A notion of reducibility ≤r between sets is specified by giving a set of procedures for computing one set from another. We say that a set A is r-reducible to a set B, A ≤rB, if one of the procedures applied to B gives A. Associated with any such reducibility notion is the structure of r-degrees, the equivalence classes of sets with respect to this reducibility, with the induced ordering. The most general notion of a computable reducibility is that of Turing, ≤T. Here we say that A ≤TB if there is a Turing machine φe which, when equipped with an oracle for B, computes A: φeB = A. Such Turing degree computations are characterized by the phenomenon that only during the computation itself do we discover which questions about B need to be answered to compute A(x). In contrast, for nearly all other computable reducibilities the set of questions needed is given in advance by a recursive procedure. Perhaps the most common example of such a procedure is many-one reducibility, ≤m: A ≤mB if there is a recursive function f such that x ∈ A ⇔ f(x) ∈ B.

Reducibilities with the property that the output, A(x), is determined by the answers that B gives to a set of questions calculated recursively from x are said to be of tabular type. The most general tabular reducibility is called truth-table reducibility, ≤tt. The procedures [e] associated with this reducibility are specified by a recursive function f (= {e}) which, for each x, gives a set of n questions about the oracle and, for each of the possible 2n sets of answers, gives the corresponding output. As usual this defines A ≤ttB as “there is an e such that [e]B = A”. It is with this notion of reducibility and the associated tt-degrees that we shall be concerned in this paper. Basic information on several such strong reducibilities can be found in Rogers [28]. For more information we recommend the survey articles by Odifreddi [25] and Degtev [2] as well as Odifreddi's book [26].

The canonicity and compactness of the KM system are problems historically important in the development of our understanding of intensional logic (as explained in Goldblatt's paper, The McKinsey axiom is not canonical). The problems, however, were unsolved for years in modal logic. In the beginning of 1990, Goldblatt showed that KM is not canonical in The McKinsey axiom is not canonical. The remaining task is to solve the problem of the compactness of KM. In this paper we present a proof showing that the KM system is not compact.

The symbols of the language of propositional modal logic are as follows:

1. A denumerably infinite set of sentence letters, for example, {p0, P1, p2, …};

2. The Boolean connectives &, ⋁, ¬, →, ↔ and parentheses;

3. The modal operators L and M where M is defined as ¬L¬.

The formation rules of well-formed propositional modal formulae are the formation rules of formulae in classic propositional logic plus the following rule:

If A is a well-formed formula, so is LA.

A normal modal system is a set of formulae that contains all tautologies and the formula

and is closed under the following transformation rules:

Uniform substitution. If A is a theorem, so is every substitution-instance of A.

Modus ponens. If A and A → B are theorems, so is B.

Necessitation. If A is a theorem, so is LA.

Let L be a normal system. Then a set S of formulae is L-consistent if and only if for any formula B, which is the conjunction of some formulae in S, B is not included in L. A set S of formulae is maximal if and only if for every formula A, S either contains A or contains ¬A. A set S of formulae is maximal L-consistent if and only if it is both maximal and L-consistent.

In recent years higher recursion theory has experienced a deep interaction with other areas of logic, particularly set theory (fine structure, forcing, and combinatorics) and infinitary model theory. In this paper we wish to illustrate this interaction by surveying the progress that has been made in two areas: the global theory of the κ-degrees and the study of closure ordinals.

We show that there is a complete, consistent Borel theory which has no “Borel model” in the following strong sense: There is no structure satisfying the theory for which the elements of the structure are equivalence classes under some Borel equivalence relation and the interpretations of the relations and function symbols are uniformly Borel.

We also investigate Borel isomorphisms between Borel structures.

We show that the 0-1 law does not hold for the class (∀∃∀ without = ) by finding a sentence in this class which almost surely expresses parity. We also show that every recursive real in the unit interval is the asymptotic probability of a sentence in this class. This expands a result by Lidia Tendera, who in 1994 proved that every rational number in the unit interval is the asymptotic probability of a sentence in the class ∀∃∀ with equality.

We investigate which part of Brouwer’s Intuitionistic Mathematics is finitistically justifiable or guaranteed in Hilbert’s Finitism, in the same way as similar investigations on Classical Mathematics (i.e., which part is equiconsistent with $\textbf {PRA}$ or consistent provably in $\textbf {PRA}$ ) already done quite extensively in proof theory and reverse mathematics. While we already knew a contrast from the classical situation concerning the continuity principle, more contrasts turn out: we show that several principles are finitistically justifiable or guaranteed which are classically not. Among them are: (i) fan theorem for decidable fans but arbitrary bars; (ii) continuity principle and the axiom of choice both for arbitrary formulae; and (iii) $\Sigma _2$ induction and dependent choice. We also show that Markov’s principle MP does not change this situation; that neither does lesser limited principle of omniscience LLPO (except the choice along functions); but that limited principle of omniscience LPO makes the situation completely classical.

