Paul Bernays

A System of Axiomatic Set Theory--Part I

For the formulation of the remaining axioms we need the notions of a function and of a one-to-one correspondence . We define a function to be a class of pairs in which different elements always have different first members; or, in other words, a class F of pairs such that, to every element a of its domain there is a unique element b of its converse domain determined by the condition 〈 a, b 〉 ηF . We shall call the set b so determined the value of F for a , and denote it (following the mathematical usage) by F ( a ). A set which represents a function—i.e., a set of pairs in which different elements always have different first members—will be called a functional set . If b is the value of the function F for a , we shall say that F assigns the set b to the set a ; and if a functional set f represents F , we shall say also that f assigns the set b to the set a . A class of pairs will be called a one-to-one correspondence if both it and its converse class are functions. We shall say that there exists a one-to-one correspondence between the classes A and B (or of A to B) if A and B are domain and converse domain of a one-to-one correspondence. Likewise we shall say that there exists a one-to-one correspondence between the sets a and b (or of a to b ) if a and b respectively represent the domain and the converse domain of a one-to-one correspondence. In the same fashion we speak of a one-to-one correspondence between a class and a set, or a set and a class.

A System of Axiomatic Set Theory

Publisher Summary The aim of this chapter is to discuss a system of axiomatic set theory. It is a modification of the axiom system due to von Neumann. It adopts the principal idea of von Neumann that the elimination of the undefined notion of a property that occurs in the original axiom system of Zermelo can be accomplished in a way so as to make the resulting axiom system elementary, in the sense of being formalizable in the logical calculus of first order, which contains no other bound variables than individual variables and no accessory rule of inference. The purpose of modifying the von Neumann system is to remain nearer to the structure of the original Zermelo system and at the same time, utilize some of the set-theoretic concepts of the Schroder Logic and of Principia Mathematica that have become familiar to logicians. A considerable simplification results from this arrangement. The theory is not set up as a pure formalism but rather in the usual manner of elementary axiom theory, where one has to deal with propositions that are understood to have a meaning and the reference to the domain of facts to be axiomatized is suggested by the names for the kinds of individuals and for the fundamental predicates.

A System of Axiomatic Set Theory: Part III. Infinity and Enumerability. Analysis

The foundation of analysis does not require the full generality of set theory but can be accomplished within a more restricted frame. Just as for number theory we need not introduce a set of all finite ordinals but only a class of all finite ordinals, all sets which occur being finite, so likewise for analysis we need not have a set of all real numbers but only a class of them, and the sets with which we have to deal are either finite or enumerable. We begin with the definitions of infinity and enumerability and with some consideration of these concepts on the basis of the axioms I—III, IV, V a, V b, which, as we shall see later, are sufficient for general set theory. Let us recall that the axioms I—III and V a suffice for establishing number theory, in particular for the iteration theorem, and for the theorems on finiteness.

Die Bedeutung Hilberts für die Philosophie der Mathematik

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On the Original Gentzen Consistency Proof for Number Theory

Publisher Summary This chapter investigates the original Gentzen consistency proof for number theory. Gentzen consistency proof for the formal system of first order number theory, including standard logic, the Peano axioms and recursive definitions is considered. The chapter describes Gentzens original proof. Gentzen admits arithmetical function symbols (with the restriction that for numerical arguments the value of the function must be computable), thus, there is no loss of generality in assuming that all prime formulas are equations between terms. A reduction process for a sequent is a procedure consisting of a terminating sequence of successive reduction steps by which the sequent is brought into a final form (Endform). It means to a sequent satisfying at least one of the two conditions: (1) that the succedent is a true numerical equation or (2) that some antecedent formula is a false numerical equation.

On the Problem of Schemata of Infinity in Axiomatic Set Theory

Publisher Summary This chapter describes Zermelos set theory that was substantially extended in 1921 when Fraenkel added his axiom of replacement to the original axioms; for, it is this axiom that enables Cantors general summation process to be carried out. This process is not restricted to forming the union of the elements of a set, and its use is essential in proving the existence of cardinalities; together with the axiom of union, general summation provides the limiting process that is used to produce cardinalities. In this way, the addition of the axiom of replacement to Zermelos set theory leads to that immense rise of cardinalities that can be bounded only by inaccessible ordinals. It is known that those initial ordinals, which are called inaccessible, can be completely characterized within the framework of the theory of ordinals. The chapter also focuses on Zermelos concept of a Grenzzahl that represents a strengthening of the concept “inaccessible ordinal.” A Grenzzahl can be defined as an inaccessible ordinal α with the property that for each smaller ordinal the cardinality of its set of subsets is less than that of α.

What Do Some Recent Results in Set Theory Suggest

Publisher Summary This chapter explores the suggestions made by some recent results of set theory. The first essential thing that emerges from the results on the independence of the continuum hypothesis do not directly concern set theory itself but rather the axiomatization of set theory and a sharper axiomatization, which allows for strict formalization. A formal system of set theory is subject to the Skolem paradox, which means that the axioms are satisfiable in a denumerable model. The possibility of nonstandard models of an axiom system is because of the presence of a principle in which a concept of set, or sequence, or predicate occurs. The model theory of an axiom system containing such a principle has a standard character; however, the corresponding concept of the model theory is identified. The independence of the continuum hypothesis is essentially tied to the formalization of set theory. It is a fact of a similar kind to the existence of nonstandard models for formalized number theory.

Scope and Limits of Axiomatics

When today one speaks on axiomatics to a public familiar with mathematics, there often seems to be not so much a need for recommending axiomatics as to warn against an overestimation of it.

Mathematics as a Domain of Theoretical Science and of Mental Experience

Publisher Summary This chapter highlights mathematics as a domain of theoretical science and mental experience. The theory of algebraic functions has been a central domain; it includes function theory, algebra, algebraic geometry, theory of Riemann surfaces, and topology. There are various embracing theories, and axiomatic set theory itself is extended by model theory, where set theoretic concepts are used independently from the axiomatization. The modified situation is especially clear in the general theory of mappings, which is called “the theory of categories.” The chapter explains that mathematicians have different opinions about the suitability of stronger or only weaker methods of idealizations.

Zur Rolle der Sprache in Erkenntnistheoretischer Hinsicht

In der Philosophie von Rudolf Carnap nimmt sein Werk ‚Logische Syntax der Sprache‘ eine markante Stellung ein. Die hier entwickelte Konzeption der Wissenschaftslogik als Studium der Wissenschaftssprache, mit den sich an sie knupfenden Begriffen, bildet sozusagen den Ausgangsrahmen fur Carnaps weitere Untersuchungen. Im Laufe dieser Untersuchungen hat er die Auffassungen, die er in der Logischen Syntax vertritt, erheblich revidiert, und auch jener Rahmen der Betrachtung selbst mit den zugehorigen Begriffsbildungen hat starke Wandlungen erfahren, wozu die Diskussionen mit den Philosophen verwandter Forschungsrichtung Wesentliches beigetragen haben.