This note arose out of my efforts to understand results of van den Dries, Denef, and Weispfenning on definable Skolem functions in the elementary theory of Qp. The first person to prove their existence was van den Dries, who devised and applied a model-theoretic criterion for theories, admitting elimination of quantifiers, which also admit definable Skolem functions [3]. The proof, though elegant, does not describe how one defines the Skolem functions. In the particular case of Qp, Denef found an ingenious, easily described method for writing out the definitions [2, pp. 14–15]. Unfortunately, his argument directly applies only in the following special case: if

and there is a fixed m ≥ 1 such that

for all , then can be given as a definable function of . While this special case includes many of interest, van den Dries' theorem seems more general. Weispfenning suggested how his results on primitive-recursive quantifier elimination could produce algorithms yielding definitions of Skolem functions in the specific theories van den Dries considered [10, pp. 470–471]. Though these algorithms provide a more concrete version of van den Dries' theorem, and do not suffer the lack of generality of Denef's result, Weispfenning's argument is extremely subtle and applies only to certain theories of valued fields.

In this paper we prove three theorems about first-order theories that are categorical in a higher power. The first theorem asserts that such a theory either is totally categorical or there exist prime and minimal models over arbitrary base sets. The second theorem shows that such theories have a natural notion of dimension that determines the models of the theory up to isomorphism. From this we conclude that I(T,ℵα,) = ℵ0 + ∣α∣ where ℵα = the number of formulas modulo T-equivalence provided that T is not totally categorical. The third theorem gives a new characterization of these theories.

Let K be an algebraically closed field and let L be its canonical language; that is, L consists of all relations on K which are definable from addition, multiplication, and parameters from K. Two sublanguages L 1 and L 2 of L are definably equivalent if each relation in L 1 can be defined by an L 2-formula with parameters in K, and vice versa. The equivalence classes of sublanguages of L form a quotient lattice of the power set of L about which very little is known. We will not distinguish between a sublanguage and its equivalence class.

Let L m denote the language of multiplication alone, and let L a denote the language of addition alone. Let f ∈ K [X, Y] and consider the algebraic function defined by f (x, y) = 0 for x, y ∈ K. Let L f denote the language consisting of the relation defined by f. The possibilities for L m ∨ L f are examined in §2, and the possibilities for L a ∨ L f are examined in §3. In fact the only comprehensive results known are under the additional hypothesis that f actually defines a rational function (i.e., when f is linear in one of the variables), and in positive characteristic, only expansions of addition by polynomials (i.e., when f is linear and monic in one of the variables) are understood. It is hoped that these hypotheses will turn out to be unnecessary, so that reasonable generalizations of the theorems described below to algebraic functions will be true. The conjecture is that L covers L m and that the only languages between L a and L are expansions of L a by scalar multiplications.

The aim of this article is to show that a basis for combinatory logic [2] must contain at least one combinator with rank strictly greater than two. We use notation of [1].

Let Q be a primitive combinator given by its reduction rule Qx 1 … xn → C, where C is a pure combination of the variables x 1,…, xn . n is called the rank of the combinator.

A set {Q 1,…,Qn } of combinators is a basis for combinatory logic if for every finite set {x 1,…,xm } of variables and every pure combination C of these variables, there exists a pure combinator Q of Q 1,…,Qn such that Qx 1…xm ↠C.

Property. The Church-Rosser theorem and the (quasi-)normalization theorem are valid for the combinatory reduction system under consideration.

Proof. See [4], [5], and [6].

Any basis must contain at least one combinator with rank strictly greater than two.

Let us assume a basis B with combinators of rank strictly less than 3; then there exists a pure combination X of the combinators in B such that: XABC ↠ CAB.

(*) First of all, we remark that if XABC ↠ M ↠ CAB, M must contain at least one occurrence of each of the variables A, B and C.

Notation. We denote by E[X 1,…,Xn ] expressions that contain at least one occurrence of every term X 1,…,Xn .

Consider k((G)) in the language of valued fields enriched with a unary predicate for the set of constants and another one for the cross-section. For perfect k, this structure is undecidable if it does not satisfy Kaplansky's conditions.

Three classes of decidable discrete linear orders with varying degrees of effectiveness are investigated. We consider how a classical order type may lie in relation to these three classes, and we characterize by their order types elements of these classes that have effective nontrivial self-embeddings.

