A more abstract example of an uncountable set is the set of all countable ordinal numbers, 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).
A more absTraCt eXampLE OF An UncouNtable sET is tHe set oF alL coUNtaBLe OrDInAl NUmbErS, 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).