Buchholz [4] simplified the system of ordinal notations of the Schütte school (cf. [12]), by using the notion of collapsing functions (cf. [5]). In this paper we give a simple relationship between Buchholz's new system of ordinal notations and Takeuti's system of ordinal diagrams. From this simple relationship it turns out that the structures of these two systems are very close.

We give two systems OT(I) (§1) and OT(I, A) (§2) of ordinal notations which are considered generalizations of Buchholz's original system, where I and A are well-ordered sets. The original system OT of Buchholz [4] is OT(ω + 1, {0}) in our sense. Here the set OT(I) of ordinal notations is defined as a subset of the set Od(I) of ordinal diagrams in [6], and the set OT(I, A) of ordinal notations as a subset of the set O(I, A) of ordinal diagrams in [14].

Classes of forcings which add a real by forcing with branching conditions are examined, and conditions are found which guarantee that the generic real is of minimal degree over the ground model. An application is made to almost-disjoint coding via a real of minimal degree.

Dominical categories are categories in which the notions of partial morphisms and their domains become explicit, with the latter being endomorphisms rather than subobjects of their sources. These categories form the basis for a novel abstract formulation of recursion theory, to which the present paper is devoted. The abstractness has of course its usual concomitant advantage of generality: it is interesting to see that many of the fundamental results of recursion theory remain valid in contexts far removed from their classic manifestations. A principal reason for introducing this new formulation is to achieve an algebraization of the generalized incompleteness theorem, by providing a category-theoretic development of the concepts and tools of elementary recursion theory that are inherent in demonstrating the theorem.

Dominical recursion theory avoids the commitment to sets and partial functions which is characteristic of other formulations, and thus allows for an intrinsic recursion theory within such structures as polyadic algebras. It is worthy of notice that much of elementary recursion theory can be developed without reference to elements.

By Gödel's generalized incompleteness theorem for consistent arithmetical system T we mean any statement of the following sort:

(1) if every recursive set is definable in T, then T is essentially undecidable [41]; or

(2) if all recursive functions are definable in T, then T is essentially undecidable [41]; or

(3) if every recursive set is definable in T, then T 0 and R 0 (the sets of Gödel numbers of the theorems and refutables of T) are recursively inseparable [39]; or

(4) if all re sets are representable in T, then T 0 is creative [28], [39]; or

(5) if T is a Rosser theory (i.e., all disjoint re sets are strongly separable in T), then T 0 and R 0 are effectively inseparable [39].

We describe a first-order theory of generalized sets intended to allow a similar treatment of sets and proper classes. The theory is motivated by the iterative conception of set. It has a ternary membership symbol interpreted as membership relative to a set-building step. Set and proper class are defined notions. We prove that sets and proper classes with a defined membership form an inner model of Bernays-Morse class theory. We extend ordinal and cardinal notions to generalized sets and prove ordinal and cardinal results in the theory. We prove that the theory is consistent relative to ZFC + (∃x) [x is a strongly inaccessible cardinal].

A model of ZFC is constructed in which the distributivity cardinal h is , and in which there are no ω 2-towers in [ω]ω. As an immediate corollary, it follows that any base-matrix tree in this model has no cofinal branches. The model is constructed via a form of iterated Mathias forcing, in which a mixture of finite and countable supports is used.

Relevance logic began in an attempt to avoid the so-called fallacies of relevance. These fallacies can be in implicational form or in deductive form. For example, Lewis's first paradox can beset a system in implicational form, in that the system contains as a theorem the formula (A & ∼A) → B; or it can beset it in deductive form, in that the system allows one to deduce B from the premisses A, ∼A.

Relevance logic in the tradition of Anderson and Belnap has been almost exclusively concerned with characterizing a relevant conditional. Thus it has attacked the problem of relevance in its implicational form. Accordingly for a relevant conditional → one would not have as a theorem the formula (A & ∼A) → B. Other theorems even of minimal logic would also be lacking. Perhaps most important among these is the formula (A → (B → A)). It is also a well-known feature of their system R that it lacks the intuitionistically valid formula ((A ∨ B) & ∼A) → B (disjunctive syllogism).

But it is not the case that any relevance logic worth the title even has to concern itself with the conditional, and hence with the problem in its implicational form. The problem arises even for a system without the conditional primitive. It would still be an exercise in relevance logic, broadly construed, to formulate a deductive system free of the fallacies of relevance in deductive form even if this were done in a language whose only connectives were, say, &, ∨ and ∼. Solving the problem of relevance in this more basic deductive form is arguably a precondition for solving it for the conditional, if we suppose (as is reasonable) that the relevant conditional is to be governed by anything like the rule of conditional proof.

If σ is the order type of a recursive linear order which has a nontrivial automorphism, we let denote the least complexity in the arithmetical hierarchy such that every recursive order of type σ has a nontrivial automorphism of complexity . In Chapter 16 of his book Linear orderings [R], Rosenstein discussed the problem of determining for certain order types σ. For example Rosenstein proved that , where ζ is the order type of the integers, by constructing a recursive linear order of type ζ which has no nontrivial Σ 1-automorphism and showing that every recursive linear order of type ζ has a nontrivial Π 1-automorphism. Rosenstein also considered linear orders of order type 2 · η, where 2 is the order type of a two-element chain and η is the order type of the rational numbers. It is easily seen that any recursive linear order of type 2 · η has a nontrivial ⊿ 2-automorphism; he showed that there is a recursive linear order of type 2 · η that has no nontrivial Σ 1-automorphism. This left the question, posed in [R] and also by Lerman and Rosenstein in [LR], of whether or ⊿ 2. The main result of this article is that :

