we identify α and fα.

If B is a κ+-complete Boolean algebra and Γ is an ultrafilter on B, then we may define the Boolean ultraproduct C(κ)B/Γ in the following manner. If ϕ(x0, x1, …, xn) is a formula of Lκ, the language for C(κ) (which has symbols for all finitary functions and relations on κ), and f0, f1, …, fn−1 are elements of κB then we define the Boolean-truth value of ϕ(f0, f1, …, fn−1) as

Theorem A. Let T be a small superstable theory, A a finite set, and ψ a weakly minimal formula over A which is contained in some nontrivial type which does not have Morley rank. Then ψ is contained in some nonalqebraic isolated type over A.

As an application we prove

Theorem B. Suppose that T is small and superstable, A is finite, and there is a nontrivial weakly minimal type p ∈ S(A) which does not have Morley rank. Then the prime model over A is not minimal over A.

Theorem A. Suppose that T is superstable and p is a nontrivial type of U-rank 1. Then R(p, L, ∞)= 1.

Theorem B. Suppose that T is totally transcendental and p is a nontrivial type of U-rank 1. Then p has Morley rank 1.