We define elementary extension and elementary equivalence in Positive Logic.

Kleene [7] and Kreisel [8] defined independently the countable (continuous) functionals. Kleene [7] defined the countable functionals of type k to be total functionals of type k acting in a continuous way when restricted to countable arguments of type k − 1. He also defined the associates for countable functionals. They are functions α: N → N containing information about how the functional acts on countable arguments. Kleene [7] showed that the countable functionals are closed under the computations derived from S1–S9 of his paper [6], and that every computable functional has a recursive associate.

Kreisel defined the continuous functionals to be equivalence-classes of associates. By his definition it is meaningless to let a continuous functional act upon anything but continuous arguments.

One disadvantage of Kleene's approach is that two different functionals may have the same associates We will later see that there may be two functionals φ1 and φ2 with the same associates but such that the relations

are not the same.

In more recent papers on the countable functionals it is normal to regard the hierarchy 〈Ct(k)〉kϵN of countable functionals as a type-structure such that the functionals in Ct(k + 1) are maps from Ct(k) to N(Ct(0) = N), see e.g. Bergstra [1] and Gandy and Hyland [3].

We will then enjoy the streamlined formalism of a type-structure in which S1–S9 have meaning, but avoid the ambiguities of Kleene's original approach.

We will presuppose a brief familiarity with the theory of the countable functionals.

In this paper, we give a self-contained and quite elementary proof that the class of all dimension drop algebras together with their distinguished faithful traces forms a Fraïssé class with the Jiang–Su algebra as its limit. We also show that the UHF algebras can be realized as Fraïssé limits of classes of C*-algebras of matrix-valued continuous functions on [0,1] with faithful traces.

We present a number of results involving almost disjoint sets and Martin's axiom. Included is an example, due to K. Kunen, of a c.c.c. partial order without property K whose product with every c.c.c. partial order is c.c.c.

In classical arithmetic a natural measure for the complexity of relations is provided by the number of quantifier alternations in an equivalent prenex normal form. However, the proof of the Prenex Normal Form Theorem uses the following intuitionistically invalid rules for permuting quantifiers with propositional constants.

Each one of these schemas, when added to Intuitionistic (Heyting's) Arithmetic IA, generates full Classical (Peano's) Arithmetic. Schema (3) is of little interest here, since one can obtain a formula intuitionistically equivalent to A ∨ ∀xBx, which is prenex if A and B are:

For the two conjuncts on the r.h.s. (1) may be successively applied, since y = 0 is decidable.

We shall readily verify that there is no way of similarly going around (1) or (2). This fact calls for counting implication (though not conjunction or disjunction) in measuring in IA the complexity of arithmetic relations. The natural implicational measure for our purpose is the depth of negative nestings of implication, defined as follows. I(F): = 0 if F is atomic; I(F ∧ G) = I(F ∨ G): = max[I(F), I(G)]; I(∀xF) = I(∃xF): = I(F); I(F → G):= max[I(F) + 1, I(G)].

We prove in particular that, in a large class of dp-minimal theories including the p-adics, definable types are dense amongst nonforking types.

In this paper we try to apply universal algebraic geometry to MV algebras, that is, we study “MV algebraic sets” given by zeros of MV polynomials, and their “coordinate MV algebras”. We also relate algebraic and geometric objects with theories and models taken in Łukasiewicz many valued logic with constants. In particular we focus on the structure of MV polynomials and MV polynomial functions on a given MV algebra.

We show that the following results in the classical theory of unbounded linear operators on Hilbert spaces can be proved within the framework of Bishop's constructive mathematics: the Kato-Rellich theorem, the spectral theorem. Stone's theorem, and the self-adjointness of the most common quantum mechanical operators, including the Hamiltonians of electro-magnetic fields with some general forms of potentials.

Varieties of the Fan Theorem have recently been developed in reverse constructive mathematics, corresponding to different continuity principles. They form a natural implicational hierarchy. Some of the implications have been shown to be strict, others strict in a weak context, and yet others not at all, using disparate techniques. Here we present a family of related Kripke models which separates all of the as yet identified fan theorems.

§0. Introduction and material background. The present paper is devoted to the study of intermediate propositional logics, and it is based on [Be, §§1 and 2].

§2 (§§0 and 1 are introductory) concerns the axiomatization of finite logics. In the literature several effective procedures to axiomatize finite logics are present (cf., for instance, [MK] and [Wr]), but, in each case, the number of propositional variables which are used is redundant. In this direction, Theorem 2.2 provides (a) a criterion to determine, given a finite logic L, the least n such that L is axiomatizable by formulas in n variables, and (b) an effective axiomatization by an n-formula. As a corollary we obtain a negative answer to Problem 7.10 of [Ho/On], showing that there is no connection between the slice to which L belongs and the number of propositional variables necessary to axiomatize L.

The principal results of the paper are in §3. In fact, a great deal of research has been done on the correspondence between conditions on the relation of Kripke-structures from one side, and axioms added to Int from the other. In this section we (a) introduce the concept of finitely separable class of Kripke-frames, and show, by means of several examples, that this concept is “wide”, in the sense that all the most studied classes of frames determined by semantical conditions are finitely separable; (b) show that each finitely separable class is axiomatizable, and that the axioms can be found by means of semantical considerations only; and (c) establish the finite model property for all the finitely separable logics.

