scholarly journals A system of axiomatic set theory. Part V. General set theory (continued)

1943 ◽  
Vol 8 (4) ◽  
pp. 89-106 ◽  
Author(s):  
Paul Bernays
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Transfinite Induction ◽  
Recursive Definition ◽  
Axiomatic Set Theory

We have still to consider the extension of the methods of number theory to infinite ordinals—or to transfinite numbers as they may also, as usual, be called.The means for establishing number theory are, as we know, recursive definition, complete induction, and the “principle of the least number.” The last of these applies to arbitrary ordinals as well as to finite ordinals, since every nonempty class of ordinals has a lowest element. Hence immediately results also the following generalization of complete induction, called transfinite induction: If A is a class of ordinals such that (1) ΟηA, and (2) αηA → α′ηA, and (3) for every limiting number l, (x)(xεl → xηA) → lηA, then every ordinal belongs to A.

scholarly journals A system of axiomatic set theory. Part III. Infinity and enumerability. Analysis

1942 ◽  
Vol 7 (2) ◽  
pp. 65-89 ◽  
Author(s):  
Paul Bernays
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Real Numbers ◽  
Full Generality ◽  
Axiomatic Set Theory

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.

A system of axiomatic set theory—Part VI

1948 ◽  
Vol 13 (2) ◽  
pp. 65-79 ◽  
Author(s):  
Paul Bernays
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Axiom Of Choice ◽  
The Axiom Of Choice ◽  
Axiomatic Set Theory ◽  
Present Section

Comparability of classes. Till now we tried to get along without the axioms Vc and Vd. We found that this is possible in number theory and analysis as well as in general set theory, even keeping in the main to the usual way of procedure.For the considerations of the present section application of the axioms Vc, Vd is essential. Our axiomatic basis here consists of the axioms I—III, V*, Vc, and Vd. From V*, as we know, Va and Vb are derivable. We here take axiom V* in order to separate the arguments requiring the axiom of choice from the others. Instead of the two axioms V* and Vc, as was observed in Part II, V** may be taken as well.

Remarks on Peano Arithmetic

2000 ◽  
Vol 20 (1) ◽  
Author(s):  
Charles Sayward
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Peano Arithmetic ◽  
The Other ◽  
Natural Numbers ◽  
Recursive Definition ◽  
Explicit Definition

<p>Russell held that the theory of natural numbers could be derived from three primitive concepts: number, successor and zero. This leaves out multiplication and addition. Russell introduces these concepts by recursive definition. It is argued that this does not render addition or multiplication any less primitive than the other three. To this it might be replied that any recursive definition can be transformed into a complete or explicit definition with the help of a little set theory. But that is a point about set theory, not number theory. We have learned more about the distinction between logic and set theory than was known in Russell's day, especially as this affects logicist aspirations.</p>

scholarly journals A system of axiomatic set theory - Part VII

1954 ◽  
Vol 19 (2) ◽  
pp. 81-96 ◽  
Author(s):  
Paul Bernays
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Basic Set ◽  
Set Up ◽  
Axiomatic Set Theory

The reader of Part VI will have noticed that among the set-theoretic models considered there some models were missing which were announced in Part II for certain proofs of independence. These models will be supplied now.Mainly two models have to be constructed: one with the property that there exists a set which is its own only element, and another in which the axioms I–III and VII, but not Va, are satisfied. In either case we need not satisfy the axiom of infinity. Thereby it becomes possible to set up the models on the basis of only I–III, and either VII or Va, a basis from which number theory can be obtained as we saw in Part II.On both these bases the Π0-system of Part VI, which satisfies the axioms I–V and VII, but not VI, can be constructed, as we stated there. An isomorphic model can also be obtained on that basis, by first setting up number theory as in Part II, and then proceeding as Ackermann did.Let us recall the main points of this procedure.For the sake of clarity in the discussion of this and the subsequent models, it will be necessary to distinguish precisely between the concepts which are relative to the basic set-theoretic system, and those which are relative to the model to be defined.

The General Idea Behind Gödel’s Proof

1992 ◽  
Author(s):  
Raymond M. Smullyan
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
The Other ◽  
General Idea ◽  
Original Proof ◽  
Extensive Class ◽  
The One ◽  
Axiomatic Set Theory ◽  
Gödel’S Theorem ◽  
Mathematical Systems

In the next several chapters we will be studying incompleteness proofs for various axiomatizations of arithmetic. Gödel, 1931, carried out his original proof for axiomatic set theory, but the method is equally applicable to axiomatic number theory. The incompleteness of axiomatic number theory is actually a stronger result since it easily yields the incompleteness of axiomatic set theory. Gödel begins his memorable paper with the following startling words. . . . “The development of mathematics in the direction of greater precision has led to large areas of it being formalized, so that proofs can be carried out according to a few mechanical rules. The most comprehensive formal systems to date are, on the one hand, the Principia Mathematica of Whitehead and Russell and, on the other hand, the Zermelo-Fraenkel system of axiomatic set theory. Both systems are so extensive that all methods of proof used in mathematics today can be formalized in them—i.e. can be reduced to a few axioms and rules of inference. It would seem reasonable, therefore, to surmise that these axioms and rules of inference are sufficient to decide all mathematical questions which can be formulated in the system concerned. In what follows it will be shown that this is not the case, but rather that, in both of the cited systems, there exist relatively simple problems of the theory of ordinary whole numbers which cannot be decided on the basis of the axioms.” . . . Gödel then goes on to explain that the situation does not depend on the special nature of the two systems under consideration but holds for an extensive class of mathematical systems. Just what is this “extensive class” of mathematical systems? Various interpretations of this phrase have been given, and Gödel’s theorem has accordingly been generalized in several ways. We will consider many such generalizations in the course of this volume. Curiously enough, one of the generalizations that is most direct and most easily accessible to the general reader is also the one that appears to be the least well known.

Resolution in type theory

1971 ◽  
Vol 36 (3) ◽  
pp. 414-432 ◽  
Author(s):  
Peter B. Andrews
Keyword(s):  
Set Theory ◽  
Type Theory ◽  
Higher Order ◽  
Order Logic ◽  
First Order Logic ◽  
Higher Order Logic ◽  
First Order ◽  
Different Types ◽  
Axiomatic Set Theory ◽  
Order Predicate Calculus

In [8] J. A. Robinson introduced a complete refutation procedure called resolution for first order predicate calculus. Resolution is based on ideas in Herbrand's Theorem, and provides a very convenient framework in which to search for a proof of a wff believed to be a theorem. Moreover, it has proved possible to formulate many refinements of resolution which are still complete but are more efficient, at least in many contexts. However, when efficiency is a prime consideration, the restriction to first order logic is unfortunate, since many statements of mathematics (and other disciplines) can be expressed more simply and naturally in higher order logic than in first order logic. Also, the fact that in higher order logic (as in many-sorted first order logic) there is an explicit syntactic distinction between expressions which denote different types of intuitive objects is of great value where matching is involved, since one is automatically prevented from trying to make certain inappropriate matches. (One may contrast this with the situation in which mathematical statements are expressed in the symbolism of axiomatic set theory.).

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