The meaning of GCH in ZF

Question

I heard one can prove that GCH implies AC(the axiom of choice) in ZF. But I am confused with the meaning of GCH in ZF. Under AC, we can define cardinal exponentiation, so the cardinal 2 to the aleph is well defined. However, Without AC, How can we restate the GCH?

Answer 1

You can formulate GCH in two ways:

1. For every ordinal $\alpha$, $2^{\aleph_\alpha}=\aleph_{\alpha+1}$. Arguably, cardinals do not make a lot of sense outside of realm of ordinals (I am not advocate of this philosophy, though), and since this is the meaning of the statement in $\sf ZFC$, this should be the meaning in $\sf ZF$ as well.

2. For every infinite set $x$, there is no set $y$ such that $|x|<|y|<|\mathcal P(x)|$. That is to say, there is no intermediate cardinalities between an infinite set and its power set.

Cantor's original formulation of CH was in line with (2), and only a decade later when the $\aleph$ notation came about that $2^{\aleph_0}=\aleph_1$ was stated.

It turns out that both of these are equivalent over $\sf ZF$, as they both imply the axiom of choice. So at the end we shouldn't worry about this too much. Nevertheless, from a local perspective, it is certainly possible that there are no intermediate cardinalities between $\omega$ and $\mathcal P(\omega)$, while at the same time $2^{\aleph_0}\neq\aleph_1$. This holds, for example, in Solovay's model.

Answer 2

In ZF, the GCH can be stated as thus: for any ordinal $\alpha$, a bijection exists between $\omega_\alpha$ and $\beth_\alpha$. Here we can define$$\beth_0:=\omega,\,\beth_{\alpha+1}:=\mathcal{P}(\beth_\alpha),\,\beth_{\gamma}:=\bigcup_{\beta\in\gamma}\beth_\beta,$$where the last equation is for nonzero limit ordinals $\gamma$ only. All this can be done; it's certainly equivalent to the usual formulation. But the usual formulation uses alephs instead of ordinals.

Answer 3

In Sierpiński's book Cardinal and Ordinal Numbers, second edition revised, 1965, on p. 88 he defines it as follows:

The assumption that, no matter what an infinite set $A$ is like, there is no set which would be of greater power than $A$ and of less power than the set of all subsets of $A$ is called the generalized Continuum Hypothesis.

In short, $|A|\lt|X|\lt|\mathcal P(A)|$ never happens when $A$ is an infinite set. It is this form of the GCH that is meant in Sierpiński's theorem that the GCH implies the axiom of choice.