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.

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.

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].

An r-normal function is a strictly increasing continuous function from r to r where r is a regular ordinal > ω (identify an ordinal with the set of smaller ordinals). Given an r-normal function f one can form a sequence {f(x, −)} x<r of r-normal functions—the Veblen hierarchy [33] on f—as follows: f(0, −) = f and, for x > 0, f(x, −) enumerates in order {z ∣ f(y, z) = z for all y < x}, the common fixed points of the f(y, −)'s for y < x. In this paper we give as readable an exposition as we can of Veblen hierarchies and of Bachmann's and Isles's techniques in [3] and [15] of using higher finite number classes for forming sequences {f(x, −)} x<y where y > r of r-normal functions which extend the Veblen hierarchy on f. We will show how these sequences—Bachmann hierarchies—yield extremely natural constructive notations for ordinals in various initial segments of the second number class. We will also consider various other techniques for obtaining constructive ordinal notations and relate them to the notations obtained by Bachmann's and Isles's techniques. In particular, we will use these notations to characterize as directly and as usefully as we can various of Takeuti's systems of constructive ordinal notations, which he calls ordinal diagrams ([31], [32]).

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.

The standard classes of a first-order theory T are certain classes of prenex T-sentences defined by restrictions on prefix, number of monadic, dyadic, etc. predicate variables, and number of monadic, dyadic, etc. operation variables. In [3] it is shown that, for any theory T, (1) the decision problem for any class of prenex T-sentences specified by such restrictions reduces to that for the standard classes, and (2) there are finitely many standard classes K 1, …, Kn such that any undecidable standard class contains one of K 1, …, Kn . These results give direction to the study of the decision problem.

Below T is predicate logic with identity and operation variables. The Main Theorem solves the decision problem for the standard classes admitting at least one operation variable.

The set theory AFC was introduced by Perlis in [2] and he noted that it both includes and is stronger than Ackermann's set theory. We shall give a relative consistency result for AFC.

AFC is obtained from Ackermann's set theory (see [2]) by replacing Ackermann's set existence schema with the schema

(where ϕ, ψ, are ∈-formulae, x is not in ψ, w is not in ϕ, y is y 1, …, yn, z is z 1, …, zm and all free variables are shown) and adding the axiom of choice for sets. Following [1], we say that λ is invisible in Rκ if λ < κ and we have

holding for every ∈-formula θ which has exactly two free variables and does not involve u or υ. The existence of a Ramsey cardinal implies the existence of cardinals λ and κ with λ invisible in Rκ, and Theorem 1.13 of [1] gives some further indications about the relative strength of the notion of invisibility.

Theorem. If there are cardinals λ and κ with λ invisible in Rκ, then AFC is consistent.

Proof. Suppose that λ is invisible in Rκ and we will show that 〈Rκ, Rλ, ∈〉 ⊧ AFC (Rλ being the interpretation of V, of course).

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).

In [4] Curry raised the possibility that his system proposed in ξ15C of [3] might be inconsistent. In this paper this inconsistency is proved using a method also employed in [1].

From Curry's axiom ⊦LH, it follows that

holds for arbitrary X.

The other results from that are required are

Modus Ponens, and the Deduction Theorem for implication:

Assuming ⊦HA, we define as in [1]:

and let

where Y is the paradoxical (or fixed point) combinator.

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].

We are concerned here with recursive function theory analogs of certain problems in chromatic graph theory. The motivating question for our work is: Does there exist a recursive (countably infinite) planar graph with no recursive 4-coloring? We obtain the following results: There is a 3-colorable, recursive planar graph which, for all k, has no recursive k-coloring; every decidable graph of genus p ≥ 0 has a recursive 2(x(p) − 1)-coloring, where x(p) is the least number of colors which will suffice to color any graph of genus p; for every k ≥ 3 there is a k-colorable, decidable graph with no recursive k-coloring, and if k = 3 or if k = 4 and the 4-color conjecture fails the graph is planar; there are degree preserving correspondences between k-colorings of graphs and paths through special types of trees which yield information about the degrees of unsolvability of k-colorings of graphs.

The theory of computational complexity deals with those functions which can be computed subject to certain restrictions on the resources (for instance, time or memory) available for computation. Blum [5] gave an axiomatic characterization of some of the properties which should be possessed by a measure of computational complexity and established the existence of speed-upable functions—computable functions which fail to possess optimal programs in a particularly strong sense. Recursion theorists tend to like such functions, and people concerned with the specifics of real computing tend to consider such functions somewhat pathological. In Theorem 2 we show that such pathology is rampant: there is a great diversity of behavior among the collections of “run-times” of different functions which do not possess optimal programs, where such diversity is gauged by certain algebraic criteria which have computational significance. Roughly speaking, these algebraic criteria concern the ways in which various functions can be intermixed to satisfy requirements that certain functions can or cannot be computed more easily than certain other functions. (More detailed motivation for the relevance of these algebraic notions is given later.) Specifically, in Theorem 2 we generalize the embeddability theorem of McCreight and Meyer [12] (discussed below) by making speed-upable functions responsible for the embedding.