denoted by Ω or ω1. The cardinality of Ω is denoted (aleph-one). It can be shown, using the axiom of choice, that is the smallest uncountable cardinal number. Thus either , the cardinality of the reals, is equal to or it is strictly larger. Georg Cantor was the first to propose the question of whether is equal to . In 1900, David Hilbert posed this question as the first of his 23 problems. The statement that is now called the continuum hypothesis, and is known to be independent of the Zermelo–Fraenkel axioms for set theory (including the axiom of choice). Without the axiom of choice Without the axiom of choice, there might exist cardinalities incomparable to (namely, the cardinalities of Dedekind-finite infinite sets). Sets of these cardinalities satisfy the first three characterizations above, but not the fourth characterization. Since these sets are not larger than the natural numbers in the sense of cardinality, some may not want to call them uncountable. If the axiom of choice holds, the following conditions on a cardinal are equivalent: and , where and is the least initial ordinal greater than However, these may all be different if the axiom of choice fails. So it is not obvious which one is the appropriate generalization of "uncountability" when the axiom fails. It may be best to avoid using the word in this case and specify which of these one means. See also Aleph number Beth number First uncountable ordinal Injective function References Bibliography Halmos, Paul, Naive Set Theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. (Springer-Verlag edition). Reprinted by Martino Fine Books, 2011. (Paperback edition). External links Proof that R is uncountable Basic concepts in infinite set theory Infinity Cardinal numbers