A Leibniz class is a class of logics closed under the formation of term-equivalent logics, compatible expansions, and non-indexed products of sets of logics. We study the complete lattice of all Leibniz classes, called the Leibniz hierarchy. In particular, it is proved that the classes of truth-equational and assertional logics are meet-prime in the Leibniz hierarchy, while the classes of protoalgebraic and equivalential logics are meet-reducible. However, the last two classes are shown to be determined by Leibniz conditions consisting of meet-prime logics only.

In what follows, a coset is a subset of a group G of the form aH, where H is a subgroup of G; H can be recovered from the coset C: it is the only subgroup which is obtained from C by a left translation; we note in passing that these cosets, that we write systematically with the group to the right, are also of the form Ka, since aH = aHa−1a. A classical combinatorial lemma involving cosets appears in Neumann [1952]: If the coset C = aH is the union of the finite family of cosets C1 = a1H1,…,Cn = anHn, then it is the union of those Ci whose corresponding Hi has finite index in H.

In a structure where a group G is defined, Boolean combinations of cosets modulo its definable subgroups form a family of definable sets (by definable, we mean “definable with parameters”). The situation when any definable set is of that kind has been characterized model-theoretically in Hrushovski and Pillay [1987]: A group G is one-based if and only if, for each n, every definable subset of the cartesian power Gn is a(finite!) Boolean combination of cosets modulo definable subgroups. One side is given by a beautiful lemma of Pillay, stating that, in a one-based group which is saturated enough, every type is a right translate of the generic of its left stabilizer.

A seminal theorem due to Weyl [14] states that if (an) is any sequence of distinct integers, then, for almost every x ∈ ℝ, the sequence (anx) is uniformly distributed modulo one. In particular, for almost every x in the unit interval, the sequence (anx) is uniformly distributed modulo one for every computable sequence (an) of distinct integers. Call such an x UD random. Here it is shown that every Schnorr random real is UD random, but there are Kurtz random reals that are not UD random. On the other hand, Weyl's theorem still holds relative to a particular effectively closed null set, so there are UD random reals that are not Kurtz random.

In [6], G. Metakides and the author introduced a general model theoretic setting in which to study the lattice of r.e. substructures of a large class of recursively presented models . Examples included , the natural numbers with equality, 〈 Q, ≤ 〉, the rationals under the usual ordering, and a large class of n-dimensional partial orderings. In this setting, we were able to generalize many of the constructions of classical recursion theory so that the constructions yield the classical results when we specialize to the case of and new results when we specialize to other models. Constructions to generalize Myhill's Theorem on creative sets [8], Friedberg's Theorem on the existence of maximal sets [3], Dekker's Theorem on the degrees of hypersimple sets [2], and Martin's Theorem on the degrees of maximal sets [5] were produced in [6]. In this paper, we give constructions to generalize the Morley-Soare Splitting Theorem [7] and Lachlan's characterization of hyperhypersimple sets [4] in §2, constructions to generalize Lachlan's theorems on the existence of major subsets and r-maximal sets contained in maximal sets [4] in §3, and constructions to generalize Robinson's construction of r-maximal sets that are not contained in any maximal sets [11] and second-order maximal sets [12] in §4.

In §1 of this paper, we give the precise definitions of our model theoretic setting and deal with other preliminaries. Also in §1, we define the notions of “uniformly nonrecursive”, “uniformly maximal”, etc. which are the key notions involved in the generalizations of the various theorems that occur in §§2, 3 and 4.

We prove the elimination of Magidor-Malitz quantifiers for R-modules relative to certain -core sentences and positive primitive formulas. For complete extensions of the elementary theory of R-modules it follows that all Ramsey quantifiers (ℵ0-interpretation) are eliminable. By a result of Baldwin and Kueker [1] this implies that there is no R-module having the finite cover property.

We study cut-elimination in first-order classical logic. We construct a sequence of polynomial-length proofs having a non-elementary number of different cut-free normal forms. These normal forms are different in a strong sense: they not only represent different Herbrand-disjunctions but also differ in their prepositional structure.

This result illustrates that the constructive content of a proof in classical logic is not uniquely determined but rather depends on the chosen method for extracting it.

The model companions of the theories of n-dimensional partial orderings and n-dimensional distributive lattices are found for each finite n. Each model companion is given as the theory of a structure which is specified. The model companions are model completions only for n = 1. The structure of the model companion of the theory of n-dimensional partial orderings is a lattice only for n = 1. Each of the model companions is seen to be finitely axiomatizable, and a set of basic formulas, each of which is existential, is specified for each model companion. Finally a topolo-gically natural notion of dense n-dimensional partial ordering is introduced and shown to have a finitely axiomatizable undecidable theory.

