In 1922 Skolem delivered an address before the Fifth Congress of Scandinavian Mathematicians in which he pointed out what he called a “relativity of set-theoretic notions”. This “relativity” has frequently been regarded as paradoxical; but today, although one hears the expression “the Löwenheim-Skolem Paradox”, it seems to be thought of as only an apparent paradox, something the cognoscenti enjoy but are not seriously troubled by. Thus van Heijenoort writes, “The existence of such a ‘relativity’ is sometimes referred to as the Löwenheim-Skolem Paradox. But, of course, it is not a paradox in the sense of an antinomy; it is a novel and unexpected feature of formal systems.” In this address I want to take up Skolem's arguments, not with the aim of refuting them but with the aim of extending them in somewhat the direction he seemed to be indicating. It is not my claim that the “Löwenheim-Skolem Paradox” is an antinomy in formal logic ; but I shall argue that it is an antinomy, or something close to it, in philosophy of language . Moreover, I shall argue that the resolution of the antinomy—the only resolution that I myself can see as making sense—has profound implications for the great metaphysical dispute about realism which has always been the central dispute in the philosophy of language. The structure of my argument will be as follows: I shall point out that in many different areas there are three main positions on reference and truth: there is the extreme Platonist position, which posits nonnatural mental powers of directly “grasping” forms (it is characteristic of this position that “understanding” or “grasping” is itself an irreducible and unexplicated notion); there is the verificationist position which replaces the classical notion of truth with the notion of verification or proof, at least when it comes to describing how the language is understood; and there is the moderate realist position which seeks to preserve the centrality of the classical notions of truth and reference without postulating nonnatural mental powers.

The purpose of this paper is to show that arithmetically minimal systems of notations can be constructed which provide notations for all ramified analytical ordinals (all the ordinals in the minimum β-model for analysis). This is a much larger section of the second number class than the Church-Kleene constructive ordinals (although still only an initial segment of the ordinals). Arithmetic minimality means that if H is an “H-set” associated with an ordinal α in our system and H′ is an H-set associated with the same ordinal α in an arbitrary system of notations S, then H is arithmetical in H′. Thus the arithmetical degrees associated with ordinals in our system are as low as possible. In order to clarify the structure of degrees of unsolvability and, more generally, to gain a deeper insight into the power set of the integers, coarser but neater classifications than the structure of Turing degrees have been sought. Several hierarchies of sets of integers have been studied, each of which organizes a certain class of sets (or their degrees of unsolvability) into a well-ordering of levels with increasing complexity of nonrecursiveness appearing at each new level. The best known of these hierarchies is the Kleene hierarchy of arithmetical sets.

The hyperarithmetical hierarchy assigns a degree of unsolvability hγ to each constructive ordinal γ. This assignment has the properties that h0 is the recursive degree and hγ+1 is the jump h′γ of hγ. For a limit ordinal λ < ω1 it is not so easy to define hγ. The original definitions used systems of notations for ordinals, see Spector [6]. There are also later notation-free definitions due to Shoenfield (unpublished) and to Hensel and Putnam [4].

Why the Post-Kleene arithmetical hierarchy of degrees of (recursive) unsolvability was extended into the transfinite is not clear. Perhaps it was thought that if a hierarchy of sufficiently fine structure could be described that would include all sets of integers, some light might be thrown on the Continuum Hypothesis, and its truth or falsity possibly even ascertained. There is also some evidence in the 1955 papers of Kleene (cf. Kleene [2], [3], [4]) that it was once hoped that a theorem for the analytical hierarchy analogous to the result of Post and Kleene

From the work of Kleene, Post and Davis it is well-known that the arithmetic sets can be characterized as those sets recursive in∅(n)for some natural numbern, where∅(0)=∅and . Actually the arithmetic sets which can be expressed in prenex form withnalternating quantifiers (applied to recursive predicates) are recursive in∅(n). Hence, starting with∅, the “jump operation”, which takes a set A into the set , serves to increase the “complexity” of sets in a uniform way. As far as extending the ω-sequence of degrees of unsolvability (i.e. the degrees represented by the∅(n)) into the 2nd number class, there is an immediate problem. One knows from Spector [17], corollary 2, p. 585, that there is no l.u.b. for the ω-sequence. So, at ω one must pick a degree in some other “natural” way. Unfortunately, what has seemed “natural” to some mathematicians has not seemed natural to others. Kleene and Davis (cf. [10] and [3] respectively) extended this arithmetic hierarchy of degrees of unsolvability by making use of natural number notations for a certain segment of ordinals of the 1st and 2nd number classes, the “constructive” ordinals. Using the Church-Kleene system S3, one can “sum up” previously obtained sets at the limit notations in a way that is certainly natural from a notational point of view.

