In set theory, a continuum is usually defined to be any set for which there exists a one to one map from there onto the open unit interval (0, 1), which is a set that can be shown to be uncountable, and having the cardinality ℵ. Many sets that seem to have "more" elements than (0, 1) can actually be shown to have such a one-to-one onto mapping, meaning they all have the same cardinality. For instance, (0, 1) can be one to one mapped onto the whole real line (-∞, +∞) by using the mapping: f:x -> tan(πx - π/2). By applying Bernstein's Theorem, it can be shown that R^{2} is also a continuum, and by extending the argument, R^{n} for any positive integer n is also readily seen to be a continuum as well. The argument can even be extended into infinite-dimensional spaces, these can all be shown to be continua.

The power set of any countable infinite set can also be shown to be a continuum. Since all countable infinite sets have a one-to-one onto mapping to the set of natural numbers N, it suffices to show that the power set of N, P(N), is a continuum. It is easy to see that every point in the unit interval (0,1) can be expressed as an infinite sum of the form 2^{-n1} + 2^{-n2} + .... The (possibly infinite) set of natural numbers used to produce that sum {n_{1}, n_{2}, ... } is of course an element of P(N). To each element in P(N) there corresponds the point 3^{-n1} + 3^{-n2} + ... which is in (0, 1); this point cannot be produced by any other element of P(N). Thus we have created a one to one mapping of P(N) onto (0, 1), showing that it is a continuum, so ℵ = 2^{ℵ0}.

However, just as the power set of a countably infinite set is not countably infinite but a continuum, the power set of a continuum is not itself a continuum but something even more uncountably infinite than that.

The continuum hypothesis conjectures that there exist no sets intermediate (in the sense of existence of one to one onto maps, or put another way, of cardinality) between a countable infinite set and a continuum, or put another way, ℵ = ℵ_{1} I believe that it has been shown that this proposition is formally undecidable from the axioms of set theory.

This is an example of the kinds of pitfalls in reasoning you get into when you try to reason about infinity!