Use of forcing to real line to make elements countable

Question

Can we use forcing techniques to force the set of elements of the real line to be countable? If not can anyone show why it is not possible?

Answer 1

No and yes.

It is a theorem (of ZF) that $\mathbb{R}$ is an uncountable set. Since forcing produces new models of ZF(C) from old ones, you cannot use forcing to produce models in which theorems of ZF(C) are false.

However...

Starting with a model $V$ of (a sufficiently large fragment of) ZFC, there is a set $\mathbb{R}^V$ in $V$ which corresponds to $V$'s version of the real numbers. You can force to construct a new model $V^* \supseteq V$ of ZFC such that $$V^* \models \text{"} \mathbb{R}^V \text{ is countable."}$$ Again, $V^*$'s version of the real numbers will still be an uncountable set from $V^*$'s perspective.

Answer 2

When we collapse the contiuum to be countable we only make the "old" real line countable. In doing so we add a lot if real numbers that weren't there before.

Internally the real numbers can never be countable, as this would contradict Cantor's theorem.