Nach allgemeiner Ueberzeugung können gewisse Sätze der Analysis nicht konstruktiv bewiesen werden. Man denke etwa an den folgenden Satz: Eine monotone und beschränkte Folge a(n) von rationalen Zahlen konvergiert, d.h. es gibt eine solche ganzzahlige Funktion k(m), daß

Unter einem konstruktiven Beweis dieses Satzes hätte man etwa die Angabe eines Verfahrens zu verstehen, das gestattet, die Funktion k(m) auf Grand der Folge a(n) rekursiv zu definieren. Wenn ein solcher Beweis vorläge, wäre insbesondere gezeigt, daß es zu einer rekursiv definierten Folge a(n) mit den verlangten Eigenschaften stets eine rekursiv definierbare Konvergenzfunktion k(m) gibt.

Es soil nun im folgenden von einigen grundlegenden Sätzen der reellen Analysis gezeigt werden, daß sie nicht konstruktiv bewiesen werden können—konstruktiv in dem Sinn, der eben angedeutet wurde. Wir stützen uns dabei auf die Begriffe der rekursiven und der berechenbaren Funktion; rekursive und berechenbare Funktionen sind zahlentheoretische Funktionen. Unter einer rekursiven Funktion möge vorläufig eine primitiv rekursive Funktion verstanden werden (vgl. etwa D. Hilbert und P. Bernays [3], S.286); berechenbar meint berechenbar im Sinne von A. Church [1], S. C. Kleene [4], A. M. Turing [6]. Der Begriff der “konstruktiv definierten Funktion” wird durch die Bestimmung als “rekursive Funktion” eher eng, durch die Bestimmung als “berechenbare Funktion” eher weit gefaßt.

Definition I. Eine Folge rationaler Zahlen Φ(n) heißt rekursiv,(berechenbar), wenn es solche rekursiven (berechenbaren) Funktionen ϕ{n), ψ(n) und χ(n) gibt, daß χ(n) ≧1 und

.

Definition II. Eine Folge Φ(n) von rationalen Zahlen heißt rekursiv (berechenbar) konvergent, wenn es eine solche rekursive (berechenbare) Funktion v(m) gibt, daß

Wir werden zeigen (Satz I): Es gibt eine rekursive, monotone und beschränkte Folge rationaler Zahlen, die nicht berechenbar konvergiert. Bei unserer Auffassungsweise ist darin enthalten, daß der zu Beginn erwähnte Satz nicht konstruktiv bewiesen werden kann.

Although several proofs have been published showing the completeness of the propositional calculus (cf. Quine (1)), for the first-order functional calculus only the original completeness proof of Gödel (2) and a variant due to Hilbert and Bernays have appeared. Aside from novelty and the fact that it requires less formal development of the system from the axioms, the new method of proof which is the subject of this paper possesses two advantages. In the first place an important property of formal systems which is associated with completeness can now be generalized to systems containing a non-denumerable infinity of primitive symbols. While this is not of especial interest when formal systems are considered as logics—i.e., as means for analyzing the structure of languages—it leads to interesting applications in the field of abstract algebra. In the second place the proof suggests a new approach to the problem of completeness for functional calculi of higher order. Both of these matters will be taken up in future papers.

The system with which we shall deal here will contain as primitive symbols

and certain sets of symbols as follows:

(i) propositional symbols (some of which may be classed as variables, others as constants), and among which the symbol “f” above is to be included as a constant;

(ii) for each number n = 1, 2, … a set of functional symbols of degree n (which again may be separated into variables and constants); and

(iii) individual symbols among which variables must be distinguished from constants. The set of variables must be infinite.

We make certain notational conventions. These are referred to as CL (Computational Logic). The relation of interchangeability is introduced as the basic connection between logical formulas. This approach lends perspicuity to the results of sentential calculus. With its technical devices CL is able to rephrase logical theorems in rather succinct manner. Our exposition tries to steer a middle course between informality and strict rigor.

It is felt that the method of CL offers advantages for the teaching of logic. Proofs are algorithmic and resemble those of elementary algebra. All inferences are reversible and practically non-tentative.

