The Continuum Hypothesis

Recall that sets A and B are said to be of the same cardinality iff there is a bijection from A to B. A set is finite iff it is of the same cardinality as some set $\{0, 1, \ldots, n\}$. A set is countable iff it's either finite or of the same cardinality of N = $\{0, 1, 2, \ldots\}$, the natural numbers. As you now know from seeing earlier proofs, both 2$^{\bf N}$ (the power set of the set of natural numbers) and R (the reals) are uncountable.

Now, the conintuum hypothesis is:

(CH)

Every infinite subset of R is either countable or of the same cardinality as R.

In 1938 Gödel proved that

• If ZFC is consistent, then ZFC $\not\vdash \neg$CH.

In 1963 Cohen proved that

• If ZFC is consistent, then ZFC $\not\vdash$CH.

So, if we assume that ZFC is consistent, then neither CH nor $\neg$CH is derivable from it. As we will see later, it's a consequence of other theorems proved by Gödel that for every set $\Psi$ of axioms for set theory, there exists an assertion $\psi$ about sets that is such that

• $\Psi \not\vdash \psi$
• $\Psi \not\vdash \neg\psi$