In [3] Myhill has constructed a complete system K which allows in it the development of a large and important section of classical mathematics. Completeness is achieved essentially by sacrificing universal quantification and introducing instead the proper ancestral as a primitive idea.

In the following we are presenting a system K0 which will be shown to be equivalent to K (i.e. the primitive operators of both systems are mutually definable in terms of one another). K0 is also complete and covers the same ground as K. K0, however, differs from K by the introduction of the limited universal quantifier instead of the proper ancestral and of concatenation instead of the ordered-pair function as primitive operators. By a further reduction K0 will be shown to be equivalent to the system K1 not containing the abstraction-operator and the class-membership relation.

The purpose of this paper is to prove two theorems and a conjecture (Conjecture II) announced in section 15 an earlier paper of the author's (cited as “CT”), and to compare them briefly with related results of Specker. Familiarity with both papers is assumed; the terminology of the former is used throughout. On two points however clarification of the usage of CT is in order, and to this chore we must first proceed.

A half-section is the lower half of a Dedekind cut; if the cut is rational, the half section is to include the rational corresponding to the real defined by the cut. A whole-section is the relation which holds between any member of the lower and any member of the upper half of some Dedekind cut. If the cut is rational the corresponding rational is to be a member of both halves.

A real number α is said to be approximate in K to any required number of decimal places if it is possible to define the predicates ‘x < α’, ‘x ≤ α’, ‘x > α’, and ‘x ≥ α’ (x rational) in K. In view of section 7 of CT this will mean that every true inequation between α and a terminating decimal will be provable in K.

If we think of denotation in a broad sense, we may say not only that, e.g., ‘Marni’ denotes the actual dog Marni but also that ‘dog’ denotes severally the dogs Marni, Fido, etc. What are ordinarily called class names thus come to denote severally the members of the class but not the class itself. This kind of a multiple denotation relation is a very simple as well as a very powerful semantical relation. In fact it can be taken as the fundamental relation of formal (extensional) semantics.

Conceptually this denotation relation is akin to the relation of satisfaction studied by Tarski, although the two must be carefully distinguished. Satisfaction (in its simplest form) is a relation between objects and sentential functions of one variable. Denotation is taken as a relation holding, on the one hand, between what is ordinarily called a proper name and the object which it names, and, on the other, between what may be called a virtual class denotator and the objects which are members of the virtual class in question. (This relation will be characterized more precisely below.) Clearly the two relations, satisfaction in its simplest form and multiple denotation, are interdefinable in an appropriate formalism.

In two papers appearing in successive issues of this Journal, Quine and Goodman, impelled by a philosophical standpoint which forswears all traffic in abstract entities, have explored the possibility of reinterpreting the language of classical mathematics so as to give it meaning (consistent with the observed overt pattern of standard usage) in terms of the narrow domain of entities which they recognize. The present paper offers several comments on this program.

Let C be the closure of a recursively enumerable set B under some relation R. Suppose there is a primitive recursive relation Q, such that Q is a symmetric subrelation of R (i.e. if Q(m, n), then Q(n, m) and R(m, n)), and such that, for each m ϵ B, Q(m, n) for infinitely many n. Then there exists a primitive recursive set A, such that C is the closure under R of A. For proof, note that , where f is a primitive recursive function which enumerates B, has the required properties. For each m ϵ B, there is an n ϵ A, such that Q(m, n) and hence Q(n, m); therefore the closure of A under Q, and hence that under R, includes B. Conversely, since Q is a subrelation of R, A is included in C. Finally, that A is primitive recursive follows from [2] p. 180.

This observation can be applied to many formal systems S, by letting R correspond to the relation of deducibility in S, so that R(m, n) if and only if m is the Gödel number of a formula of S, or of a sequence of formulas, from which, together with axioms of S, a formula with the Gödel number n can be obtained by applications of rules of inference of S.

It is an immediate consequence of results of Church and Gödel that there exist arithmetical recursively unsolvable problems, that is, recursively unsolvable problems of the form [M](P = Q) where P and Q are polynomials and [M] is some finite sequence of existential and universal quantifiers. A question which is immediately raised by this result is whether there exist unsolvable problems of this form where [M] is some finite sequence of existential quantifiers only. As a matter of fact this question is easily seen to be closely related to the tenth problem in the famous list proposed by Hilbert in 1900.

In this paper, we prove the existence of recursively unsolvable problems of the form

where P and Q are polynomials with non-negative integral coefficients. As a matter of fact we show that every recursively enumerable predicate is of the form (1), and conversely that every predicate of the form (1) is recursively enumerable. While our result does not yield the recursive unsolvability of Hilbert's tenth problem, it is easily seen that any considerable improvement of our result would yield this unsolvability.

The author wishes to take this opportunity to express his gratitude to Professors Alonzo Church and E. L. Post with whom he has had the privilege of discussing some of the questions involved in this paper. He also wishes to thank his friends Melvin Hausner and Jacob Schwartz who have made valuable suggestions.

Im Folgenden wird eine Vereinfachung der Beweisanordnung für Henkin's Theorem [2] p. 162 angegeben (Abschnitt I). – Die Vereinfachung zeigt sich u.a. darin, daß für den abgeänderten Beweis eine Abschätzung der “projektiven Klasse” (im Sinne der Kleene-Mostowski'schen Theorie der projektiven Mengen von natürlichen Zahlen) für das zu konstruierende Modell gelingt (Abschnitte II–IV). Die Kenntnis der Henkin'schen Arbeit wird vorausgesetzt.

