General Continuum Hypothesis

The assertion that for any Ordinal :

Lemma

Let be the Constructible Hierarchy. Then

Lemma

It is consistent that (i.e. The Continuum Hypothesis) and at the same time .

Proof

Define

Clearly the Finite Function Forcing has

If we compare and , we note that the former does not completely embed into the latter e.g. take the constant function , then many values are determined to be .

The problem is that does not have c.c.c. anymore. The small countable functions above create an Antichain of size

By Delta System Lemma we can show has the -Chain Condition Thus in our concrete case, we start from a model of so

and force with which has the -Chain Condition, so that’s -Chain Condition. Then all cardinals are preserved. Because is a -Closed Forcing, we can see that preserves . Thus