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Transfinite ordinals in recursive number theory

Published online by Cambridge University Press:  12 March 2014

R. L. Goodstein
R. L. Goodstein*
Affiliation:
The University, Reading, England
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Extract

The possibility of constructing a numerical equivalent of a system of trans-finite ordinals, in recursive number theory, was briefly indicated in a previous paper, where consideration was confined to ordinals less than ε (the first to satisfy ε = ω). In the present paper we construct a representation, by functions of number-theoretic variables, for ordinals of any type.

In addition to definite numerals, and numeral variables, we introduce majorant variables σ, ω, ωr , r ≧ 1. A relation containing a single majorant variable σ is required to hold, not necessarily for all non-negative integral values of σ, but for all values greater than some assigned constant.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 12 , Issue 4 , December 1947 , pp. 123 - 129
Copyright
Copyright © Association for Symbolic Logic 1947

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References

1 In this Journal, vol. 9(1944), pp. 33–41.

2 Cf. Ackermann, W., Zum Hibertschen Aufbau der reelen Zahlen, Mathematische Annalen, vol. 99(1928), p. 120.CrossRefGoogle Scholar