In [5], we studied the relational systems /Ā obtained from the recursive functions of one variable by identifying two such functions if they are equal for all but finitely many х ∈ Ā, where Ā is an r-cohesive set. The relational systems /Ā with addition and multiplication defined pointwise on them, were once thought to be potential candidates for nonstandard models of arithmetic. This, however, turned out not to be the case, as was shown by Feferman, Scott, and Tennenbaum [1]. We showed, letting A and B be r-maximal sets, and letting denote the complement of X, that /Ā and are elementarily equivalent (/Ā ≡ ) if there are r-maximal supersets C and D of A and B respectively such that C and D have the same many-one degree (C =mD). In fact, if A and B are maximal sets, /Ā ≡ if, and only if, A =mB. We wish to study the relationship between the elementary equivalence of /Ā and , and the Turing degrees of A and B.

The material presented here belongs to the model theory of the Lκ, λ languages. Our results are either infinitary analogs of important theorems in finitary model theory, or else show that such analogs do not exist.

For example, it is well known that whenever i, and , have the same true Lω, ω sentences (i.e., are elementarily equivalent) for i = 1, 2, then the cardinal sums 1 + 2 and + have the same true Lω, ω sentences, and the direct products 1 · 2 and · have the same true Lω, ω sentences [3]. We show that this is true when ‘Lω, ω’ is replaced by ‘Lκ, λ’ if and only if κ is strongly inaccessible. For Lω1, ω this settles a question, posed by Lopez-Escobar [7].

In §3 we give a complete description of the expressive power of those sentences of Lκ, λ in which the identity symbol is the only relation symbol which occurs. This extends a result by Hanf [4].

The aim of this paper is to study tag systems as defined by Post [Post 1943, pp. 203–205 and Post, 1965, pp. 370–373]. The existence of a tag system with unsolvable halting problem was proved by Minsky by constructing a universal tag system [Minsky 1961, see also Cocke and Minsky 1964, Wang 1963, and Minsky 1967, pp. 267–273]. Hence the halting problem of a tag system can be of the complete degree 0′. We shall prove that the halting problem for a tag system can have an arbitrary (recursively enumerable) degree of undecidability (Corollary III).

A related problem arises when we ask if there exists a uniform procedure for determining, given a tag system, whether or not there is any word on which the tag system does not halt, an “immortal” word in the system. The alternative, of course, being that the system eventually halts on every (finite) word. It is shown here that this problem, the immortality problem for tag systems, is recursively unsolvable of degree 0″ (Corollary II).

A theory T is said to be categorical in power κ iff T has a model of power κ and any two models of power κ are isomorphic.

It was conjectured by Morley [4] that if T is a theory in a language with κ > ω symbols and T is categorical in power κ, then T has a model of power < κ. The aim of this paper is to prove the following theorem.

Theorem A. Let κ be a regular cardinal such that ω < κ < 2ω. Let T be a theory in a language with κ symbols such that T is categorical in power κ. Then:

(a) T has a model of power < κ.

(b) T is categorical in all powers μ ≥ κ.

The notion of a recursive function with type-2 arguments was introduced by Kleene [4]. Soon E, the type-2 object that introduces numerical quantification, began to play a central role in the development of the theory. First, Kleene [4, XLVIII] proved that the hyperarithmetical functions are exactly the functions recursive in E. Later, Gandy [1, p. 14] proved the existence of a selection operator recursive in E (that is, some ψ(х, ), partial recursive in E, having the property that, if {х}(y, ) converges for some y, then ψ(х, ) converges and is such a y).

S. A. Kripke has given [6] a very simple notion of model for intuitionistic predicate logic. Kripke's models consist of a quasi-ordering (C, ≤) and a function ψ which assigns to every c ∈ C a model of classical logic such that, if c ≤ c′, ψ(c′) is greater or equal to ψ(c). Grzegorczyk [3] described a class of models which is still simpler: he takes, for every ψ(c), the same universe. Grzegorczyk's semantics is not adequate for intuitionistic logic, since the formula

where х is not free in α. holds in his models but is not intuitionistically provable. It is a conjecture of D. Klemke that intuitionistic predicate calculus, strengthened by the axiom scheme (D), is correct and complete with respect to Grzegorczyk's semantics. This has been proved independently by D. Klemke [5] by a Henkinlike method and me; another proof has been given by D. Gabbay [1]. Our proof uses lattice-theoretical methods.