The purpose of this paper is to present two groups of results which have turned out to have a surprisingly close interconnection. The first two results (Theorems 1 and 2) were inspired by the following question: we know what sets are “decidable” — namely, the recursive sets (according to Church's Thesis). But what happens if we modify the notion of a decision procedure by (1) allowing the procedure to “change its mind” any finite number of times (in terms of Turing Machines: we visualize the machine as being given an integer (or an n-tuple of integers) as input. The machine then “prints out” a finite sequence of “yesses” and “nos”. The last “yes” or “no” is always to be the correct answer.); and (2) we give up the requirement that it be possible to tell (effectively) if the computation has terminated? I.e., if the machine has most recently printed “yes”, then we know that the integer put in as input must be in the set unless the machine is going to change its mind; but we have no procedure for telling whether the machine will change its mind or not. The sets for which there exist decision procedures in this widened sense are decidable by “empirical” means — for, if we always “posit” that the most recently generated answer is correct, we will make a finite number of mistakes, but we will eventually get the correct answer. (Note, however, that even if we have gotten to the correct answer (the end of the finite sequence) we are never sure that we have the correct answer.)

The hope that mathematical methods employed in the investigation of formal logic would lead to purely computational methods for obtaining mathematical theorems goes back to Leibniz and has been revived by Peano around the turn of the century and by Hilbert's school in the 1920's. Hilbert, noting that all of classical mathematics could be formalized within quantification theory, declared that the problem of finding an algorithm for determining whether or not a given formula of quantification theory is valid was the central problem of mathematical logic. And indeed, at one time it seemed as if investigations of this “decision” problem were on the verge of success. However, it was shown by Church and by Turing that such an algorithm can not exist. This result led to considerable pessimism regarding the possibility of using modern digital computers in deciding significant mathematical questions. However, recently there has been a revival of interest in the whole question. Specifically, it has been realized that while no decision procedure exists for quantification theory there are many proof procedures available—that is, uniform procedures which will ultimately locate a proof for any formula of quantification theory which is valid but which will usually involve seeking “forever” in the case of a formula which is not valid—and that some of these proof procedures could well turn out to be feasible for use with modern computing machinery. Hao Wang [9] and P. C. Gilmore [3] have each produced working programs which employ proof procedures in quantification theory. Gilmore's program employs a form of a basic theorem of mathematical logic due to Herbrand, and Wang's makes use of a formulation of quantification theory related to those studied by Gentzen. However, both programs encounter decisive difficulties with any but the simplest formulas of quantification theory, in connection with methods of doing propositional calculus. Wang's program, because of its use of Gentzen-like methods, involves exponentiation on the total number of truth-functional connectives, whereas Gilmore's program, using normal forms, involves exponentiation on the number of clauses present. Both methods are superior in many cases to truth table methods which involve exponentiation on the total number of variables present, and represent important initial contributions, but both run into difficulty with some fairly simple examples. In the present paper, a uniform proof procedure for quantification theory is given which is feasible for use with some rather complicated formulas and which does not ordinarily lead to exponentiation. The superiority of the present procedure over those previously available is indicated in part by the fact that a formula on which Gilmore's routine for the IBM 704 causes the machine to computer for 21 minutes without obtaining a result was worked successfully by hand computation using the present method in 30 minutes. Cf. §6, below. It should be mentioned that, before it can be hoped to employ proof procedures for quantification theory in obtaining proofs of theorems belonging to “genuine” mathematics, finite axiomatizations, which are “short,” must be obtained for various branches of mathematics. This last question will not be pursued further here; cf., however, Davis and Putnam [2], where one solution to this problem is given for ele

This paper presents several results whose common element is their connection with Hubert's tenth problem.1 The one that seems most striking (perhaps because it can be stated with a minimum of technical apparatus) is this: there does not exist an algorithm for determining whether or not a polynomial (in n variables) represents every integer.2 In addition, the present paper (i) gives a simple characterization of the diophantine sets (of positive, non-negative, and arbitrary integers), and (ii) gives a rigorous proof of Myhill's theorem [6] that there is no decision method for statements of the form

Hilbert's tenth problem is to find an algorithm for determining whether or not a diophantine equation possesses solutions. A diophantine predicate (of positive integers) is defined to be one of the form where P is a polynomial with integral coefficients (positive, negative, or zero). Previous work has considered the variables as ranging over nonnegative integers; but we shall find it more useful here to restrict the range to positive integers, no essential change being thereby introduced. It is clear that the recursive unsolvability of Hilbert's tenth problem would follow if one could show that some non-recursive predicate were diophantine. In particular, it would suffice to show that every recursively enumerable predicate is diophantine. Actually, it would suffice to prove far less.

There are a number of open problems involving the concepts of decidability and essential undecidability. This paper will present solutions to some of these problems. Specifically: (1) Can a decidable theory have an essentially undecidable, axiomatizable extension (with the same constants)? (2) Are all the complete extensions of an undecidable theory ever decidable? We shall show that the answer to both questions is in the affirmative. In answering question (1), the decidable theory for which an essentially undecidable axiomatizable extension will be constructed is the theory of the successor function and a single one-place predicate. It will also be shown that the decidability of this theory is a “best possible” result in the following direction: the theory of either of the common diadic arithmetic functions and a one-place predicate; i.e., of addition and a one-place predicate, or of multiplication and a one-place predicate, is undecidable. Before establishing the main result, it is convenient to give a simple proof that a decidable theory can have an axiomatizable (simply) undecidable extension. This is, of course, an immediate consequence of the main result; but the proof is simple and illustrates the methods that we are going to use in this paper.