Alonzo Church

A Formulation of the Simple Theory of Types

The purpose of the present paper is to give a formulation of the simple theory of types which incorporates certain features of the calculus of λ-conversion. A complete incorporation of the calculus of λ-conversion into the theory of types is impossible if we require that λ x and juxtaposition shall retain their respective meanings as an abstraction operator and as denoting the application of function to argument. But the present partial incorporation has certain advantages from the point of view of type theory and is offered as being of interest on this basis (whatever may be thought of the finally satisfactory character of the theory of types as a foundation for logic and mathematics). For features of the formulation which are not immediately connected with the incorporation of λ-conversion, we are heavily indebted to Whitehead and Russell, Hilbert and Ackermann, Hilbert and Bernays, and to forerunners of these, as the reader familiar with the works in question will recognize. The class of type symbols is described by the rules that i and o are each type symbols and that if α and β are type symbols then ( αβ ) is a type symbol: it is the least class of symbols which contains the symbols i and o and is closed under the operation of forming the symbol ( αβ ) from the symbols α and β .

An Unsolvable Problem of Elementary Number Theory

Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academic journals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers, and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community take advantage of advances in technology. For more information regarding JSTOR, please contact support@jstor.org.

A Note on the Entscheidungsproblem

In a recent paper the author has proposed a definition of the commonly used term “effectively calculable” and has shown on the basis of this definition that the general case of the Entscheidungsproblem is unsolvable in any system of symbolic logic which is adequate to a certain portion of arithmetic and is ω-consistent. The purpose of the present note is to outline an extension of this result to the engere Funktionenkalkul of Hilbert and Ackermann. In the authors cited paper it is pointed out that there can be associated recursively with every well-formed formula a recursive enumeration of the formulas into which it is convertible. This means the existence of a recursively defined function a of two positive integers such that, if y is the Godel representation of a well-formed formula Y then a(x, y) is the Godel representation of the x th formula in the enumeration of the formulas into which Y is convertible. Consider the system L of symbolic logic which arises from the engere Funktionenkalkul by adding to it: as additional undefined symbols, a symbol 1 for the number 1 (regarded as an individual), a symbol = for the propositional function = (equality of individuals), a symbol s for the arithmetic function x +1, a symbol a for the arithmetic function a described in the preceding paragraph, and symbols b 1 , b 2 , …, b k for the auxiliary arithmetic functions which are employed in the recursive definition of a ; and as additional axioms, the recursion equations for the functions a , b 1 , b 2 , …, b k (expressed with free individual variables, the class of individuals being taken as identical with the class of positive integers), and two axioms of equality, x = x , and x = y →[F(x)→F(y)] .

On the concept of a random sequence

ly the Kollektiv may be represented by an infinite sequence of points of an appropriate space, the Merkmalraum. Or if the number of possible outcomes of a trial is finite (and it may well be argued that this is always the case for any actual physical observation), it is sufficient to employ an infinite sequence of natural numbers which are less than a fixed natural number. This infinite sequence—of points or of natural numbers—satisfies certain conditions which correspond to those appearing in the description of a Kollektiv as just given, and which we shall express by saying that it is a random sequence (regel-

The constructive second number class

The existence of at least a vague distinction between what I shall call the constructive and the non-constructive ordinals of the second number class, that is, between the ordinals which can in some sense be effectively built up to step by step from below and those for which this cannot be done (although there may be existence proofs), is, I believe, somewhat generally recognized. My purpose here is to propose an exact definition of this distinction and of the related distinction between constructive and non-constructive functions of ordinals in the second number class; where, again to speak vaguely, a function is constructive if there is a rule by which, whenever a value of the independent variable (or a set of values of the independent variables) is effectively given, the corresponding value of the dependent variable can be effectively obtained, effectiveness in the case of ordinals of the second number class being understood to refer to a step by step process of building up to the ordinal from below. Much of the interest of the proposed definition lies, of course, in its absoluteness, and would be lost if it could be shown that it was in any essential sense relative to a particular scheme of notation or a particular formal system of logic. I t is my present belief that the definition is absolute in this way—towards those who do not find this convincing the definition may perhaps be allowed to stand as a challenge, to find either a less inclusive definition which cannot be shown to exclude some ordinal which ought reasonably to be allowed as constructive, or a more inclusive definition which cannot be shown to include some ordinal of the second class which cannot be seen to be constructive. I t is believed that the distinction which it is proposed to develop between constructive and non-constructive ordinals (and functions of ordinals) should be of interest generally in connection with applications of the transfinite ordinals to mathematical problems. The relevance of the distinction is especially clear, however, in the case of applications of the ordinals to certain questions of symbolic logic (for example, various questions more or less closely related to the well known theorem of Gödel on undecidable propositions) f—this is

