A class b is well-ordered by a dyadic relation R if it is ordered by R (see order) and, for every class a such that a ? b, there is a member x of a, such that xRy holds for every member y of a; and R is then called a well-ordering relation. The ordinal number of a class b well-ordered by a relation R, or of a well-ordering relation R, is defined to be the relation-number (q. v.) of R.<\/p>\n
The ordinal numbers of finite classes (well-ordered by appropriate relations) are called finite ordinal numbers. These are 0, 1, 2, … (to be distinguished, of course, from the finite cardinal numbers 0, 1, 2, . . .).<\/p>\n
The first non-finite (transfinite or infinite) ordinal number is the ordinal number of the class of finite ordinal numbers, well-ordered in their natural order, 0, 1, 2, . . .; it is usually denoted by the small Greek letter omega. — A.C.<\/p>\n
G. Cantor, Contributions to the Founding of the Theory of Transfinite Numbers, translated and with an introduction by P. E. B. Jourdain, Chicago and London, 1915. (new ed. 1941); Whitehead and Russell, Princtpia Mathematica. vol. 3.<\/p>\n
Ordinal number A class b is well-ordered by a dyadic relation R if it is ordered by R (see order) and, for every class a such that a ? b, there is a member x of a, such that xRy holds for every member y of a; and R is then called a well-ordering relation. … Continue reading “Ordinal number”<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-72125","post","type-post","status-publish","format-standard","hentry","category-encyclopedic-dictionary"],"_links":{"self":[{"href":"https:\/\/www.biblia.work\/dictionaries\/wp-json\/wp\/v2\/posts\/72125","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.biblia.work\/dictionaries\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.biblia.work\/dictionaries\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.biblia.work\/dictionaries\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.biblia.work\/dictionaries\/wp-json\/wp\/v2\/comments?post=72125"}],"version-history":[{"count":0,"href":"https:\/\/www.biblia.work\/dictionaries\/wp-json\/wp\/v2\/posts\/72125\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.biblia.work\/dictionaries\/wp-json\/wp\/v2\/media?parent=72125"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.biblia.work\/dictionaries\/wp-json\/wp\/v2\/categories?post=72125"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.biblia.work\/dictionaries\/wp-json\/wp\/v2\/tags?post=72125"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}