To eliminate quantifiers in the first-order theory of the p-adic field Qp, Ax and Kochen use a language containing a symbol for a cross-section map n → pn from the value group Z into Qp [1, pp. 48–49]. The primitive-recursive quantifier eliminations given by Cohen [2] and Weispfenning [10] also apply to a language mentioning the cross-section, but none of these authors seems entirely happy with his results. As Cohen says, “all the operations… introduced for our simple functions seem natural, with the possible exception of the map n → pn” [2, p. 146]. So all three authors show that various consequences of quantifier elimination—completeness, decidability, model-completeness—also hold for a theory of Qp not employing the cross-section [1, p. 453; 2, p. 146; 10, §4]. Macintyre directs a more specific complaint against the cross-section [5, p. 605]. Elementary formulae which use it can define infinite discrete subsets of Qp; yet infinite discrete subsets of R are not definable in the language of ordered fields, and so certain analogies between Qp and R suggested by previous model-theoretic work seem to break down.

To avoid this problem, Macintyre gives up the cross-section and eliminates quantifiers in a theory of Qp written just in the usual language of fields supplemented by a predicate V for Qp's valuation ring and by predicates Pn for the sets of nth powers in Qp (for all n ≥ 2).

The purpose of this paper is to study logical implications which are much weaker than the implication of intuitionistic logic.

In §1 we define the system SI (system of Simple Implication) which is obtained from intuitionistic logic by restricting the inference rules of intuitionistic implication. The implication of the system SI is called the “simple implication” and denoted by ⊃, where the simple implication ⊃ has the following properties:

(1) The simple implication ⊃ is much weaker than the usual intuitionistic implication.

(2) The simple implication ⊃ can be interpreted by the notion of provability, i.e., we have a very simple semantics for SI so that a sentence A ⊃ B is interpreted as “there exists a proof of B from A”.

(3) The full-strength intuitionistic implication ⇒ is definable in a weak second order extension of SI; in other words, it is definable by help of a variant of the weak comprehension schema and the simple implication ⊃. Therefore, though SI is much weaker than the intuitionistic logic, the second order extension of SI is equivalent to the second order extension of the intuitionistic logic.

(4) The simple implication is definable in a weak modal logic MI by the use of the modal operator and the intuitionistic implication ⇒ with full strength. More precisely, A ⊃ B is defined as the strict implication of the form ◽(A ⇒ B).

In §1, we show (3) and (4). (2) is shown in §2 in a more general setting.

Semantics by introduction rules of logical connectives has been studied from various points of view by many authors (e.g. Gentzen [4], Lorentzen [5], Dummett [1], [2], Prawitz [8]. Martin-Löf [7], Maehara [6]). Among them Gentzen (in §§10 and 11 of [4]) introduced such a semantics in order to justify logical inferences and the mathematical induction rule. He observed that all of the inference rules of intuitionistic arithmetic, except for those on implication and negation, are justified by means of his semantics, but justification of the inference rules on implication and negation contains a circular argument for the interpretation by introduction rules, where the natural interpretation of A ⊃ B by ⊃-introduction rule is “there exists a proof of B from A ” (cf. §11 of Gentzen [4]).

We show that the existence of a recursively enumerable set whose Turing degree is neither low nor complete cannot be proven from the basic axioms of first order arithmetic (P −) together with Σ2 -collection (BΣ2). In contrast, a high (hence, not low) incomplete recursively enumerable set can be assembled by a standard application of the infinite injury priority method. Similarly, for each n, the existence of an incomplete recursively enumerable set that is neither lown nor high n-1, while true, cannot be established in P− + BΣn+1 . Consequently, no bounded fragment of first order arithmetic establishes the facts that the highn and lown jump hierarchies are proper on the recursively enumerable degrees.

§0. Introduction. We started our study of filters of partitions in [15]. We shall here restrict ourselves to the consideration of filters on (ω)ω, the set of all infinite partitions of ω. §1 is an attempt to elucidate the connection between filters on (ω)ω and filters over ω. Given a filter H over ω, we define two filters FH and GH on (ω)ω, and we characterize p-points, rare ultrafilters and Ramsey ultrafilters in terms of properties of the associated filters of partitions.

The remainder of the paper is devoted to the study of those filters that can be associated with Hindman's theorem and its extensions. Let us introduce some notation. Suppose * is an associative operation on ω, and let a subset A of ω and an ordinal α with 0 < α ≤ ω be given.

