The degree spectrum of a countable structure is the set of all Turing degrees of presentations of that structure. We show that every nonlow Turing degree lies in the spectrum of some differentially closed field (of characteristic 0, with a single derivation) whose spectrum does not contain the computable degree 0. Indeed, this is an equivalence, for we also show that if this spectrum contained a low degree, then it would contain the degree 0. From these results we conclude that the spectra of differentially closed fields of characteristic 0 are exactly the jump-preimages of spectra of automorphically nontrivial graphs.

Let T be a set of axioms for a classical theory TC (e.g. abelian groups, linear order, unary function, algebraically closed fields, etc.). Suppose we regard T as a set of axioms for an intuitionistic theory TH (more precisely, we regard T as axioms in Heyting's predicate calculus HPC).

Question. Is TH decidable (or, more generally, if X is any intermediate logic, is TX decidable)? In [1] we gave sufficient conditions for the undecidability of TH. These conditions depend on the formulas of T (different axiomatization of the same TC may give rise to different TH) and on the classical model theoretic properties of TC (the method did not work for model complete theories, e.g. those of the title of the paper). For details see [1]. In [2] we gave some decidability results for some theories: The problem of the decidability of theories TH for a classically model complete TC remained open. An undecidability result in this direction, for dense linear order was obtained by Smorynski [4]. The cases of algebraically closed fields and real closed fields and divisible abelian groups are treated in this paper. Other various decidability results of the intuitionistic theories were obtained by several authors, see [1], [2], [4] for details.

One more remark before we start. There are several possible formulations for an intuitionistic theory of, e.g. fields, that correspond to several possible axiomatizations of the classical theory. Other formulations may be given in terms of the apartness relation, such as the one for fields given by Heyting [5]. The formulations that we consider here are of interest as these systems occur in intuitionistic mathematics. We hope that the present methods could be extended to the (more interesting) case of Heyting's systems [5].

This paper is concerned with the hyderdegrees of elements of uncountable Borel subsets of ωω. The Borel subsets of ωω are the so-called Δ11 subsets of ωω, which are the subsets of ωω that are Δ11 in some parameter f: ω → ω.

The results of this paper were inspired by two earlier results about the hyperdegrees of elements of Σ11 subsets of ωω. The first is known as the Gandy basis theorem, and asserts that the collection of functions of strictly lower hyperdegree than the hyperjump form a basis for the Σ11 subsets of ωω (see Rogers [3, p. 421]). The second is that there exists an uncountable Σ11 subset of ωω no element of which is hyperarithmetic in any other. The latter is credited to Feferman in Harrison [1], and appears there as Corollary 2.7, p. 537.

Two questions arise. Can we find other basis theorems if we replace Σ11 by Δ11 in Gandy's result? Can we replace Σ11 by Δ11 in Feferman's result?

We simultaneously answer the first positively, and the second negatively by proving that any hyperdegree, in which the hyperjump is hyperarithmetic, forms a basis for the Δ11 subsets of ωω with no hyperarithmetic elements. We conjecture that this holds also for uncountable Δ11 subsets of ωω. This conjecture is open despite the recent claim in Notices of the American Mathematical Society, vol. 19 (1972), p. A616, Abstract 696-02-10.

In this paper we prove that if L is a set of modal propositional formulas then FR(L) (the class of all frames in which every formula of L holds) is elementary, Δ-elementary or not ΣΔ-elementary. For single modal formulas the second of these cases does not occur.

The model theoretic terminology and results used here are from [1]. (The underlying first order language contains only one, binary, predicate letter in addition to the identity symbol.) We presuppose familiarity with the usual notions and notations of propositional modal logic. A structure for our first order language is called a frame. (So a frame is an ordered couple 〈W, R〉 with domain W and R a binary predicate on W, i.e. a subset of W × W.) A valuation V on F is a function from the set of proposition letters to the power set of W. Using the well-known Kripke truth definition V can be extended to a function from the set of all modal propositional formulas to the power set of W. A modal propositional formula φ holds in a frame F (= 〈W, R〉) if, for all V on F, V(φ) = W. Notation: FR(φ) for the class of all frames in which φ holds. For a set L of modal propositional formulas we define FR(L) as ⋂φ∈L FR(φ). Obviously both FR(L) and cFR(L) (the complement of FR(L)) are closed under isomorphisms.

