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Natural well-orderings

Published online by Cambridge University Press:  12 March 2014

David Isles
David Isles*
Affiliation:
Tufts University Medford, Massachusetts 02155
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Extract

There seem to be at least two approaches to the problem of generating notations for ordinals: the papers of Bachmann [2], Gerber [5], Feferman [3], and Isles [6] provide examples of one and those of Ackermann [1], Takeuti [10], Kino [7], Schütte [9], and Pfeiffer [8] the other. In both of these the normal forms produced can be regarded as being finite trees “based on” some ordinal π.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 36 , Issue 2 , June 1971 , pp. 288 - 300
Copyright
Copyright © Association for Symbolic Logic 1971

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References

[1]Ackermann, W., Konstruktiver Aufbau eines Abschnitts der zweiten Cantorschen Zahlenklasse, Mathematische Zeitschrift, vol. 53 (1951), pp. 403–413.CrossRefGoogle Scholar
[2]Bachmann, H., Die Normalfunktionen und das Problem der ausgezeichneten Folgen von Ordnungszahlen, Vitrteljahrschrift der Naturforsckenden Gesellschaft in Zürich, vol. 95 (1950), pp. 115–147.Google Scholar
[3]Feferman, S., Systems of predicative analysis, II: representations of ordinals, this Journal, vol. 33 (1968), pp. 193–220.Google Scholar
[4]Gaifman, H., A generalization of Mahlo's method for obtaining large cardinal numbers, Israel Journal of Mathematics, vol. 5 (1967), pp. 188–199.CrossRefGoogle Scholar
[5]Gerber, H., Brouwer's bar theorem and a system of ordinal notations, Conference on Intuitionism and Proof Theory, Buffalo, 1968.Google Scholar
[6]Isles, D., Regular ordinals and normal forms, I, Conference on Intuitionism and Proof Theory, Buffalo, 1968.Google Scholar
[7]Kino, A., On ordinal diagrams, Journal of the Mathematical Society of Japan, vol. 13 (1961), pp. 346–356.CrossRefGoogle Scholar
[8]Pfeiffer, H., Ein Bezeichnungs-system für Ordinalzahlen, Archiv für mathematische Logik und Grundlagenforschung, vol. 12 (1969), pp. 12–17.CrossRefGoogle Scholar
[9]Schütte, K., Ein konstruktives system von Ordinalzahlen I, II, Archiv für mathematische Logik und Grundlagenforschung, vol. 11 (1968), pp. 126–137 and vol. 12 (1969), pp. 3–11.CrossRefGoogle Scholar
[10]Takeuti, G., Ordinal diagrams, Journal of the Mathematical Society of Japan, vol. 9 (1957), pp. 386–394.CrossRefGoogle Scholar
[11]Takeuti, G., Consistency proofs of subsystems of classical analysis, Annals of Mathematics, vol. 86 (1967), pp. 299–348.CrossRefGoogle Scholar