Sign is a primitive term of CL. Its denning property is a capacity for entering into binary combination with other signs, or with itself, according to this convention:

x and y are signs if, and only if, (x y) is a sign. Bracketing is looked upon as an operation on signs in terms of which other operations are definable. The practice of some authors in classifying brackets under the heading of symbols, seems to us questionable; for unlike symbols or signs brackets are never used to denote anything. Brackets enter into the composition of signs, not to denote a grouping, but rather to exhibit it, in the manner of a diagram.

An unending list of letters, with or without subscripts,

serve to denote arbitrary signs. An arbitrary sign may or may not have other signs as parts. The numeral “2” is used as a constant. Bracketing abbreviation is as follows:

and so on. Outermost brackets will ordinarily be omitted. This kind of bracketing may be termed left-associative. For convenience, these bracketing conventions are crystallized into a rule.

This note should be considered as an appendix to a paper by McKinsey. Familiarity with this paper is assumed; its terminology and notation will be used without explanation. McKinsey's main result (theorem 1) is that every class of sentences fulfilling P1–P5, with a set of substitutions satisfying A1–A4, is a system of modal logic in the sense that the “associated” set, T2, contains all theorems of S4. It is also proved (theorem 2) that for T2 to contain all theorems of S5, it is sufficient that the elements of S form a group. The basic idea of the construction is the formalization, by means of D5, of our intuitive notion of possibility.

In view of the great generality of S, it may be of some interest to give a condition that is both necessary and sufficient for T2 to contain S5 after some restriction suggested by our intuitive notions about modalities has first been imposed upon S. The following postulate is related to the idea that if a sentence is possible, then its negation is not necessary. Informally speaking, it requires that S be sufficiently “comprehensive” for this idea to become formalizable in terms of S.

P. If ♢α is in T1, then there exists an element sm of S such that ♢∼sm(α) is not in T1.

I shall now prove that if P is satisfied, the following condition is both necessary and sufficient for T2 to contain all theorems of S5.

C. If α is in T1, then ♢s(α) is in T1 for all elements s of S.

The concept of a recursively definite predicate of natural numbers was introduced by F. B. Fitch in his An extension of basic logic as follows:

Every recursive predicate is recursively definite. If R(x1, …, xn) is recursively definite so is (Ey)R(x1, …, xn−1, y) and (y)R(x1, …, xn−1, y). If R is recursively definite and S is the proper ancestral of R, then S is recursively definite, where the proper ancestral of a relation is defined as follows: if R is of even degree, say 2m, then the proper ancestral of R is the relation S such that for all x1, …, xm, y1, …, ym, S(x1, …, xm, y1, …, ym) is true if and only if there is a finite sequence of sequences (z11, …, zm1), (z12, …, Zm2), …, (z1k, …, Zmk) such that R(Z11, …, Zm1, z12, …, zm2), R(z12, …, zm2, z13, …, zm3), …, R(z1,k−1, z1k, …, Zmk) are all true, where (z11, …, zm1) is (x1, …, xm) and (z1k, …, zmk) is (y1, …, ym).

An arithmetic predicate is one which is definable in terms of the operations ‘+’ and ‘·’ of elementary arithmetic, the connectives of the classical prepositional calculus, and quantifiers.

An (m + n)-valued propositional calculus2 may happen to be a subsystem of an m-valued propositional calculus , though the converse is never true. This fact may give us the impression that, as m grows, the content of becomes meagre. The present treatment is intended to remove this impression by constructing a complete, m-valued sub-system of any (m + n)-valued propositional calculus.

In the following we adopt the customary, autonymous mode of speech according to which symbols belonging to the object calculi or languages are used in the syntactic language as names for themselves, and juxtaposition serves to denote juxtaposition.

2.1 = df stands for definational identity in the syntactic language.

2.11 ≡ stands for definational identity in the object calculi.

2.2 ∊, ⊂, ∩, {x1, …, xn} are used in their meanings as customarily employed in the theory of sets—∊ for class membership, ⊂ for proper inclusion, ∩ for the product operation of classes, {x1, …, xn} for the class with x1, …, xn as its only elements.

2.3 x, y, z are used as unspecified natural numbers including 0. m, n, i, j are used as unspecified natural numbers other than 0.

2.401 Definition. δ = df as the function of two variables defined for any x, y such that

2.41 Definition. For m ≧ 2, ιm = df the function of two variables denned on the set {0, …, m − 1} such that ιm (x, y) = y − x for x ≦ y and ιn(x, y) = 0 for x > y.