The family of readability toposes, of which the effective topos is the best known, was discovered by Martin Hyland in the late 1970's. Since then these toposes have been used for several purposes. The effective topos itself was originally intended as a category in which various recursion-theoretic or effective constructions would live as natural parts of the higher-order type structure. For example the hereditary effective operators become the higher types over N (Hyland [1982]), and effective domains become the countably-based domains in the topos (McCarty [1984], Rosolini [1986]). However, following the discovery by Moggi and Hyland that it contained nontrivial small complete categories, the effective topos has also been used to provide natural models of polymorphic type theories, up to and including the theory of constructions (Hyland [1987], Hyland, Robinson and Rosolini [1987], Scedrov [1987], Bainbridge et al. [1987]).

Over the years there have also been several different constructions of the topos. The original approach, as in Hyland [1982], was to construct the topos by first giving a notion of Pω-valued set. A Pω-valued set is a set X together with a function =x: X × X → Pω. The elements of X are to be thought of as codes, or as expressions denoting elements of some “real underlying” set in the topos. Given a pair (x,x′) of elements of X, the set =x (x,x′) (generally written ) is the set of codes of proofs that the element denoted by x is equal to the element denoted by x′.

Hereditarily finite sets are sets which are finite, whose members are finite, the members of whose members are finite, and so on. In ZF there are but countably many such sets; they constitute Vω. Were ZF to lose its axiom of regularity, however, one could not guarantee that the number of hereditarily finite sets would remain countable.

In Mostowski set theory, in which atomic sets are permissible, each atom, in isolation, would form a hereditarily finite collection. The number of hereditarily finite sets could be anything one should choose.

Even in a world that did not permit the free adjunction of arbitrary, meaningless atoms, the number of hereditarily finite sets could remain large. In Finsler set theory, it is shown as Theorem 22, below, that there are uncountably many hereditarily finite sets.

The reader who is interested in this paradoxical sounding fact can turn directly to §4 after grasping these introductory concepts. §3 is an exhaustive list of the smallest Finsler sets; it is hoped that this list will prove useful in checking future attempts to classify the finite Finsler sets.

Finsler set theory is not a firmly axiomatized theory. It is, at its present stage, a family of theories undergoing evolution. It permits the usual mathematical operations with sets. One can employ ordinal numbers, cardinal numbers, and the usual methods of Cantorian set theory freely. But there is a somewhat different interpretation attached to the concept “set” than one is used to in Zermelo-Fraenkel set theory, ZF.

In this work we describe a new approach to the notions of relevance and paraconsistency. Unlike the works of Anderson and Belnap or da Costa (see [2], [8] and [7]) we shall mainly be guided in it by semantical intuitions. In the first two sections we introduce and investigate the algebraic structures that reflect those intuitions. The corresponding formal systems are briefly described in the third section (a more detailed treatment of these systems, including full proofs, will be given in another paper).

Our basic intuitive idea is that of “domains of discourse” or “relevance domains”. Classical logic, so we think, is valid in as much as sentences get values inside one domain; limitations on its use can be imposed only with respect to inferences in which more than one domain is involved. There are two basic binary relations over the collection of domains. One is relevance. It is reflexive and symmetric (but not necessarily transitive). Under a given interpretation two sentences are relevant to each other when their values are in relevant domains. Another basic relation between domains, no less important, is that of grading according to “degrees of reality”. The idea behind it is not new. Gentzen, for example, divided in [9] the world of mathematics into three grades, representing three “levels of reality”. The elementary theory of numbers has the highest degree or level of reality; set theory has the smallest degree and mathematical analysis occupies the intermediate level. In the theory of types, or in the accumulative von Neumann universe for set theory, we can find indication of a richer hierarchy.

Let ω be the set of natural numbers, i.e. {0,1,2,…}. A set A (≤ω) is called n-generic if it is Cohen-generic for n-quantifier arithmetic. As characterized by Jockusch [4], this is equivalent to saying that for every set of strings S, there is a σ < A such that σ ∈ S or ∀ν ≥ σ(ν ∉ S). When we say degree, we mean Turing degree (of unsolvability). We call a degree n-generic if it has an n-generic representative. A nonrecursive degree a is called minimal if there is no nonrecursive degree b with b < a. Jockusch [4] exhibited various properties of generic degrees, and he showed that any 2-generic degree bounds no minimal degree. Chong and Jockusch [1] showed that any 1-generic degree below 0′ bounds no minimal degree. Haught [3] refuted one of the conjectures in [1] and showed that if a is a 1-generic degree and 0 < b < a < 0′ then b is also 1-generic. We show here that there is a 1-generic degree which bounds a minimal degree. This gives an affirmative answer to questions in [1] and [4], As any 1-generic degree below 0′ bounds no minimal degree, we see that our 1-generic degree which bounds a minimal degree is not below 0′, but can be constructed recursively in 0″. Furthermore we see that the initial segments below 1-generic degrees are not order isomorphic.

An important program in the study of the structure the lattice of r.e. sets modulo finite sets, is the classification of the orbits under Aut(), the automorphism group of , If Φ ∈ Aut() and Φ(We) =* Wh(e) for all e ∈ ω, then h is called a presentation of Φ. Define Autχ() to be the class of those elements of Aut() that have a presentation in the class of functions X, where for instance X might be the class of Δn functions for n ∈ ω. If Φ ∈ Autx() then we will say that Φ is an X-automorphism. Note that we only need to consider Δn- and Πn-orbits and automorphisms since if f is a Σn-presentation, then f is a total function and therefore Δn.