Alternatives to Zermelo’s assumption

1. The axiom of choice. The object of this paper is to consider the possibUity of setting up a logic in which the axiom of choice is false. The way of approach is through the second ordinal class, in connection with which there appear certain alternatives to the axiom of choice. But these alternatives have consequences not only with regard to the second ordinal class but also with regard to other classes, whose definitions do not involve the second ordinal class, in particular with regard to the continuum. And therefore it is possible to consider these alternatives as, in some sense, postulates of logic. In what follows we proceed, after certain introductory considerations, to state these postulates, to inquire into their character, and to derive as many as possible of their consequences. The axiom of choice, which is also known as Zermelos assumption,f and, in a weakened form, as the multiplicative axiom,f is a postulate of logic which may be stated in the following way: Given any set X of classes which does not contain the null class, there exists a one-valued function, F, such that if x is any class of the set X then F(x) is a member of the class x. An equivalent statement is that there exists an assignment to every class x belonging to the set X of a unique element p such that p is contained in x. The important case is that in which the set X contains an infinite number of classes, because the assertion of the postulate is obviously capable of proof when the number of classes is finite. Accordingly a convenient, although not quite precise, characterization of the axiom of choice is obtained by saying that it is a postulate which justifies the employment of an infinite number of acts of arbitrary choice.

Correction to a Note on the Entscheidungsproblem

In A note on the Entscheidungsproblem the author gave a proof of the unsolvability of the general case of the Entscheidungsproblem of the engere Funktionenkalkul. This proof, however, contains an error, in order to correct which it is necessary to modify the “additional axioms” of the system L so that they contain no free variables (either free individual variables or free propositional function variables). The additional axioms of L other than x = y →[ F(x) → F(y) ] contain no free propositional function variables, and hence it is sufficient to replace each one by an expression obtained from it by quantifying all the individual variables by means of universal quantifiers initially placed—thus, in particular, x = x is replaced by (x)[x=x ]. Moreover the axiom x = y →[ F(x) → F(y) ] may be replaced by the following set of axioms: and similar axioms for each of the functions b 1 , b 2 , …, b k .

Some Theorems on Definability and Decidability

In this paper a theorem about numerical relations will be established and shown to have certain consequences concerning decidability in quantification theory, as well as concerning the foundation of number theory. The theorem is that relations of natural numbers are reducible in elementary fashion to symmetric ones; i.e.: Theorem I. For every dyadic relation R of natural numbers there is a symmetric dyadic relation H of natural numbers such that R is definable in terms of H plus just truth-functions and quantification over natural numbers . To state the matter more fully, there is a (well-formed) formula ϕ of pure quantification theory, or first-order functional calculus, which meets these conditions: (a) ϕ has ‘ x ’ and ‘ y ’ as sole free individual variables; (b) ϕ contains just one predicate letter, and it is dyadic; (c) for every dyadic relation R of natural numbers there is a symmetric dyadic relation H of natural numbers such that, when the predicate letter in ϕ is interpreted as expressing H , ϕ comes to agree in truth-value with ‘ x bears R to y ’ for all values of ‘ x ’ and ‘ y ’.

Mathematics and Logic

Publisher Summary The purpose of this chapter is to regard a language as a set of primitive symbols and formation rules and—in some sense that it is not necessary to make definite—meanings for the expressions of the language. \The chapter concerns an old question relative to which developments have come to a conclusion or at least a pause. It is not true that opinions are agreed now. But the cessation of active development means that the matter can be summed up and even that some attempt may be made at adjudication. There are two senses in which it is maintained that logic is prior to mathematics. One of these, the stronger sense, is the doctrine that has come to be known as “logicism.” And the other, the weak sense, is the sense in which the standard postulational or axiomatic view of the nature of mathematics requires the priority of logic as being the means by which the consequences of a particular system of mathematical postulates are determined.