Ordinal Numbers View Full Text


Ontology type: schema:Chapter     


Chapter Info

DATE

1982

AUTHORS

Gaisi Takeuti , Wilson M. Zaring

ABSTRACT

The theory of ordinal numbers is essentially a theory of well-ordered sets. For Cantor an ordinal number was “the general concept which results from (a well-ordered aggregate) M if we abstract from the nature of its elements while retaining their order of precedence ....”It was Gottlob Frege (1848–1925) and Bertrand Russell (1872–1970), working independently, who removed Cantor’s numbers from the realm of psychology. In 1903 Russell defined an ordinal number to be an equivalence class of well-ordered sets under order isomorphism. Russell’s definition has a certain intuitive appeal. By his definition the ordinal number one is the class of all well-ordered singleton sets, the ordinal number two is the class of all well-ordered double-ton sets, etc. But this definition has a serious defect from the point of view of ZF set theory because the class of all singleton sets is a proper class, as is the class of all doubleton sets, etc. For our purposes we would like ordinal numbers to be sets and to acheive this we take a different approach from that of Russell. More... »

PAGES

35-55

Book

TITLE

Introduction to Axiomatic Set Theory

ISBN

978-1-4613-8170-9
978-1-4613-8168-6

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/978-1-4613-8168-6_7

DOI

http://dx.doi.org/10.1007/978-1-4613-8168-6_7

DIMENSIONS

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