We characterize, in terms of determinacy, the existence of the least inner model of “every object of type k has a sharp.” For k ∈ ω, we define two classes of sets, and , which lie strictly between and . Let #k be the (partial) sharp function on objects of type k. We show that the determinacy of follows from

L[#k] ⊨ “every object of type k has a sharp”,

and we show that the existence of indiscemibles for L[#k] is equivalent to a slightly stronger determinacy hypothesis, the determinacy of .

Storage operators have been introduced by J. L. Krivine in [5] they are closed λ-terms which, for a data type, allow one to simulate a “call by value” while using the “call by name” strategy. In this paper, we introduce the directed λ-calculus and show that it has the usual properties of the ordinary λ-calculus. With this calculus we get an equivalent—and simple—definition of the storage operators that allows to show some of their properties:

• the stability of the set of storage operators under the β-equivalence (Theorem 5.1.1);

• the undecidability (and semidecidability) of the problem “is a closed λ-term t a storage operator for a finite set of closed normal λ-terms?” (Theorems 5.2.2 and 5.2.3);

• the existence of storage operators for every finite set of closed normal λ-terms (Theorem 5.4.3);

• the computation time of the “storage operation” (Theorem 5.5.2).

Both syntactic and semantic solutions are given for the entailment problem of duality theory. The test algebra theorem provides both a syntactic solution to the entailment problem in terms of primitive positive formula and a new derivation of the corresponding result in clone theory, viz. the syntactic description of Inv(Pol(R)) for a given set R of unitary relations on a finite set. The semantic solution to the entailment problem follows from the syntactic one, or can be given in the form of an algorithm. It shows, in the special case of a purely relational type, that duality-theoretic entailment is describable in terms of five constructs, namely trivial relations, intersection, repetition removal, product, and retractive projection. All except the last are concrete, in the sense that they are described by a quantifier-free formula. It is proved that if the finite algebra M generates a congruence-distributive variety and all subalgebras of M are subdirectly irreducible, then concrete constructs suffice to describe entailment. The concept of entailment appropriate to strong dualities is also introduced, and described in terms of coordinate projections, restriction of domains, and composition of partial functions.

We show that it is not possible to construct a Fraenkel-Mostowski model in which the axiom of choice for well-ordered families of sets and the axiom of choice for sets of well-orderable sets are both true, but the axiom of choice is false.

We show that the Π4-theory of the partial order of recursively enumerable weak truth-table degrees is undecidable, and give a new proof of the similar fact for r.e. T-degrees. This is accomplished by introducing a new coding scheme which consists in defining the class of finite bipartite graphs with parameters.

Some methods of constructing nonstandard models work only for particular theories, such as ZFC, or CA + AC (which is second order number theory with the choice scheme). The examples of this which motivated the results of this paper occur in the main theorems of [5], which state that if T is any consistent extension of either ZFC0 (which is ZFC but with only countable replacement) or CA + AC and if κ and λ are suitably chosen cardinals, then T has a model which is κ-saturated and has the λ-Bolzano-Weierstrass property. (Compare with Theorem 3.5.) Another example is a result from [12] which states that if T is any consistent extension of CA + AC and cf (λ) > ℵ0, then T has a natural λ-Archimedean model. (Compare with Theorem 3.1 and the comments following it.) Still another example is a result in [6] in which it is shown that if a model of Peano arithmetic is expandable to a model of ZF or of CA, then so is any cofinal extension of . (Compare with Theorem 3.10.) Related types of constructions can also be found in [10] and [11].

A reflection principle will be proved here, allowing these constructions to be extended to models of many other theories, among which are some exceedingly weak theories and also all of their completions.

The present article, which is a revised version of part of [Hu1], deals with various relations between models which might serve as exact formulations for the vague concept “similar” or “almost isomorphic”. One natural class of such formulations is equivalence in a given logic. Another way to express similarity is by potential isomorphism, i.e., isomorphism in some extension of the set-theoretic universe. The class of extensions may be restricted to give different notions of potential isomorphism. A third method is to study the winning strategies for an Ehrenfeucht-Fraïssé-game played between the two models, and the properties of the resulting equivalence and nonequivalence trees.

The basic question studied here is whether one such notion of similarity implies another. Some implications and counterexamples listed in this part are previously known or trivial, but all are mentioned for completeness' sake. Only models of cardinality ℵ1 are considered. Some results are therefore connected with the Continuum Hypothesis.

We define the ideal with the property that a real omits all Borel sets in the ideal which are coded in a transitive model if and only if it is an amoeba real over this model. We investigate some other properties of this ideal. Strolling through the "amoeba forest" we gain as an application a modification of the proof of the inequality between the additivities of Lebesgue measure and Baire category.

We establish a course-of-values induction principle for K-finite sets in intuitionistic type theory. Using this principle, we prove a pigeonhole principle conjectured by Bénabou and Loiseau. We also comment on some variants of this pigeonhole principle.

Let be a finitely presented variety with operations Ω and let be the variety having the same set of operations Ω but defined by the empty set of identities. A partial-algebra is a set P with a set of mappings containing for each n-ary operation f of Ω a mapping , where D ⊆ Pn. An incomplete -algebra is a partial -algebra which satisfies the defining identities of , insofar as they can be applied to the partial operations of (Trevor Evans [4, p. 67]). We call an incomplete -algebra a partial Evans-algebra if it can be embedded in a member of the variety .

If the class of all partial Evans -algebras is (first-order) finitely axiomatizable, then the word problem for the variety is solvable. (Evans [4, 5]). In 1953 Evans [5, p. 79] raised the question of whether the converse is true. In this paper we show that the answer is in the negative.

