Let α be an admissible ordinal. In this paper we study the structure of the upper semilattice of α-recursively enumerable degrees. Various results about the structure which are of fundamental importance had been obtained during the past two years (Sacks-Simpson [7], Lerman [4], Shore [9]). In particular, the method of finite priority argument of Friedberg and Muchnik was successfully generalized in [7] to an α-finite priority argument to give a solution of Post's problem for all admissible ordinals. We refer the reader to [7] for background material, and we also follow closely the notations used there.

Whereas [7] and [4] study priority arguments in which the number of injuries inflicted on a proper initial segment of requirements can be effectively bounded (Lemma 2.3 of [7]), we tackle here priority arguments in which no such bounds exist. To this end, we focus our attention on the fine structure of Lα , much in the fashion of Jensen [2], and show that we can still use a priority argument on an indexing set of requirements just short enough to give us the necessary bounds we seek.

The question of continuity of functions defined on Baire space NN or on the reals has been of great interest to constructivists. We have little hope of answering the question without a fundamental philosophical analysis of the notion of constructive function. However, one can ask whether the continuity of functions can be derived in known intuitionistic formal systems, and answer these questions by mathematical means.

The constructivist does not generally wish to restrict himself to reals (or members of NN ) given by a recursive law; therefore the most natural formal systems have variables for reals, sequences and functions. The axioms of these systems turn out to be compatible with the assumption that every real or sequence is given by a recursive law, and every function by an effective operation. An underivability result, therefore, is only strengthened if this assumption is taken as an axiom; and once this is done, we may as well work with (indices of) effective operations, within arithmetic.

By KLS is meant the assertion that each effective operation on Baire space NN is continuous. In [B1] it is shown that this statement cannot be proved in various intuitionistic formal systems. The continuity of functions on the reals has been of even greater interest to constructivists than the continuity of functions on NN ; by KLS(R) is meant the assertion that each effective operation from the real numbers R to R is continuous. It is the purpose of the present note to show that KLS(R) is also underivable in intuitionistic formal systems.

In [7] it is shown that if Σ is a type omitted in the structure = ω, +, ·, < and complete with respect to Th() then Σ is omitted in models of Th() of all infinite powers. The proof given there extends readily to other models of P. In this paper the result is extended to models of ZFC. For pre-tidy models of ZFC, the proof is a straightforward combination of the methods in [7] and in Keisler and Morley ([9], [6]). For other models, the proof involves forcing. In particular, it uses Solovay and Cohen's original forcing proof that GB is a conservative extension of ZFC (see [2, p. 105] and [5, p. 77]).

The method of proof used for pre-tidy models of set theory can be used to obtain an alternate proof of the result for This new proof yields more information. First of all, a condition is obtained which resembles the hypothesis of the “Omitting Types” theorem, and which is sufficient for a theory T to have a model omitting a type Σ and containing an infinite set of indiscernibles. The proof that this condition is sufficient is essentially contained in Morley's proof [9] that the Hanf number for omitting types is so the condition will be called Morley's condition.

If T is a pre-tidy theory, Morley's condition guarantees that T will have models omitting Σ in all infinite powers.

In this article, unpublished methods of Solovay and Kunen are applied to describe conditions under which an uncountable regular κ can satisfy weak (κ, λ)-compactness (see 1.1(3) below), yet lie below 2μ for some μ < κ. The argumentation is in informal ZFC, and general set-theoretic notation is standard. The lower-case Greek letters κ, λ, μ, ν are reserved for cardinals in the sense of some transitive or inner model of a reasonable set theory, φ, Ψ, θ, are (arithmetizations of) formulas in some extension of the first-order language of set theory, and other lower-case Greek letters except Є are metavariables for arbitrary ordinals. If M is transitive, M ⊨ φ abbreviates 〈M, Є 〉 ⊨ φ. [2], [3], [9] and [1] provide more information about large-cardinal theory for those who wish it.

1.1. Definitions. (1) κ is inaccessible iff κ is regular and ℵκ = κ; strongly inaccessible iff κ is regular and ℶκ = κ, i.e., λ < κ for all λ < κ; weakly inaccessible iff κ is inaccessible but not strongly inaccessible.

(2) L κλ is the infinitary language with conjunctions and disjunctions of length < κ and quantification over sequences of length < λ, and PL κ the prepositional language with κ letters and conjunctions and disjunctions of length < κ.

A constructive proof is given which shows that every nonrecursive r.e. many-one degree is represented by the family of decision problems for partial implicational propositional calculi whose well-formed formulas contain at most two distinct variable symbols.

A new reduction class is presented for the satisfiability problem for well-formed formulas of the first-order predicate calculus. The members of this class are closed prenex formulas of the form ∀x∀yC. The matrix C is in conjunctive normal form and has no disjuncts with more than three literals, in fact all but one conjunct is unary. Furthermore C contains but one predicate symbol, that being unary, and one function symbol which symbol is binary.

