Independence of the Continuum Hypothesis

Let be a countable Transitive Model of . Consider , the Finite Function Forcing (in ) Let be a -Generic Filter over and

Then is a binary matrix, and each row is a subset of . For with define

This is a set and it is Dense in . Therefore

is an injection from into . Thus we have proved that in the Model Extension by Finite Function Forcing:

We are not done yet! We need (which needs ). By the lemma below, has c.c.c. Also, Chain Condition forcing preserves Regular Cardinals. Thus we are done (as both and are successors, thus Regular)

Lemma

For any , has the c.c.c.

Proof

Suppose is uncountable. Fix and consider

This is an uncountable set of finite sets, so by Delta System Lemma find an uncountable Delta System with root . By pigeonhole, find in such that

But now so and are compatible. Thus is not an Antichain, so has c.c.c.

Remark

Same proof works for any i.e. we can make the continuum as large as we like. Note that we have not yet proven that

is consistent, but merely that is consistent.

Theorem

In the setup as above, if then

Proof

By the Number of Nice Names we can find

By Hausdorff’s Formula calculate

So there is at most Nice Names of subsets of in . Thus we conclude

but we have already shown