Let CSg denote the variety of commutative semigroups. We call an incomplete CSg-algebra an incomplete commutative semigroup and we call a partial Evans CSg-algebra a partial Evans commutative semigroup. It is known (A. I. Malcev [9] see also Evans [6]) that the variety of commutative semigroups has solvable word problem. We show (Theorem 1) that the class of all partial Evans commutative semigroups is not finitely axiomatizable. Therefore the solvability of the word problem for the variety of commutative semigroups does not imply finite axiomatizability of the class of all partial Evans commutative semigroups.

In the presence of large cardinals, or sufficient determinacy, every equivalence relation in either admits a wellordered separating family or continuously reduces E0.

A new method of “minimal” readability is proposed and applied to show that the definable functions of Heyting arithmetic (HA)—functions f such that HA ⊢ ∀x∃!yA(x, y) ⇒ for all m, A(m, f(m)) is true, where A(x, y) may be an arbitrary formula of ℒ(HA) with only x,y free—are precisely the provably recursive functions of the classical Peano arithmetic (PA), i.e., the < ε0-recursive functions. It is proved that, for prenex sentences provable in HA, Skolem functions may always be chosen to be < ε0-recursive. The method is extended to intuitionistic finite-type arithmetic, , and elementary analysis. Generalized forms of Kreisel's characterization of the provably recursive functions of PA and of the no-counterexample-interpretation for PA are consequently derived.

By an equational theory we mean a set of equations from some fixed language which is closed with respect to logical consequences. We regard equations as universal sentences whose quantifier-free parts are equations between terms. In our notation, we suppress the universal quantifiers. Once a language has been fixed, the collection of all equational theories for that language is a lattice ordered by set inclusion The meet in this lattice is simply intersection; the join of a collection of equational theories is the equational theory axiomatized by the union of the collection. In this paper we prove, for languages with only finitely many fundamental operation symbols, that any nontrivial finitely axiomatizable equational theory covers some other finitely axiomatizable equational theory. In fact, our result is a little more general.

There is an extensive literature concerning lattices of equational theories. These lattices are always algebraic. Compact elements of these lattices are the finitely axiomatizable equational theories. We also call them finitely based. The largest element in the lattice is compact; it is the equational theory based on the single equation x ≈ y. The smallest element of the lattice is the trivial theory consisting of tautological equations. For all but the simplest languages, the lattice of equational theories is intricate. R. McKenzie in [6] was able to prove in essence that the underlying language can be recovered from the isomorphism type of this lattice.

The notion of CM-triviality was introduced by Hrushovski, who showed that his new strongly minimal sets have this property. Recently Baudisch has shown that his new ω1-categorical group has this property. Here we show that any group of finite Morley rank definable in a CM-trivial theory is nilpotent-by-finite, or equivalently no simple group of finite Morley rank can be definable in a CM-trivial theory.

We study how equivalent nonisomorphic models of unsuperstable theories can be. We measure the equivalence by Ehrenfeucht-Fraïssé games. This paper continues [HS].

Our main goal in this paper is to establish a Glimm-Effros type dichotomy for arbitrary analytic equivalence relations.

The original Glimm-Effros dichotomy, established by Effros [Ef], [Ef1], who generalized work of Glimm [G1], asserts that if an Fσ equivalence relation on a Polish space X is induced by the continuous action of a Polish group G on X, then exactly one of the following alternatives holds:

(I) Elements of X can be classified up to E-equivalence by “concrete invariants” computable in a reasonably definable way, i.e., there is a Borel function f: X → Y, Y a Polish space, such that xEy ⇔ f(x) = f(y), or else

(II) E contains a copy of a canonical equivalence relation which fails to have such a classification, namely the relation xE0y ⇔ ∃n∀m ≥ n(x(n) = y(n)) on the Cantor space 2ω (ω = {0,1,2, …}), i.e., there is a continuous embedding g: 2ω → X such that xE0y ⇔ g(x)Eg(y).

Moreover, alternative (II) is equivalent to:

(II)′ There exists an E-ergodic, nonatomic probability Borel measure on X, where E-ergodic means that every E-invariant Borel set has measure 0 or 1 and E-nonatomic means that every E-equivalence class has measure 0.

Let p be a fixed prime integer, other than 2, Qp the field of p-adic numbers, and Ωp the completion of the algebraic closure of Qp. Let Rp be the ring of entire functions in one variable t over Ωp; that is, Rp is the ring of functions f: Ωp → Qp such that f(t) is given by a power series around 0, of infinite radius of convergence:

and where ∣a∣p is the p-adic norm of a in Ωp. We prove:

Theorem A. The positive existential theory of Rp in the language L = {0, 1, t, +, ·} is undecidable.

Theorem A gives a negative answer to the analogue of Hilbert's tenth problem for Rp in the language L. Related results include those of [2] where it is shown that the first-order theory of entire functions on the complex plane is undecidable and the similar result for analytic functions on the open unit disk (this is due to Denef and Gromov, communicated to us by Cherlin and is as of now unpublished).

It would be desirable to have a similar result in the language which, instead of the variable t, has a predicate for the transcendental (that is, nonconstant) elements of Rp. A related problem is the similar problem for meromorphic functions on the real or p-adic plane or on the unit open or closed disk. These problems seem for the moment rather hard in view of the fact that the analogue of Hilbert's Tenth Problem for the field of rational functions over the complex numbers (or any algebraically closed field of characteristic zero) is an open problem.