Beginning with Ramsey's theorem of 1930, combinatorists have been intrigued with the notion of large cardinals satisfying partition relations. Years of research have established the smaller ones, weakly ineffable, Erdös, Jonsson, Rowbottom and Ramsey cardinals to be among the most interesting and important large cardinals in set theory. Recently, cardinals satisfying more powerful infinite-exponent partition relations have been examined with growing interest. This is due not only to their inherent qualities and the fact that they imply the existence of other large cardinals (Kleinberg [2], [3]), but also to the fact that the Axiom of Determinacy (AD) implies the existence of great numbers of such cardinals (Martin [5]).

That these properties are more often than not inconsistent with the full Axiom of Choice (Kleinberg [4]) somewhat increases their charm, for the theorems concerning them tend to be a little odd, and their proofs, circumforaneous. The properties are, as far as anyone knows, however, consistent with Dependent Choice (DC).

Our basic theorem will be the following: If k > ω and k satisfies k→(k)k then the least cardinal δ such that has a δ-additive, uniform ultrafilter. In addition, if ACω is assumed, we will show that δ is greater than ω, and hence a measurable cardinal. This result will be strengthened somewhat when we prove that for any k, δ, if then .

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+.

Sometimes it is said that mathematics talks only about truths that are invariant under isomorphisms. Thus, a syntactical characterization of sentences preserved under isomorphisms seems to be important to understand what mathematics is.

The importance of such sentences is underlined, for instance, by Bourbaki's requirement in [1, Chapter IV] that all axioms for mathematical structures satisfy the condition of being what they call transportable, that is, preserved under isomorphisms. In [2, §1], a formalization of Bourbaki's notion of mathematical structure and of the notion of transportable sentence was given and discussed. We adopt here, as in [2], Bourbaki's terminology, and call a sentence φ transportable if φ is true in a structure E if and only if φ is true in any structure F isomorphic to E.

An alternative formulation for Bourbaki's notion of structure is offered in [2, §3], which is also intended as a formalization of Suppes' notion of set-theoretical predicate, as formulated, for instance, in [3]. It is this alternative formulation that we adopt here.

In [2, p. 106], it was conjectured that a set-theoretical sentence is transportable if and only if it is equivalent to a sentence of type theory. One direction of the conjecture is obvious, namely, that any sentence of type theory is transportable. It is also clear that there are sentences of set theory which are not transportable. The main purpose of the present paper is to prove the rest of the conjecture. That is, we prove here that any transportable set-theoretical sentence is equivalent to a sentence of type theory.

One of the most fundamental results of higher-type recursion theory was first proved by Thomas Grilliot [3]. He showed that the functional embodying number quantification, 2E, is computable from a type-2 object F if and only if F has an effective discontinuity. For the background to this work we recommend Normann's book [9] or his summary [10], and for an example of its many applications Normann-Wainer [11].

The purpose of §1 of this paper is to give an effective proof of the “only if” part of this result: i.e. a construction of a discontinuity of F from the information 2E = {e}F. This contrasts with Grilliot's argument which is topological and nonconstructive, and shows that “Grilliot's trick” for computing 2E from a discontinuity of F is essentially the only method.

We then apply the method to two more general situations in which it gives new results. In §2 we treat type-3 objects in which 3E is computable (assuming AC and CH), and obtain a strengthening of the result of Hartley [5, §5]. In §3 we consider the superjump and characterise it as the least among a class of functionals with a particular sort of family of discontinuities.

We prove that, in the choiceless Solovay model, every set of reals is H-Ramsey for every happy family H that also belongs to the Solovay model. This gives a new proof of Törnquist’s recent theorem that there are no infinite mad families in the Solovay model. We also investigate happy families and mad families under determinacy, applying a generic absoluteness result to prove that there are no infinite mad families under $A{D^ + }$.