Definition. If X is a class of functions and ⊆ {Wi: i < ω}, then is an X-orbit iff is an orbit under Aut() and for all A, B ∈ there is a Φ ∈ Autx() satisfying Φ(A) =* B. (Here Φ: → .)

Harrington proved that the creative sets form a Δ0-orbit and Soare proved that the maximal sets form Δ3-orbit. We will show that there is no Boolean algebra such that {A: A is r.e. and ℒ*(A) ≈ ] forms a Δ2-orbit, where ℒ*(A) is the principal filter of A in ; ℒ*(A) = {B: B ⊆* A & B ∈ }. The idea behind the proof of this theorem is very similar to the proof by Soare that maximal sets do not form a Δ2-orbit (see Soare [1974] or Soare [1987]).

We show that the modal propositional logic G, originally introduced to describe the modality “it is provable that”, is also sound for various interpretations using filters on ordinal numbers, for example the end-segment filters, the club filters, or the ineffable filters. We also prove that G is complete for the interpretation using end-segment filters. In the case of club filters, we show that G is complete if Jensen's principle □κ holds for all κ < ℵω; on the other hand, it is consistent relative to a Mahlo cardinal that G be incomplete for the club filter interpretation.

We consider IOpen, the subsystem of PA (Peano Arithmetic) with the induction scheme restricted to quantifier-free formulas.

We prove that each model of IOpen can be embedded in a model where the equation has a solution. The main lemma states that there is no polynomial f{x,y) with coefficients in a (nonstandard) DOR M such that ∣f(x,y) ∣ < 1 for every (x,y) Є C, where C is the curve defined on the real closure of M by C: x2 + y2 = a and a > 0 is a nonstandard element of M.

The aim of the present paper is to give some general conditions for constructions by finitely many workers. Constructions using infinitely many workers will not be considered here, although there are examples of such constructions. The original construction using the method of workers, due to Harrington [H], has a worker n for each n ∈ ω, as do the constructions in [K1] and [K2]. Marker [M] obtains a result using three workers. In [AJK], there are two constructions that use three workers. There are also two constructions that have, for an arbitrary recursive ordinal α, one worker for each β < α.

The main result here is a metatheorem, which is patterned after Proposition 1 of Ash [A]. As in [A], the object of the construction is to attach “labels” to the nodes in a highly nonrecursive path through a tree, while recursively enumerating neighborhoods of an “adherent” point in a metric space. There is a family of relations associated with the labels, and the metatheorem here and the one in [A] both say that the construction will succeed if these relations satisfy a list of properties. There are significant differences between the result here and that in [A]. One difference is that certain relations which in [A] were required to be r.e. need not be r.e. here. Another difference is that there are extra relations here, and as a result, the list of properties to be satisfied is longer and more horrible than that in [A].

V. Lifschitz defined in 1979 a variant of realizability which validates Church's thesis with uniqueness condition, but not the general form of Church's thesis. In this paper we describe an extension of intuitionistic arithmetic in which the soundness of Lifschitz' realizability can be proved, and we give an axiomatic characterization of the Lifschitz-realizable formulas relative to this extension. By a “q-variant” we obtain a new derived rule. We also show how to extend Lifschitz' realizability to second-order arithmetic. Finally we describe an analogous development for elementary analysis, with partial continuous application replacing partial recursive application.

It is consistent that, for every n ≥ 2, every stationary subset of ωn consisting of ordinals of cofinality ωκ, where κ = 0 or κ ≤ n − 3, reflects fully in the set of ordinals of cofinality ωn−1. We also show that this result is best possible.

This paper investigates the algebraic structure of the Medvedev lattice . We prove that is not a Heyting algebra. We point out some relations between and the Dyment lattice and the Mučnik lattice. Some properties of the degrees of enumerability are considered. We give also a result on embedding countable distributive lattices in the Medvedev lattice.

In this paper we discuss cylindric algebras with terms. The setting is two—sorted algebras—one sort for terms and one for Boolean elements. As with cylindric algebras, a cylindric algebra with terms has its roots in first order predicate logic [HMT1].

Let Σ be a set of sentences in a first order language with terms, equality and variables u0,u1,u2, …, Define a relation ≡Σ on Fm, the set of formulas, by φ ≡Σθ if and only if Σ ⊢ φ ↔ θ, and on Tm, the set of terms, by τ ≡Σσ if and only if Σ ⊢ τ ≈ σ. The operations +, ·, cκ, 0, 1 are defined as usual on equivalence classes. Define , where is σ with τ substituted for all occurrences of uκ. That the operation *κ, for κ < α, is well defined follows from the first order axioms of equality. Let vκ = [uκ]. To establish the link between terms and Booleans, define operations oκ as follows: , where φ' is a variant of φ such that uκ is free for τ in φ′ and is φ′ with τ substituted for all free occurrences of uκ in φ′. From the first order axioms it follows that oκ, for κ < α, is well defined. Finally, instead of diagonal elements, we define a Boolean-valued operation on terms as follows: [τ] e [σ] = [τ ≈ σ].