Let (*X, *T) be the nonstandard extension of a Hausdorff space (X, T). After Wattenberg [6], the monad m(x) of a near-standard point x in *X is defined as m{x) = μT (st(x)). Consider the relation

Frank Wattenberg in [6] and [7] investigated the possibilities of extending the domain of R ns to the whole of *X. Wattenberg's extensions of R ns were required to be equivalence relations, among other things. Because the nontrivial ways of constructing such extensions usually produce monadic relations, the said condition practically limits (to completely regular spaces) the class of spaces for which such extensions are possible. Since symmetry and transitivity are not, after all, characteristics of the kind of nearness that is obtained in a general topological space, it may be expected that if these two requirements are relaxed, then a monadic extension of R ns to *X should be possible in any topological space. A study of such extensions of R ns is the purpose of the present paper. We call a binary relation W ⊆ *X × *X an infinitesimal on *X if it is monadic and reflexive on *X. We prove, among other things, that the existence of an infinitesimal on *X that extends R ns is equivalent to the condition that the space (X, T) be regular.

Let T be a complete first-order theory over a finite relational language which is axiomatized by universal and existential sentences. It is shown that T is almost trivial in the sense that the universe of any model of T can be written . where F is finite and I 1, I 2, …, In are mutually indiscernible over F. Some results about complete theories with ∃∀-axioms over a finite relational language are also mentioned.

The purpose of this paper is to define and investigate a new (equational) class of algebras, which we call semi-De Morgan algebras, as a common abstraction of De Morgan algebras and distributive pseudocomplemented lattices. We were first led to this class of algebras in 1979 (in Brazil) as a result of our attempt to extend both the well-known theorem of Glivenko (see [4, Theorem 26]) and Lakser's characterization of principal congruences to a setting more general than that of distributive pseudocomplemented lattices. In subsequent years, our work in [20] on a subvariety of Ockham algebras, first considered by Berman [3], renewed our interest in semi-De Morgan algebras by providing new examples. It seems worth mentioning that these new algebras may also turn out to be useful in resolving a conjecture made in [22] to unify certain strikingly similar results on Heyting algebras with a dual pseudocomplement (see [21]) and Heyting algebras with a De Morgan negation (see [22]).

In §2 we introduce semi-De Morgan algebras and prove the main theorem, which, roughly speaking, states that certain elements of a semi-De Morgan algebra form a De Morgan algebra. Several applications then follow, including new axiomatizations of distributive pseudocomplemented lattices, Stone algebras and De Morgan algebras.

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).

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.

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.

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.

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 .

In the paper we prove the following theorem:

Theorem. There is a model N of open induction in which the set of primes is bounded and N is such that its field of fractions 〈N*, +, ·, <〉 is elementarily equivalent to 〈Q, +, ·, <〉 (the standard rationals).

We fix an ω 1-saturated model 〈M, +, ·, <〉 of PA. Let 〈M*, +, ·, <〉 denote the field of fractions of M. The model N that we are looking for will be a substructure of 〈M*, +, ·, <〉.

If A ⊆ M* then let Ā denote the ring generated by A within M*, Ậ the real closure of A, and A* the field of fractions generated by A. We have

Let J ⊆ M. Then 〈M*, +, ·〉 is a linear space over J*. If x 1,…,xk ∈ M*, we shall say that x 1,…,xk are J-independent if 〈1, x 1,…, xk 〉 are J*-independent in the usual sense. As usual, we extend the notion of J-independence to the case of infinite sets.

If A ⊆ M* and X ⊆ A, then we say that X is a J-basis of A if X is a maximal subset of A which is J-independent.

We give a completeness theorem for a logic with probability quantifiers which is equivalent to the logics described in a recent survey paper of Keisler [K]. This result improves on the completeness theorems in [K] in that it works for languages with function symbols and produces a model whose universe is an analytic subset of the real line, and whose relations and functions are Borel relative to this universe.

A subset of the Cantor space ω 2 is called monotone iff it is closed upward under the partial ordering ≤ defined by x ≤ y iff x(n) ≤ y(n) for all n ∈ ω. A set is -positive ( -positive) iff it is monotone and -positive set is a countable union of -positive sets; a -positive set is a countable intersection of -positive sets. (See Cenzer [2] for background information on these concepts.) It is clear that any -positive set is and monotone; the converse holds for n ≤ 2 [2] and was conjectured by Dyck to hold for greater n. In this note, we will disprove this conjecture by giving examples of monotone sets (for n ≥ 3) which are not even -positive.

First we note a few isomorphisms. The space ( ω 2, ≤) is isomorphic to the space ( ω 2 ≥), so instead of monotone and positive sets we may construct hereditary and negative sets (the analogous notions with “closed upward” replaced by “closed downward”). Also, ( ω 2, ≤) is isomorphic to ((ω), ⊆), where denotes the power set operator, or to ((S), ⊆) for any countably infinite set S.

In order to remove extraneous notation from the proofs, we state the results in an abstract form (whose generality is deceptive).

The principal aim of this paper is to present a construction method for nontotal extensional combinatory algebras. This is done in §2. In §0 we give definitions of some basic notions for partial combinatory algebras from which the corresponding notions for (total) combinatory algebras are obtained as specializations. In §1 we discuss some properties of nontotal extensional combinatory algebras in general. §2 describes a “partial” variant of reflexive complete partial orders yielding nontotal extensional combinatory algebras. Finally, §3 deals with properties of the models constructed in §2, such as incompletability, having no total submodel and the pathological behaviour with respect to the interpretation of unsolvable λ-terms.

In much the same way that it is possible to construct a model of hyperbolic geometry in the Euclidean plane, it is possible to model quantum logic within the classical propositional calculus.