The aim of this paper is to study tag systems as defined by Post [Post 1943, pp. 203–205 and Post, 1965, pp. 370–373]. The existence of a tag system with unsolvable halting problem was proved by Minsky by constructing a universal tag system [Minsky 1961, see also Cocke and Minsky 1964, Wang 1963, and Minsky 1967, pp. 267–273]. Hence the halting problem of a tag system can be of the complete degree 0′. We shall prove that the halting problem for a tag system can have an arbitrary (recursively enumerable) degree of undecidability (Corollary III).

A related problem arises when we ask if there exists a uniform procedure for determining, given a tag system, whether or not there is any word on which the tag system does not halt, an “immortal” word in the system. The alternative, of course, being that the system eventually halts on every (finite) word. It is shown here that this problem, the immortality problem for tag systems, is recursively unsolvable of degree 0″ (Corollary II).

We show that from large cardinals it is consistent to have the tree property simultaneously at ${\aleph _{{\omega ^2} + 1}}$ and ${\aleph _{{\omega ^2} + 2}}$ with ${\aleph _{{\omega ^2}}}$ strong limit.

We construct a model of Peano arithmetic in an uncountable language which has no elementary end extension. This answers a question of Gaifman and contrasts with the well-known theorem of MacDowell and Specker which states that every model of Peano arithmetic in a countable language has an elementary end extension. The construction employs forcing in a nonstandard model.

In this paper we present a new proof of a decidability result for the firstorder theories of certain subvarieties of Heyting algebras. By a famous result of Grzegorczyk, the full first-order theory of Heyting algebras is undecidable. In contrast, the first-order theory of Boolean algebras and of many interesting subvarieties of Boolean algebras is decidable by a result of Tarski [8]. In fact, Kozen [6] gives a comprehensive quantitative classification of the complexities of the first-order theories of various subclasses of Boolean algebras (including the full variety).

This stark contrast may be reconciled from the standpoint of universal algebra as arising out of the byplay between structure and decidability: A good structure theory entails positive decidability results. Boolean algebras have a well-developed structure theory [5], while the corresponding theory for Heyting algebras is quite meagre. Viewed in this way, we may hope to obtain decidability results if we focus attention on subclasses of Heyting algebras with good structural properties.

K. Idziak and P. M. Idziak [4] have considered an interesting subvariety of Heyting algebras, , which is the variety generated by all linearly-ordered Heyting algebras. This variety is shown to be the largest subvariety of Heyting algebras with a decidable theory of its finite members. However their proof is rather indirect, proceeding via semantic interpretation into the monadic second order theory of trees. The latter is a powerful theory—it interprets many other theories—but is computationally highly infeasible. In fact, by a celebrated theorem of Rabin, its complexity is not bounded by any elementary recursive function. Consequently, the proof of [4], besides being indirect, also gives no information on the quantitative computational complexity of the theory of .

Here we pursue the theme of structure and decidability. We isolate the indecomposable algebras in and use this to prove a theorem on the structure of if -algebras. This theorem relates the -algebras structurally to Boolean algebras. This enables us to bootstrap the known decidability results for Boolean algebras to the variety if .

The present note is conceived as a sequel to Mostowki's paper, Über die Unabhängigkeit des Wohlordnungssatzes vom Ordnungsprinzip (which will be quoted as “M.”).

In that paper Mostowski considers a system of axioms , very close to the system of axiomatic set theory of Bernays, but where the axiom of choice is not supposed to hold. Mostowski proves that the well-ordering theorem (Wohlordnungssatz) cannot be derived from the system plus the principle of simple ordering (Ordnungsprinzip).

We prove that all known examples of weakly o-minimal nonvaluational structures have no definable Skolem functions. We show, however, that such structures eliminate imaginaries up to definable families of cuts. Along the way we give some new examples of weakly o-minimal nonvaluational structures.

We study large cardinal properties associated with Ramseyness in which homogeneous sets are demanded to satisfy various transfinite degrees of indescribability. Sharpe and Welch [25], and independently Bagaria [1], extended the notion of $\Pi ^1_n$ -indescribability where $n<\omega $ to that of $\Pi ^1_\xi $ -indescribability where $\xi \geq \omega $ . By iterating Feng’s Ramsey operator [12] on the various $\Pi ^1_\xi $ -indescribability ideals, we obtain new large cardinal hierarchies and corresponding nonlinear increasing hierarchies of normal ideals. We provide a complete account of the containment relationships between the resulting ideals and show that the corresponding large cardinal properties yield a strict linear refinement of Feng’s original Ramsey hierarchy. We isolate Ramsey properties which provide strictly increasing hierarchies between Feng’s $\Pi _\alpha $ -Ramsey and $\Pi _{\alpha +1}$ -Ramsey cardinals for all odd $\alpha <\omega $ and for all $\omega \leq \alpha <\kappa $ . We also show that, given any ordinals $\beta _0,\beta _1<\kappa $ the increasing chains of ideals obtained by iterating the Ramsey operator on the $\Pi ^1_{\beta _0}$ -indescribability ideal and the $\Pi ^1_{\beta _1}$ -indescribability ideal respectively, are eventually equal; moreover, we identify the least degree of Ramseyness at which this equality occurs. As an application of our results we show that one can characterize our new large cardinal notions and the corresponding ideals in terms of generic elementary embeddings; as a special case this yields generic embedding characterizations of $\Pi ^1_\xi $ -indescribability and Ramseyness.