In this note we shall be interested in the following problems.

Problem 1. Can IΔ0 ⊢ ∀x∃y > x(y is prime)?

Here I Δ0 is Peano arithmetic with the induction axiom restricted to bounded (i.e. Δ0) formulae.

Problem 2. Can IΔ0 ⊢ Δ0 PHP?

Here Δ0 PHP (Δ0 pigeonhole principle) is the schema

for θ ∈ Δ0, or equivalently in IΔ0, for a Δ0 formula F(x,y)

written .

By obtaining partial solutions to Problem 2 we shall show that Problem 1 has a positive solution if IΔ0 is replaced by IΔ0 + ∀xx log(x) exists.

Our notation will be entirely standard (see for example [3] and [4]). In particular all logarithms will be to the base 2 and in expressions like log(x), (1 + ε)x, etc. we shall always mean the integer part of these quantities.

Concerning Problem 2 we remark that it is shown in [5] that for k ∈ N and F ∈ Δ0,

As far as we know this is the best result of this form, in that we do not know how to replace log(z) k by anything larger. However, as we shall show in Theorem 1, we can do much better if we increase the difference between the sizes of the domain and range of F.

In what follows let M be a countable nonstandard model of IΔ0, and let be those subsets of M defined by Δ0 formulae with parameters from M.

Theorem 1. For k ∈ N and F ∈ Δ0,

Here log0(x) = x, log k + 1(x) = log(log k (x)).

Proof. To simplify matters, consider first the case k = 1. So assume M ⊨ a log(a) exists and with and a > 1. The idea of the proof is the following.

The notion of a higher level ordering is a generalization of the usual notion of an order introduced by Becker in the study of sums of even powers in a field; see [1] for a general reference. A precise definition of an ordering of level 2n (level n in the terminology of [1]) is given in Definition 1.1(II) below.

In [1] Becker worked out the extension theory of fields with a higher level ordering and introduced the notion of a (generalized) real closure for such fields. For a survey of (the analog of) the Artin-Schreier theory of fields with a higher level ordering, see [2]. In [10] Jacob proved decidability, completeness and model-completeness (in a suitable language) for the theory of generalized real-closed fields. Jacob's results are in a sense optimal insofar as the nonuniqueness of generalized real closures (see [1, Chapter IV, Theorems 12 and 13]) prevents quantifier elimination results from holding in languages natural from an algebraic point of view.

The following remarkable fact stems from the work of Becker and Harman: a field having a proper ordering (i.e. one which is not just an order) of any level necessarily has an ordering of (exact) level 2n for each integer n ≥ 2, plus two usual orders, and, moreover, a tight connection holds between orderings of two consecutive levels (cf. [9, Corollary 1.4] and [1, Theorem 15, p. 37]).

This paper is meant to be a comment on Beth's definability theorem. In it we shall make the following points.

Implicit definability as mentioned in Beth's theorem for first-order logic is a special case of a more general notion of uniqueness. If α is a nonlogical constant, T α a set of sentences, α* an additional constant of the same syntactical category as α and T α, a copy of T α with α* instead of α, then for implicit definability of α in T α one has, in the case of predicate constants, to derive α(x 1,…,xn ) ↔ α*(x 1,…,xn ) from T α ∪ T α*, and similarly for constants of other syntactical categories. For uniqueness one considers sets of schemata S α and derivability from instances of S α ∪ S α* in the language with both α and α*, thus allowing mixing of α and α* not only in logical axioms and rules, but also in nonlogical assumptions. In the first case, but not necessarily in the second one, explicit definability follows. It is crucial for Beth's theorem that mixing of α and α* is allowed only inside logic, not outside. This topic will be treated in §1.

Let the structural part of logic be understood roughly in the sense of Gentzen-style proof theory, i.e. as comprising only those rules which do not specifically involve logical constants. If we restrict mixing of α and α* to the structural part of logic which we shall specify precisely, we obtain a different notion of implicit definability for which we can demonstrate a general definability theorem, where a is not confined to the syntactical categories of nonlogical expressions of first-order logic. This definability theorem is a consequence of an equally general interpolation theorem. This topic will be treated in §§2, 3, and 4.

The price is failure on a class of inductive inference problems that are easily solved, in contrast, by nonBayesian mechanical learners. By “mechanical” is meant “simulable by Turing machine”.