In this paragraph we shall define the notion of model companion (cf. [4]) and indicate the way in which we shall demonstrate that one theory is the model companion of another in this paper. For T and T* theories in a common language, T* is called a model companion of T if and only if the following two conditions are satisfied: first, Tand T* are mutually model consistent, which means that every model of either is embeddable in some model of the other; secondly, T* is model complete, which means that if and are both models of T* and is a substructure of , then is an elementary substructure of . A definition of model completion may be obtained by strengthening the notion of model companion to also require that T* admit elimination of quantifiers. In all of our examples the model companion will have only one countable model. Although the ℵ0-categoricity of the model companions follows from Saracino [8], we give specific proofs since these proofs fit so naturally in our analyses.

The notations and the definitions used here are identical to those in [B.L.]. We also assume that the reader is familiar with the properties of the Rudin-Keisler order on types [L.].

Let T denote an ℵ0-stable countable theory. The main result in [B.L.] is: All countable models of T are almost homogeneous if and only if T satisfies

(*) For all models M of T and all ā ∈ M, if p ∈ S1(ā) is strongly regular multidimensional, then Dim(p; M) ≥ ℵ0.

In this paper we investigate the case of a theory T which does not satisfy condition (*) and, under certain additional assumptions, we construct nonisomorphic and non-almost-homogeneous countable models. The same type of construction has been used before to show that if T is multidimensional, then for all α ≥ 1, T has at least nonisomorphic models of cardinality ℵα [La.], [Sh.]. As a corollary of our main theorem (Theorem 6) and of the previous result in [B.L.], we prove Vaught's Conjecture (and, in fact, Martin's Strong Conjecture) for theories T with αT finite.

Although we are here interested in countable models, we can also note that our construction proves that theories satisfying the assumptions in Theorem 6 have at least nonisomorphic models of cardinality ℵα.

We use model-theoretic methods described in [3] to obtain ordinal analyses of a number of theories of first- and second-order arithmetic, whose proof-theoretic ordinals are less than or equal to Γ0.

In this note we show:

Theorem 1.1. Let G be a Polish group and X a Polish G-space with the induced orbit equivalence relation EG Borel as a subset of X × X. Then exactly one of the following:

(I) There is a countable languageℒand a Borel function

such that for all x1, x2 ∈ X

or

(II) there is a turbulent Polish G-space Y and a continuous G-embedding

There are various bows and ribbons which can be woven into these statements. We can strengthen (I) by asking that θ also admit a Borel orbit inverse, that is to say some Borel function

for some Borel set B ⊂ Mod(ℒ), such that for all x ∈ X

and then after having passed to this strengthened version of (I) we still obtain the exact same dichotomy theorem, and hence the conclusion that the two competing versions of (I) are equivalent. Similarly (II) can be relaxed to just asking that τ be a Borel G-embedding, or even simply a Borel reduction of the relevant orbit equivalence relations. It is in fact a consequence of 1.1 that all the plausible weakenings and strengthenings of (I) and (II) are respectively equivalent to one another.

I will not closely examine these possible variations here. The equivalences alluded to above follow from our main theorem and the results of [3]. That monograph had previously shown that (I) and (II) are incompatible, and proved a barbaric forerunner of 1.1, and gone on to conjecture the dichotomy result above.

By a variety, we mean a class of structures in some language containing only function symbols which is equationally defined or equivalently is closed under homomorphisms, submodels and products.

If K is a class of -structures then I(K, λ) denotes the number of nonisomorphic models in K of cardinality λ. When we say that K has few models, we mean that I(K,λ) < 2λ for some λ > ∣∣. If I(K,λ) = 2λ for all λ > ∣∣, then we say K has many models. In [9] and [10], Shelah has shown that for an elementary class K, having few models is a strong structural condition.

We investigate small theories of Boolean ordered o-minimal structures. We prove that such theories are ℵ0-categorical. We give a complete characterization of their models up to bi-interpretability of the language. We investigate types over finite sets, formulas and the notions of definable and algebraic closure.

This paper investigates the problem of extending the recursion theoretic construction of a minimal degree to the Kripke [2]-Platek [5] recursion theory on the ordinals less than an admissible ordinal α, a theory derived from the Takeuti [11] notion of a recursive function on the ordinal numbers. As noted in Sacks [7] when one generalizes the recursion theoretic definition of relative recursiveness to α-recursion theory for α > ω the two usual definitions give rise to two different notions of reducibility. We will show that whenever α is either a countable admissible or a regular cardinal of the constructible universe there is a subset of α whose degree is minimal for both notions of reducibility. The result is an excellent example of a theorem of ordinary recursion theory obtainable via two different constructions, one of which generalizes, the other of which does not. The construction which cannot be lifted to α-recursion theory is that of Spector [10]. We sketch the reasons for this in §3.

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.