continuum hypothesis

The continuum hypothesis is:

aleph₁ = 2^aleph₀

That is, the cardinality of the set of all countable ordinals is equal to the cardinality of the set of all subsets of the natural numbers. Note that there are strictly more countable ordinals than countable cardinals. The continuum hypothesis is, loosely speaking, a postulate as to how many more there are.

aleph₁, the cardinal corresponding to of the set of all countable ordinals, is also the smallest cardinal number that is strictly greater than aleph₀, the cardinality of the set of natural numbers. Furthermore, 2^aleph₀ is equal to c, the cardinality of the set of real numbers. Combining these facts, we see that the continuum hypothesis can be restated as:

There is no cardinal number k such that |N| < k < |R|.

The generalised continuum hypothesis, or GCH, proposed by Felix Hausdorff, states that aleph_n+1 = 2^aleph_n. Given the axioms of Zermelo-Fraenkel set theory (ZF), GCH implies the axiom of choice, but the converse is not true.

In ZFC (ZF + AC, the axiom of choice), we know the following about c:

• It is larger than aleph₀.
• It is aleph_k for some ordinal number k.
• It must be an infinite successor cardinal. It cannot be, for example, the limit cardinal aleph_ω: nonzero cardinals are either limit cardinals or successor cardinals, but not both. It can, however, be this cardinal's successor, aleph_ω+1.

So, is the continuum hypothesis (limited or general) actually true? It's not quite so clear-cut. Kurt Gödel proved that, assuming ZF is consistent, ZFC + CH (the axioms of ZF set theory, along with the axiom of choice and the continuum hypothesis) is also consistent. Conversely, Paul Cohen proved that, assuming ZF is consistent, ZFC + (c = aleph_k+1) is consistent for any countable ordinal k. Much as Lobechevsky demonstrated the existence of a number of different non-Euclidian geometries, Cohen and Gödel have demonstrated the existence of a number of different consistent versions of set theory.