One of the central tenets of Bayesianism, which is common to the heterogeneous collection of views which fall under this rubric, is that hypothesis change proceeds via conditionalization on accumulated evidence, the posterior probability of a given hypothesis on the evidence being computed using Bayes's theorem. We show that this strategy for hypothesis change precludes the solution of certain problems of inductive inference by mechanical means—problems which are solvable by mechanical means when the restriction to this Bayesian strategy is lifted. Our discussion proceeds as follows. After some technical preliminaries, the concept of (formal) learner is introduced along with a criterion of inferential success. Next we specify a class of inductive inference problems, and then define the notion of “Bayesian behavior” on those problems. Finally, we exhibit an inductive inference problem from the specified class such that (a) some nonmechanical Bayesian learner solves the problem, (b) some nonBayesian mechanical learner solves the problem, (c) some mechanical learner manifests Bayesian behavior on the problem, but (d) no mechanical Bayesian learner solves the problem.

Insofar as possible terminology and notation are drawn from Osherson, Stob, and Weinstein [1986].

In [8], we have shown the equivalence of almost strong minimality and strong unidimensionality. More precisely, we proved:

Theorem [8]. Let T be a countable stable theory. Then the following two conditions are equivalent:

(i) T is almost strongly minimal;

(ii) T can be extended to a theory such that any two nonalgebraic types are not almost orthogonal.

In the present paper, we define the notion of strong 2-dimensionality (of T). We show that if T is strongly 2-dimensional then T is ω-stable and its model has a simple structure. Roughly speaking, in a model of a strongly 2-dimensional theory, one of the following holds: (a) every element is in acl (δi is strongly minimal), or (b) every element is in acl (δ is strongly regular). Shelah's definition of 2-dimensionality does not imply even superstability. (See Exercise 5.5 in [6, Chapter V, §5].) We show also that condition (a) above implies strong 2-dimensionality of T. However condition (b) does not imply strong 2-dimensionality in general.

Our notations and conventions are standard. T is always countable and stable. We work in . A,B,… are used to denote small subsets of . , … are used to denote finite sequences of elements in . δ, φ,… are used to denote formulas (with parameters), p, q, … are used to denote types (with parameters). The fact that p is a nonforking (forking) extension of q is denoted by p ⊃nfq(p ⊃fq). If p is stationary, p∣A denotes the type in S(A) which is parallel to p. (or ) denotes the set of realizations of p (or δ). The Morley rank of p is denoted by RM(p).

A countably infinite relational structure M is called absolutely ubiquitous if the following holds: whenever N is a countably infinite structure, and M and N have the same isomorphism types of finite induced substructures, there is an isomorphism from M to N. Here a characterisation is given of absolutely ubiquitous structures over languages with finitely many relation symbols. A corresponding result is proved for uncountable structures.

We use connections between conjunctive game formulas and the theory of inductive definitions to define the notions of a coinductive formula and its approximations. Corresponding to the theory of conjunctive game formulas we develop a theory of coinductive formulas, including a covering theorem and a normal form theorem for many sorted languages. Applying both theorems and the results on “model interpolation” obtained in this paper, we prove a many-sorted interpolation theorem for ω1ω-logic, which considers interpolation with respect to the language symbols, the quantifiers, the identity, and countably infinite conjunction and disjunction.

We present a downward Löwenheim-Skolem theorem which transfers downward formulas from L∞,ω to , ω. The simplest instance is:

Theorem 1. Let λ > κ be infinite cardinals, and let L be a similarity type of cardinality κ at most. For every L-structure M of cardinality λ and every X ⊆ M there exists a model N ≺ M containing the set X of power ∣X∣ · κ such that for every pair of finite sequences a, b ∈ N

The following theorem is an application:

Theorem 2. Let λ<κ, T ∈ , ω, and suppose χ is a Ramsey cardinal greater than λ. If T has the (χ, , ω)-unsuperstability property, then T has the (χ, , ω)-unsuperstability property.

A Boolean product construction is used to give examples of existentially closed algebras in the universal Horn class ISP(K) generated by a universal class K of finitely subdirectly irreducible algebras such that Γa(K) has the Fraser-Horn property. If ⟦a ≠ b⟧ ∩ ⟦c ≠ d ⟧ = ∅ is definable in K and K has a model companion of K-simple algebras, then it is shown that ISP(K) has a model companion. Conversely, a sufficient condition is given for ISP(K) to have no model companion.