Let be a finite set of (nonlogical) predicate symbols. By an -structure, we mean a relational structure appropriate for . Let be the set of all -structures with universe {1, …, n}. For each first-order -sentence σ (with equality), let μ n (σ) be the fraction of members of for which σ is true. We show that μ n (σ) always converges to 0 or 1 as n → ∞, and that the rate of convergence is geometrically fast. In fact, if T is a certain complete, consistent set of first-order -sentences introduced by H. Gaifman [6], then we show that, for each first-order -sentence σ, μ n (σ) → n 1 iff T ⊩ ω. A surprising corollary is that each finite subset of T has a finite model. Following H. Scholz [8], we define the spectrum of a sentence σ to be the set of cardinalities of finite models of σ. Another corollary is that for each first-order -sentence a, either σ or ˜σ has a cofinite spectrum (in fact, either σ or ˜σ is “nearly always“ true).

Let be a subset of which contains for each in exactly one structure isomorphic to . For each first-order -sentence σ, let ν n (σ) be the fraction of members of which a is true. By making use of an asymptotic estimate [3] of the cardinality of and by our previously mentioned results, we show that v n(σ) converges as n → ∞, and that lim n ν n (σ) = lim n μ n (σ).If contains at least one predicate symbol which is not unary, then the rate of convergence is geometrically fast.

We suggest the name “approximation theorems” for a new kind of theorems which are strong versions of preservation theorems. A typical preservation theorem has the form:

A sentence φ is preserved by the relation R (i.e. and imply ) iff there exists φ* ϵ Φ such that ⊧φ↔ φ*

where R is a relation between structures and Φ is a class of sentences (depending, of course, on R). Usually, Φ is described in syntactical terms and it is easy to see that every element of it is, indeed, preserved by R. A typical approximation theorem has the form:

For every sentence Φ there is a sentence Φ* ϵ Φ (the “approximation” of Φ) such that, for all sentences δ which are preserved under R,

(a) if ⊧δ → φ then ⊧δ → φ* and

(b) if ⊧φ → δ then ⊧φ* → δ.

An approximation theorem obviously implies the corresponding preservation theorem.

The first approximation theorem was proved by Vaught in [14] (see Corollary 2.3 below). That paper inspired the present one. Vaught's result is stated in topological terms. It says that for each Borel set B there is an invariant Borel set B* explicitly defined by an L ω1ω sentence, such that B − ⊆ B* ⊆ B + where B − (B +) is the largest (smallest) invariant set included in (containing) B.

As has been shown by the investigations of some mathematicians, there are numerous analogies between forcing and omitting types. Evidence of that can also be found in the application of forcing to model theory. Both methods use the Rasiowa-Sikorski lemma on the existence in a Boolean algebra of an ultrafilter intersecting every dense set from a given denumerable family.

In this paper we will not use the terms Cohen forcing and omitting types in their original sense. We shall deal mainly with the Scott-Boolean kind of forcing as a transformation of Cohen's idea and with methods used in logic that consist in finding a suitable ultrafilter in the Lindenbaum-Tarski algebra for a given theory and defining canonical models—under the name of omitting types.

The two methods will be confronted to show that Cohen's forcing stems from logical methods. Attention will also be drawn to some differences between forcing in set theory and the general methods of logic.

In logic we construct a model by defining relations in a set of constants. In particular when defining a model for set theory we define a certain relation E in a set of constants. It is usually immaterial what E is, in particular whether it is the true relation of membership up to isomorphism.

Validity in recursive structures was investigated by several authors. Kreisel [10] has shown that there exists a consistent sentence of classical predicate calculus (CPC) that does not possess a recursive model. The sentence is a conjunction of the axioms of a variant of Bernays set theory, including the axiom of infinity. The language contains additional constants besides ϵ. Later Kreisel [2] and Mostowski [3] presented a sentence (not possessing recursive models) which was a conjunction of axioms of a variant of Bernays set theory without the axiom of infinity but still with additional constants besides ϵ. Later Mostowski [4] improved the result by giving a sentence which can be demonstrated in Heyting arithmetic to be consistent and to have no recursive models. Rabin [6] obtained a simple proof that some sentence of set theory with the single nonlogical constant ϵ does not have any recursively enumerable models.

More generally, Mostowski [5] has shown that the set of all sentences valid in all RE models is not arithmetical and Vaught [1] improved this result by showing that it holds for a language of one binary relation. In fact, Vaught gives· a way of translating n-place relations to 2-place ones that preserves the RE characteristic of the model. For further results pertaining to recursive models see Vaught [1].

