Today was the final exam in a course on *set-theoretical forcing*. It was one of
the hardest courses I’ve attended, but at least the exam was easy. But what the
heck is forcing anyway?

It’s a technique for independence proofs. It was originally developed by Paul Cohen for proving the independence of Continuum Hypothesis (CH) from Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC).

A theory is *consistent* if it does not allow contradictions. For example, ZFC
is thought to be consistent (although you can’t prove it in ZFC), so you can’t
derive a contradiction from the axioms of ZFC.

An axiom is *independent* of a theory if you can’t prove or disprove it from the
theory. You can prove the independence by showing that the theory is consistent
with the axiom and with the negation of the axiom. Assuming the consistency of
ZFC, you can prove that ZFC together with CH is consistent. Using forcing, you
can also prove that ZFC together with the negation of CH is consistent. Thus CH
is independent of ZFC.

How does this work in practice? We assume the existence of countable transitive
model of ZFC, $V$. Then we come up with a partially-ordered set (*forcing
poset*) that is used to construct a generic extension of the model, $V[G]$.
This model is constructed so that it witnesses whatever we want to prove. Its
existence proves the claim.^{1}

To prove that ZFC is consistent with the negation of continuum hypothesis, i.e. $2^\omega > \omega_1$, we would take a cardinal $\kappa$ that is larger than $\omega_1$ in $V$. We then construct $V[G]$ so that there are at least $\kappa$ subsets of $\omega$. Since $V$ and $V[G]$ have the same cardinals, $2^\omega > \omega$.

The tricky part is finding a suitable forcing poset. One of the ways to make it
easier is to use *iterated forcing*, where you repeat the forcing transfinite
number of times. I’d tell you how it works, but unfortunately I don’t understand
it.

- I definitely do not understand this part, but I trust the authorities. ↩