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A system of axiomatic set theory. Part V. General set theory (continued)53

Published online by Cambridge University Press:  12 March 2014

Paul Bernays
Paul Bernays*
Affiliation:
Zurich
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Extract

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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 8 , Issue 4 , December 1943 , pp. 89 - 106
Copyright
Copyright © Association for Symbolic Logic 1944

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Footnotes

53

Parts I–IV appeared in this Journal, vol. 2 (1937), pp. 65–77; vol. 6 (1941), pp. 1–17; vol. 7 (1942), pp. 65–89, 133–145.

References

54 General recursion in the frame of the Fraenkel axiomatic set theory (with the strong form of the Ersetzungsaxiom, equivalent to our axiom V*) was established by von Neumann, on the basis of his independent theory of ordinals, in his paper Über die Definition durch transfinite Induktion und verwandte Fragen der allgemeinen Mengenlehre, Mathematische Annalen, vol. 99 (1928), pp. 373–391.

55 Zorn, M., A remark on method in transfinite algebra, Bulletin of the American Mathematical Society, vol. 41 (1935), pp. 667–;670.CrossRefGoogle Scholar

56 Deutsche Mathematik, vol. 4 (1939), pp. 567–577. See the review by Rózsa Péter in this Journal, vol. 6 (1941), pp. 65–66.

57 See Part II, §4, p. 3.

58 See Part IV, §11, p. 134.

59 The following argument has been substituted for an earlier one which was more complicated, and which besides required the axiom of choice. This simplification was suggested by Gödel's proofs of the theorems 7.811 and 8.62 in his monograph The consistency of the continuum hypothesis (Princeton 1940). However, these proofs are based on the theorem 5.18 (proved by means of the power axiom), which coincides with our pair class axiom. Hence we here need to modify Gödel's method of proof.

60 §13, p. 94.

61 In order to avoid distinguishing between equality and identity, we may introduce the notion of a proper polynomial, defining a polynomial over C to be a proper polynomial if every element of its converse domain is different from zero. As is easily seen, there is for every polynomial a uniquely determined proper polynomial equal to it, which is obtained from it by omitting the elements (monomials) with the coefficient zero. The class of proper polynomials exists. Then we may define the proper algebraic sum of proper polynomials p and q as the proper polynomial which is equal to the algebraic sum of p and q; and analogously the proper algebraic difference and the proper algebraic product of proper polynomials can be defined. With respect to the operations thus defined the class of proper polynomials constitutes a ring. (Added September 12, 1943.)