The algebras studied in this paper were suggested to the author by William Craig as a possible substitute for cylindric algebras. Both kinds of algebras may be considered as algebraic versions of first-order logic. Cylindric algebras can be introduced as follows. Let ℒ be a first-order language, and let be an ℒ-structure. We assume that ℒ has a simple infinite sequence ν 0, ν 1, … of individual variables, and we take as known what it means for a sequence ν 0, ν 1, … of individual variables, and we take as known what it means for a sequence x = 〈x 0, x 1, …〉 of elements of to satisfy a formula ϕ of ℒ in . Let ϕ be the collection of all sequences x which satisfy ϕ in . We can perform certain natural operations on the sets ϕ , of basic model-theoretic significance: Boolean operations = cylindrifications diagonal elements (0-ary operations) . In this way we make the class of all sets ϕ into an algebra; a natural abstraction gives the class of all cylindric set algebras (of dimension ω). Thus this method of constructing an algebraic counterpart of first-order logic is based upon the notion of satisfaction of a formula by an infinite sequence of elements. Since, however, a formula has only finitely many variables occurring in it, it may seem more natural to consider satisfaction by a finite sequence of elements; then ϕ becomes a collection of finite sequences of varying ranks (cf. Tarski [10]). In forming an algebra of sets of finite sequences it turns out to be possible to get by with only finitely many operations instead of the infinitely many ci 's and dij 's of cylindric algebras. Let be the class of all algebras of sets of finite sequences (an exact definition is given in §1).

A standard result in topological dynamics is the existence of minimal subsystem. It is a direct consequence of Zorn's lemma: given a compact topological space X with a map f: X→X, the set of compact non empty subspaces K of X such that f(K) ⊆ K ordered by inclusion is inductive, and hence has minimal elements. It is natural to ask for a point-free (or formal) formulation of this statement. In a previous work [3], we gave such a formulation for a quite special instance of this statement, which is used in proving a purely combinatorial theorem (van de Waerden's theorem on arithmetical progression).

In this paper, we extend our analysis to the case where X is a boolean space, that is compact totally disconnected. In such a case, we give a point-free formulation of the existence of a minimal subspace for any continuous map f: X→X. We show that such minimal subspaces can be described as points of a suitable formal topology, and the “existence” of such points become the problem of the consistency of the theory describing a generic point of this space. We show the consistency of this theory by building effectively and algebraically a topological model. As an application, we get a new, purely algebraic proof, of the minimal property of [3]. We show then in detail how this property can be used to give a proof of (a special case of) van der Waerden's theorem on arithmetical progression, that is “similar in structure” to the topological proof [6, 8], but which uses a simple algebraic remark (Proposition 1) instead of Zorn's lemma. A last section tries to place this work in a wider context, as a reformulation of Hilbert's method of introduction/elimination of ideal elements.

We consider ωn-automatic structures which are relational structures whose domain and relations are accepted by automata reading ordinal words of length ωn for some integer n ≥ 1. We show that all these structures are ω-tree-automatic structures presentable by Muller or Rabin tree automata. We prove that the isomorphism relation for ω2-automatic (resp. ωn-automatic for n > 2) boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups) is not determined by the axiomatic system ZFC. We infer from the proof of the above result that the isomorphism problem for ωn-automatic boolean algebras, n ≥ 2, (respectively, rings, commutative rings, non commutative rings, non commutative groups) is neither a -set nor a -set. We obtain that there exist infinitely many ωn-automatic, hence also ω-tree-automatic, atomless boolean algebras , which are pairwise isomorphic under the continuum hypothesis CH and pairwise non isomorphic under an alternate axiom AT, strengthening a result of [14].

Non-n-ampleness as denned by Pillay [20] and Evans [5] is preserved under analysability. Generalizing this to a more general notion of Σ-ampleness, this gives an immediate proof for all simple theories of a weakened version of the Canonical Base Property (CBP) proven by Chatzidakis [4] for types of finite SU-rank. This is then applied to the special case of groups.

The aim of this paper is to correct an error in the proof in [2] of a strengthened version of Hilbert's second ε-theorem.