Let R be a commutative ring with 1 and R[X 1, …, Xn ] the polynomial ring in n variables over R. Then for any relation f(X) = 0 in R[X] there exists a conjunction of equations φ f such that f(X) = 0 holds in R[X] iff φ f holds in R; φ f is of course the formula saying that all the coefficients of f(X) vanish. Moreover, φ f is independent of R and formed uniformly for all polynomials f up to a given formal degree. In this paper we investigate first order theories T for which a similar phenomenon holds. More precisely, we let TAH be the universal Horn part of a theory T and look at free extensions of models of T in the class of models of TAH . We ask whether an atomic relation t 1(X, a) = t 2(X, a) or R(t 1(X, a), …, tn (X, a)) in can be equivalently expressed by a finite or infinitary formula φ(a) in , such that φ(y) depends only on t i{X, y) and not on or a 1, …, am ∈ A.

We will show that for a wide class of theories T “defining formulas” φ(y) in this sense exist and can be taken as infinite disjunctions of positive existential formulas.

Introduction. The purpose of this paper is to show how results from the theory of inductive definitions can be used to obtain new compactness theorems for uncountable admissible languages. These will include improvements of the compactness theorem by J. Green [9].

In [2] Barwise studies admissible sets satisfying the Σ1-compactness theorem. Our approach is to consider admissible sets satisfying what could be called the abstract extended completeness theorem, that is, sets where the consequence relation of the admissible fragment LA is Σ1-definable over A. We will call such sets Σ1-complete. For countable admissible sets, Σ1-completeness follows from the completeness theorem for LA .

Having restricted our attention to Σ1-complete sets we are led to a stronger notion also true on countable admissible sets, namely what we shall call uniform Σ1-completeness. We will see that this notion can be viewed as extending to uncountable admissible sets, properties related to both the “recursion theory” and “proof theory” of countable admissible sets.

By following Barwise's recent approach to admissible sets allowing “urelements,” we show that there is a natural connection between certain structures arising from the theory of inductive definability, and uniformly Σ1-complete admissible sets . The structures we have in mind are called uniform Kleene structures.

This paper is devoted to the study of the infinitistic rules of proof i.e. those which admit an infinite number of premises. The best known of these rules is the ω-rule. Some properties of the ω-rule and its connection with the ω-models on the basis of the ω-completeness theorem gave impulse to the development of the theory of models for admissible fragments of the language . On the other hand the study of representability in second order arithmetic with the ω-rule added revealed for the first time an analogy between the notions of re-cursivity and hyperarithmeticity which had an important influence on the further development of generalized recursion theory.

The consideration of the subject of infinitistic rules in complete generality seems to be reasonable for several reasons. It is not completely clear which properties of the ω-rule were essential for the development of the above-mentioned topics. It is also worthwhile to examine the proof power of infinitistic rules of proof and what distinguishes them from finitistic rules of proof.

What seemed to us the appropriate point of view on this problem was the examination of the connection between the semantics and the syntax of the first order language equipped with an additional rule of proof.

We prove the following extension of a result of Keisler and Morley. Suppose is a countable model of ZFC and c is an uncountable regular cardinal in . Then there exists an elementary extension of which fixes all ordinals below c, enlarges c, and either (i) contains or (ii) does not contain a least new ordinal.

Related results are discussed.

A number is a nonnegative integer, and E is the set of numbers. In [3], J. C. E. Dekker introduced the concept of a regressive set of order n as the range of a one-one function f of n arguments such that (i) domain f ⊆ En and, if (x 1, …, xn ) ∈ domain f, and yi ≤ ≤ xi for 1 ≤ i ≤ n, then (y 1, …, yn ) ∈ domain f, and (ii) if 1 ≤ i ≤ n and (x 1, …, xn ) ∈ domain f, then f(x 1 … x i−1, xi ∸ 1, x i+1 … xn ) can be found effectively from f(x 1 … xn ). (0 ∸ 1 = 0 and, for m ≥ 1, m ∸ 1 = m − 1.) Since one can take the view, as Dekker did when first introducing regressive functions in [1], that a regressive set of order one is the range of a function of the above type which is of order one and everywhere defined, it seems natural to study the n-dimensional analogue in which (i) is replaced by “domain f = En .” It is the purpose of this paper to study such sets.

In [1] Gandy established the following selection theorem for recursion in type-2 objects.

Theorem. Let F be a normal type-2 object. Then it is possible to select (uniformly and effectively in F) an integer from each nonempty set of integers semirecursive in F .

Notice that this really asserts that the predicates semirecursive in F are closed under existential quantification over type-0. Moschovakis [6] has essentially proven this theorem for F of arbitrary type.

In [2] Grilliot stated a powerful generalization of Gandy's result, namely:

Grilliot's Selection Theorem. Let F be a normal type-(n + 2) object (n an arbitrary integer). Then it is possible to select (uniformly and effectively in F) a nonempty recursive in F subset of each nonempty semirecursive in F set oftype-(n − 1) objects.

