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# Topic: Inaccessible cardinal

###### In the News (Sat 25 May 13)

 Aleph number - Wikipedia, the free encyclopedia ) is by definition the cardinality of the set of all natural numbers, and (assuming, as usual, the axiom of choice) is the smallest of all infinite cardinalities. That it is consistent with ZFC was demonstrated by Kurt GĂ¶del in 1940; that it is independent of ZFC was demonstrated by Paul Cohen in 1963. Any inaccessible cardinal is a fixed point of the aleph function as well. en.wikipedia.org /wiki/Aleph_number   (851 words)

 Encyclopedia :: encyclopedia : Field (mathematics)   (Site not responding. Last check: 2007-11-01) The surreal numbers form a Field containing the reals, and would be a field except for the fact that they are a proper class, not a set. The set of all surreal numbers with birthday smaller than some inaccessible cardinal number form a field. , the nimbers with birthday smaller than any infinite cardinal are all examples of fields. www.hallencyclopedia.com /Field_(mathematics)   (1584 words)

 Infinitary Logic Given a pair κ, λ of infinite cardinals such that λ ≤ κ, we define a class of infinitary languages in each of which we may form conjunctions and disjunctions of sets of formulas of cardinality < κ, and quantifications over sequences of variables of length < λ. It was quickly realized that a measurable cardinal must be inaccessible, but the falsity of the converse was not established until the 1960s when Tarski showed that measurable cardinals are weakly compact and his student Hanf showed that the first, second, etc. inaccessibles are not weakly compact (cf. Although the conclusion that measurable cardinals must be monstrously large is now normally proved without making the detour through weak compactness and infinitary languages, the fact remains that these ideas were used to establish the result in the first instance. plato.stanford.edu /entries/logic-infinitary   (6663 words)

 Infinity can not exist An inaccessible cardinal, if one exists, would be beyond all of those. I am looking forward to a time when infinite sets are given a basis which would require sets a and B to have the same size in order for them to have the same cardinality. Exponentiation of cardinals is enough to get you to uncountables but not of ordinals. www.forum-one.org /new-6518969-4343.html   (15694 words)

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