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Formal development of ordinal number theory1

Published online by Cambridge University Press:  12 March 2014

Steven Orey
Steven Orey*
Affiliation:
Cornell University and University of Minnesota
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Extract

In this paper we shall develop a theory of ordinal numbers for the system ML, [6]. Since NF, [2], is a sub-system of ML one could let the ordinal arithmetic developed in [9] serve also as the ordinal arithmetic of ML. However, it was shown in [9] that the ordinal numbers of [9], NO, do not have all the usual properties of ordinal numbers and that theorems contradicting basic results of “intuitive ordinal arithmetic” can be proved.

In particular it will be a theorem in our development of ordinal numbers that, for any ordinal number α, the set of all smaller ordinal numbers ordered by ≤ has ordinal number α; this result does not hold for the ordinals of [9] (see [9], XII.3.15). It will also be an easy consequence of our definition of ordinal number that proofs by induction over the ordinal numbers are permitted for arbitrary statements of ML; proofs by induction over NO can be carried through only for stratified statements with no unrestricted class variables.

The class we shall take as the class of ordinal numbers, to be designated by ‘ORN’, will turn out to be a proper subclass of NO. This is because in ML there are two natural ways of defining the concept of well ordering. Sets which are well ordered in the sense of [9] we shall call weakly well ordered; sets which satisfy a certain more stringent condition will be called strongly well ordered. NO is the set of order types of weakly well ordered sets, while ORN is the class of order types of strongly well ordered sets. Basic properties of weakly and strongly well ordered sets are developed in Section 2.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 20 , Issue 2 , June 1955 , pp. 95 - 104
Copyright
Copyright © Association for Symbolic Logic 1955

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Footnotes

1

The material of this paper was part of a Ph.D. thesis presented to the Faculty of Cornell University in September, 1953. I wish to express my sincere thanks to Prof. J. B. Rosser for his suggestions and advice in connection with the writing of that thesis.

References

BIBLIOGRAPHY

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