What is the smallest-cardinal

Extensions of Haar Measure In Compact Groups
by Gerald Itzkowitz

Mathematics Department, Queens College of C.U.N.Y.


A discussion of the problem, "Can the Haar measure on a compact group G which has character w(G) be extended to a translation and inversion invariant measure on G with a new measure character exp(exp(G)))?", will be given. Here w(G) is the smallest cardinal of a base for the open sets of G and the character c(G) of the Haar measue space is the smallest cardinal of a collection of sets S for which every Haar measurable set in G can be approximated in measure. It is an elementary fact that for compact topological groups G that c(G)=w(G) (Reference: Hewitt and Ross, Abstract Harmonic Analysis, Vol. I). This problem was solved positively for the circle by Kakutani, Kodaira, and Oxtoby in 1950, and by Itzkowitz for compact connected Abelian groups in 1964 (Thesis). Shortly afterwards the results were extended by Hewitt and Ross to all compact Abelian groups using a theorem on dense pseudocompact subgroups from my thesis. A discussion of the methods employed will be integrated with more recent results on compact group structure to indicate how research could continue toward a complete solution of the problem.

http://atlas-conferences.com/c/a/p/a/63.htm

Continuum Hypothesis and Chaos - smallest-cardinal

I didn't express myself too clearly, sorry about that.
There's a theorem that says that given any set of cardinals, there exists a smallest cardinal which is strictly greater than all cardinals in the set.
Thus, the theorem implies that there must be a least cardinal greater than aleph_0, by taking the set {aleph_0}. Then, one defines "aleph_1" to be this cardinal, whose existence is guaranteed by the theorem.