A second order formula is called Π1 if, in its prenex normal form, all second order quantifiers are universal. A sequent F1, … Fm → G1 …, Gn is called Π1 if a formula

is Π1

If we consider only Π1 sequents, then we can easily generalize the completeness theorem for the cut-free first order predicate calculus to a cut-free Π1 predicate calculus.

In this paper, we shall prove two interpolation theorems on the Π1 sequent, and show that Chang's theorem in [2] is a corollary of our theorem. This further supports our belief that any form of the interpolation theorem is a corollary of a cut-elimination theorem. We shall also show how to generalize our results for an infinitary language. Our method is proof-theoretic and an extension of a method introduced in Maehara [5]. The latter has been used frequently to prove the several forms of the interpolation theorem.

Let W0, W1 … be one of the usual enumerations of recursively enumerable (r.e.) subsets of the set N of nonnegative integers. (Background information will be given later.) Suggestions of Anil Nerode led to the following

Definitions. Let B be a subset of N and let ψ be a partial recursive function.

There seem to be at least two approaches to the problem of generating notations for ordinals: the papers of Bachmann [2], Gerber [5], Feferman [3], and Isles [6] provide examples of one and those of Ackermann [1], Takeuti [10], Kino [7], Schütte [9], and Pfeiffer [8] the other. In both of these the normal forms produced can be regarded as being finite trees “based on” some ordinal π.

The nonrecursiveness of an arithmetical set can be measured in at least two ways. One way is to consider the level of the set in the arithmetical hierarchy. Another is to consider the degree of unsolvability of the set; more precisely to look at the set of degrees recursive in the degree of the set. The Hierarchy Theorem [1] indicates a correspondence between these two ways of considering the nonrecursiveness of an arithmetical set. The results below indicate some of the contrasts between these two points of view. (It should perhaps be mentioned that, although the results are stated in terms of degrees and levels in the arithmetical hierarchy, they can also be stated, in view of the Hierarchy Theorem, strictly in terms of degrees.)

A significant portion of the study of large cardinals in set theory centers around the concept of “partition relation”. To best capture the basic idea here, we introduce the following notation: for x and y sets, κ an infinite cardinal, and γ an ordinal less than κ, we let [x]γ denote the collection of subsets of x of order-type γ and abbreviate with the partition relation for each function F frominto y there exists a subset C of κ of cardinality κ such that (such that for each α < γ) the range of F on [С]γ ([С]α) has cardinality 1. Now although each infinite cardinal κ satisfies the relation for each n and m in ω (F. P. Ramsey [8]), a connection with large cardinals arises when one asks, “For which uncountable κ do we have κ → (κ)2?” Indeed, any uncountable cardinal κ which satisfies κ → (κ)2 is strongly inaccessible and weakly compact (see [9]). As another example one can look at the improvements of Scott's original result to the effect that if there exists a measurable cardinal then there exists a nonconstructible set. Indeed, if κ is a measurable cardinal then κ → (κ)< ω, and as Solovay [11] has shown, if there exists a cardinal κ such that κ → (κ)< ω3 (κ → (ℵ1)< ω, even) then there exists a nonconstructible set of integers.

In 1936 Alonzo Church proposed the following thesis: Every effectively computable number-theoretic function is general recursive. The classical mathematician can easily give examples of nonrecursive functions, e.g. by diagonalizing a list of all general recursive functions. But since no such function has been found which is effectively computable, there is as yet no classical evidence against Church's Thesis.

The intuitionistic mathematician, following Brouwer, recognizes at least two notions of function: the free-choice sequence (or ordinary number-theoretic function, thought of as the ever-finite but ever-extendable sequence of its values) and the sharp arrow (or effectively definable function, all of whose values can be specified in advance).