Intuitive explanation of Cohen's forcing (continuum hypothesis) and how Godel proved CH was consistent within ZFC?

Question

I'm new to set theory and am really struggling to get my head around this. Can anyone give me an intuitive explanation so I can get a general grasp? Cheers

Answer 1

When you want to prove a relative independence result ("If $T$ is consistent, then $T$ doesn't prove $\varphi$") there are essentially two ways to do it:

• Analyze the structure of $T$-proofs as combinatorial objects in their own right. An example of this is the usual (Gödel-Rosser) proof of the second incompleteness theorem, that no recursively axiomatizable consistent theory extending PA can prove its own consistency (and in particular, Rosser's improvement on Gödel's original argument was removing the last vestige of an appeal to models).

• Build a model of $T$ in which $\varphi$ fails (and then apply the soundness theorem). For example, to prove that the theory of groups doesn't prove $\forall x,y(x*y=y*x$), it suffices to exhibit an example of a nonabelian group, together with a proof that it is in fact a nonabelian group.

Generally, the second approach is much easier, and this is what Gödel and Cohen each did - although models of ZFC are extremely complicated objects, so rather than build appropriate models "from scratch" they showed how appropriate models could be constructed given "starting models" of ZFC (which is fine - we're assuming that ZFC is consistent to begin with, and that means we can apply the completeness theorem). Interestingly, they went in opposite directions:

• Gödel showed that for any model $M$ of ZFC, there is a smaller structure $N\subseteq M$ which forms a model of ZFC+CH.

• Cohen showed that for any model $M$ of ZFC, there is a larger structure $M\subseteq N$ which forms a model of ZFC+$\neg$CH.

  • OK, that's not quite true: Cohen only showed that the above statement holds if $M$ is countable. But by the downward Lowenheim-Skolem theorem, if ZFC has a model then it has a countable model, so that's fine.

  • It's also worth pointing out that Cohen's method is much more flexible that Gödel's: Gödel's arguments cannot be used to show the consistency of ZFC+$\neg$CH, whereas Cohen's argument also lets us build models in which CH holds, so Cohen solves both parts of the problem while Godel solves only one. That said, Cohen's method doesn't make Gödel's obsolete, it's just superior for this particular problem.

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That describes what they're doing; now, how did they do it?

Gödel's construction (inner models) is much simpler (and so it's unsurprising that it was discovered earlier). Intuitively, we can define the constructible universe $L$ recursively by $$L_0=\emptyset,\quad L_{\alpha+1}=\mathcal{P}_{def}(L_\alpha),\quad L_\lambda=\bigcup_{\alpha<\lambda}L_\alpha\mbox{ for $\lambda$ limit}$$ where $\mathcal{P}_{def}(X)$ is the set of all subsets of $X$ definable in the structure $(X,\in)$, and let $$L=\bigcup_{\alpha\in Ord}L_\alpha.$$ It turns out that we can run this construction inside an arbitrary model of ZFC (indeed, ZF) and the result satisfies ZFC+CH (indeed, a lot more); roughly speaking, every element of $L$ is "definable" in a certain sense, so we can calculate $2^{\aleph_0}$ by counting the number of definitions of the appropriate type (although it's actually more complicated than that).

• The limitation of Gödel's argument mentioned above comes from the fact that $L$ is "minimal" in a precise sense - there's no hope in general that we can find a nice submodel of a given model in which CH fails, since maybe our starting model is $L$ itself.

Meanwhile, Cohen's method (forcing) is quite involved, and too complicated to explain here. The answers to this MSE question explain it a bit, but there's just too much to say to give a good explanation. This article by Chow may be helpful.