It had been the main trouble about the 1900 Hilbert's list of complications. The generalized continuum hypothesis asserts that there are no intermediate cardinals among each infinite established X and its electricity established.

, at times called the continuum. It truly is an infinite cardinal variety and is denoted by c displaystyle mathbf mathfrak c

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listing of 23 important unsolved complications, 10 of which he introduced to the Second Worldwide Congress of

Probably the thought of demonstrability from the continuum speculation could The Continuum Condo Official Sales have led mathematical sciences along a different path of enhancement:

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Cantor gave two proofs which the cardinality of the set of integers is strictly smaller than that of your list of serious quantities (see Cantor's initially uncountability proof and Cantor's diagonal argument).

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of Decision (AC), Gödel confirmed that CH is consistent with normal set principle and Cohen confirmed that ~CH is according to standard set theory

Having said that, this doesn't imply that all this kind of completions could well be suitable in any individual feeling, or that all would make your mind up the dimensions from the reals. What Woodin proves is that every one completions negate CH, and forcing axioms demonstrate that there is no less than a single this kind of completion.