Um eine maximale widerspruchsfreie Menge Γω von geschlossenen Formeln, in der zu jeder Existenzaussage (∃x)A(x) eine Beispielaussage A(uij) vorhanden ist, zu erhalten, führt Henkin einen Turm von Kalkülen S0, S1, …, Si, … und dessen Vereinigung Sω ein. Abwechselnd wird dann die gegebene Menge von Formeln in Si zu einer maximalen widerspruchsfreien Menge Γi erweitert und in Si+1 durch Hinzunahme von Beispielaussagen für alle Existenzaussagen aus Γi ergänzt, für die das nicht schon in einem früheren Schritt geschehen ist. Γω ist dann die Vereinigung aller Γi.

Die Vereinfachung besteht nun darin, daß – nach Erweiterung von S0 zu einem Kalkül S* durch Hinzunahme einer einfachen Folge u1, u2, … von “neuen” Konstanten – in S* zu jeder Existenzaussage eine bedingte Beispielaussage gebildet wird. K sei die Klasse dieser Aussagen, Λ die gegebene widerspruchsfreie Menge; dann wird Λ* = K ◡ Λ in S* einmal zu einer maximalen widerspruchsfreien Menge ergänzt.

Unter Verwendung der Standard-Anordnung aller geschlossenen Formeln von S* erhalten wir für diejenigen von der Form (∃x)A(x) eine Aufzählung (∃x)Ai(x), dazu eine Indexfolge ji (i =1, 2, …), definiert durch:

It is known that we can introduce in number theory (for example, the system Z of Hilbert-Bernays) by induction schemata certain predicates of natural numbers which cannot be expressed explicitly within the framework of number theory. The question arises how we can define these predicates in some richer system, without employing induction schemata. In this paper a general notion of definability by induction (relative to number theory), which seems to apply to all the known predicates of this kind, is introduced; and it is proved that in a system L1 which forms an extension of number theory all predicates which are definable by induction (hereafter to be abbreviated d.i.) according to the definition are explicitly expressible.

In order to define such predicates and prove theorems answering to their induction schemata, we have to allow certain impredicative classes in L1. However, if we want merely to prove that for each constant number the special case of the induction schema for a predicate d.i. is provable, we do not have to assume the existence of impredicative classes. A certain weaker system L2, in which only predicative classes of natural numbers are allowed, is sufficient for the purpose. It is noted that a truth definition for number theory can be obtained in L2. Consistency proofs for number theory do not seem to be formalizable in L2, although they can, it is observed, be formalized in L1.

In general, given any ordinary formal system (say Zermelo set theory), it is possible to define by induction schemata, in the same manner as in number theory, certain predicates which are not explicitly definable in the system. Here again, by extending the system in an analogous fashion, these predicates become expressible in the resulting system. The crucial predicate instrumental to obtaining a truth definition for a given system is taken as an example.

The sign ‘⊃’ (or ‘→’ or ‘C’) functions in many logical systems in a way which precludes its interpretation as either strict or material implication. For example, in the systems of Heyting, Johansson, Fitch and Bernays (positive logic), the following are theorems:

Now if ‘⊃’ were interpreted as strict implication, ⊃2 would mean ‘if p is true, then p is strictly implied by every proposition’, i.e. ‘if p is true, it is necessarily true’, which is false for contingently true p. If on the other hand ‘⊃’ were interpreted as material implication, ⊃1 would reduce to ‘~p ∨ p’, i.e. to the law of excluded middle, which is conspicuously lacking in the systems mentioned. The reader is likely in practice to veer between these two interpretations. Thus in Fitch or Heyting on realizing that ‘~p⊃▪ p⊃q’ is a theorem, one thinks of it as meaning ‘a false proposition implies everything’ and regards the implication as material; but the presence of ‘p⊃p’ as a theorem, even for choices of p which do not satisfy excluded middle, inclines one again to the strict interpretation. This vacillation, while it need not lead to the commission of any formal fallacies, tends to hamstring one's intuition and thus waste time. The purpose of this paper is to suggest an interpretation of ‘⊃’ which will prevent such havering.

Let two formulae A and B be called interdeducible if A ⊢ B and B ⊢ A.

It has been shown that the conditioned disjunction function [X, Y, Z] with the same truth-table as (X & Y) ∨ (Z & ) together with the logical constants t and f, form a complete set of independent connectives for the 2-valued propositional calculus and that these connectives are self-dual. This has since been generalised to the theorem which states that the conditioned disjunction function [Y, X1, X2, …, Xm, Y] with the same truth-table as (X1 & J1(Y)) ∨ (X2 & J2(Y)) ∨ … ∨ (Xm & Jm(Y)) together with the logical constants 1, 2, …, m form a complete set of independent connectives for the m-valued propositional calculus and that these connectives are self-dual. It has been conjectured by Church that conditioned disjunction together with the universal and existential quantifiers form a complete set of independent connectives for the 2-valued erweiterter Aussagenkalkül. The object of the present paper is to prove a theorem for the m-valued erweiterter Aussagenkalkül which reduces, in the case m = 2, to the conjecture of Church. In the m-valued propositional calculus if the propositional variable X occurs as a free variable in the formula then (∃X) and (X) are read “there exists X such that ” and “for all X, ”, respectively. If for a given assignment of truth-values to the remaining free propositional variables occurring in , takes the truth-value f(x), where x is the truth-value of X, then (∃X) and (X) take the truth-values min (f(1), f(2), …, f(m)), max(f(1), f(2), …, f(m)), respectively. We shall prove:

Theorem. The conditioned disjunction function, together with the universal and existential quantifiers, form a complete set of independent connectives for the m-valued erweiterter Aussagenkalkül.