Hilbert's original theorem says (effectively) that formulae of the form ∃xA → A(εxA) can be eliminated from deductions of ∈-free sentences B (i.e. those which do not contain the ε-symbol) from collections X of ε-free sentences. The extended version (Theorem III. 11 of [2]) says that, in addition, instances of Ackermann's schema can also be eliminated.

The error in [2] occurs in the proof of Theorem III.7 and is described as follows. If A′ is the formula obtained from A by replacing every occurrence of ∃yB by B(εyB) (or ∀yC by C(εyB) if B is of the form ¬ C), and if A is a Q3 or Q4-axiom then, contrary to the claim on p. 73 of [2], A′ is not necessarily an axiom of the same form. For example, if A is the Q3-axiom

where the rank of εx∃yD(x, y) is ≤ the rank of εyB (see p. 70) and where B is D(t,y), then A′ is

which is not a Q3-axiom. In other words, the notion of rank as defined in [2] is inadequate for the proof of Theorem III.11.

It will be shown later in this paper that the class of all Hilbert spaces is not an elementary class (in the wider sense) in the lower predicate calculus. It is not difficult to find a type and a sentence in Lω1ω1 (of that type) whose models are precisely the Hilbert spaces (slightly altered to include in their domains the real or complex numbers).

We describe a method for obtaining classical logic from intuitionistic logic which does not depend on any proof system, and show that by applying it to the most important implicational relevance logics we get relevance logics with nice semantical and proof-theoretical properties. Semantically all these logics are sound and strongly complete relative to classes of structures in which all elements except one are designated. Proof-theoretically they correspond to cut-free hypersequential Gentzen-type calculi. Another major property of all these logics is that the classical implication can faithfully be translated into them.

We prove that each Σ20 set which is hypersimple relative to ∅′ is noncuppable in the structure of the Σ20 enumeration degrees. This gives a connection between properties of Σ20 sets under inclusion and and the Σ20 enumeration degrees. We also prove that some low non-computably enumerable enumeration degree contains no set which is simple relative to ∅′.

A many-one degree is functional if it contains the index set of some class of partial recursive functions but does not contain an index set of a class of r.e. sets. We give a natural embedding of the r.e. m-degrees into the functional degrees of 0′. There are many functional degrees in 0′ in the sense that every partial-order can be embedded. By generalizing, we show there are many functional degrees in every complete Turing degree.

There is a natural tie between the studies of index sets and partial-many-one reducibility. Every partial-many-one degree contains one or two index sets.

The paper uses the formalism of indexed categories to recover the proof of a standard final coalgebra theorem, thus showing existence of final coalgebras for a special class of functors on finitely complete and cocomplete categories. As an instance of this result, we build the final coalgebra for the powerclass functor, in the context of a Heyting pretopos with a class of small maps. This is then proved to provide models for various non-well-founded set theories, depending on the chosen axiomatisation for the class of small maps.

A significant portion of the study of large cardinals in set theory centers around the concept of “partition relation”. To best capture the basic idea here, we introduce the following notation: for x and y sets, κ an infinite cardinal, and γ an ordinal less than κ, we let [x]γ denote the collection of subsets of x of order-type γ and abbreviate with the partition relation for each function F frominto y there exists a subset C of κ of cardinality κ such that (such that for each α < γ) the range of F on [С]γ ([С]α) has cardinality 1. Now although each infinite cardinal κ satisfies the relation for each n and m in ω (F. P. Ramsey [8]), a connection with large cardinals arises when one asks, “For which uncountable κ do we have κ → (κ)2?” Indeed, any uncountable cardinal κ which satisfies κ → (κ)2 is strongly inaccessible and weakly compact (see [9]). As another example one can look at the improvements of Scott's original result to the effect that if there exists a measurable cardinal then there exists a nonconstructible set. Indeed, if κ is a measurable cardinal then κ → (κ)< ω, and as Solovay [11] has shown, if there exists a cardinal κ such that κ → (κ)< ω3 (κ → (ℵ1)< ω, even) then there exists a nonconstructible set of integers.