Notice again that this actually says that predicates semirecursive in F are closed under quantification over type-(n − 1) objects.

Despite the similarity of these two results, Gandy and Grilliot proposed rather different methods of proof. Furthermore, the proof that Grilliot presented in [2] contains an error which cannot easily be corrected. (We will comment on the nature of this error at the end of §1.) Fortunately, however, Grilliot's theorem is valid. We will present a proof of Grilliot's selection theorem which is a direct generalization of the proof of Gandy's theorem given in [6]. In fact, we will prove a general result (the theorem stated in §2) which subsumes both Gandy's and Grilliot's results.

The problem of treating the semantics of intuitionistic logic within the framework of intuitionistic mathematics was first attacked by E. W. Beth [1]. However, the completeness theorem he thought to have obtained, was not true, as was shown in detail in a report by V. H. Dyson and G. Kreisel [2]. Some vague remarks of Beth's, for instance in his book, The foundations of mathematics , show that he sustained the hope of restoring his proof. But arguments by K. Gödel and G. Kreisel gave people the feeling that an intuitionistic completeness theorem would be impossible [3]. (A (strong) completeness theorem would imply

for any primitive recursive predicate A of natural numbers, and one has no reason to believe this for the usual intuitionistic interpretation.) Nevertheless, the following contains a correct intuitionistic completeness theorem for intuitionistic predicate logic. So the old arguments by Godel and Kreisel should not work for the proposed semantical construction of intuitionistic logic. They do not, indeed. The reason is, loosely speaking, that negation is treated positively.

Although Beth's semantical construction for intuitionistic logic was not satisfying from an intuitionistic point of view, it proved to be useful for the development of classical semantics for intuitionistic logic. A related and essentially equivalent classical semantics for intuitionistic logic was found by S. Kripke [4].

The constructible universe L of Gödel [2] has a natural well-ordering < L given by the order of construction; a closer look reveals that this well-ordering is definable by a Σ1 formula. Cohen's method of forcing provides several examples of models of ZF + V ≠ L which have a definable well-ordering but none is definable by a relatively simple formula.

Recently, Mansfield [7] has shown that if a set of reals (or hereditarily countable sets) has a Σ1, well-ordering then each of its elements is constructible. A question has thus arisen whether one can find a model of ZF + V ≠ L that has a Σ1 well-ordering of the universe. We answer this question in the affirmative.

The main result of this paper is

Theorem. There is a model of ZF + V ≠ L which has a Σ1 well-ordering.

The model is a generic extension of L by adjoining a branch through a Suslin tree with certain properties. The branch is a nonconstructible subset of ℵ1. Note that by Mansfield's theorem, the model must not have nonconstructible subsets of ω.

Our results can be generalized in several directions. We note that in particular, we can get a model with a Σ1 well-ordering that is not L[X] for any set X. As one might expect from a joint paper by a recursion theorist and a set theorist, the proof consists of a construction and a computation.

We use a fundamental theorem of Vaught, called the covering theorem in [V] (cf. theorem 0.1 below) as well as a generalization of it (cf. Theorem 0.1* below) to derive several known and a few new results related to the logic . Among others, we prove that if every countable model in a PCφ1ω class has only countably many automorphisms, then the class has either ≤ℵ0 or exactly nonisomorphic countable members (cf. Theorem 4.3*) and that the class of countable saturated structures of a sufficiently large countable similarity type is not among countable structures (cf. Theorem 5.2). We also give a simple proof of the Lachlan-Sacks theorem on bounds of Morley ranks (§7).

Monotone inductive definitions occur frequently throughout mathematical logic. The set of formulas in a given language and the set of consequences of a given axiom system are examples of (monotone) inductively defined sets. The class of Borel subsets of the continuum can be given by a monotone inductive definition. Kleene's inductive definition of recursion in a higher type functional (see [6]) is fundamental to modern recursion theory; we make use of it in §2.

Inductive definitions over the natural numbers have been studied extensively, beginning with Spector [11]. We list some of the results of that study in §1 for comparison with our new results on inductive definitions over the continuum. Note that for our purposes the continuum is identified with the Baire space ωω.

It is possible to obtain simple inductive definitions over the continuum by introducing real parameters into inductive definitions over N—as in the definition of recursion in [5]. This is itself an interesting concept and is discussed further in [4]. These parametric inductive definitions, however, are in general weaker than the unrestricted set of inductive definitions, as is indicated below.

In this paper we outline, for several classes of monotone inductive definitions over the continuum, solutions to the following characterization problems:

(1) What is the class of sets which may be given by such inductive definitions ?

(2) What is the class of ordinals which are the lengths of such inductive definitions ?

These questions are made more precise below. Most of the results of this paper were announced in [2].