Is the Continuum Hypothesis a definite problem?

I am not a Platonist.

In general, while I do find it entertaining to think about god, afterlife, or a concrete mathematical universe, I find more comfort in the uncertainty of existence than I do in the likelihood that my belief is wrong, or in the terrifying conviction that comes along with believing in something (and everyone else is wrong).

So in particular, I don’t believe there is a concrete mathematical universe where $\ZF$ or $\ZFC$ governs things. So I don’t believe that mathematical questions which involve anything larger than we can comprehend should have a definitive and concrete answer (e.g. what is the value of Graham’s number? What is the largest number that we can compute, experimentally? And other ill-defined questions which often have an answer depending on your coordinates within time and space).

The best example of such problem is the Continuum Hypothesis. Cantor hypothesized that $2^{\aleph_0}=\aleph_1$ in 1878, 1882 and 1895 (see this historical overview for details). It took some time, but after formalizing the notion of sets, logic, and so on, it was finally shown that under a relatively broad notion of “set” it is impossible to prove or disprove the continuum hypothesis. In fact, in the absence of the axiom of choice, Cantor’s different formulations over the years are not even equivalent (see this overview).

After being shown independent, there had been more than a handful of people who called out that a question without a yes or no answer is a wrong question. And I disagree. Vehemently. “What is for dinner” before anything is cooking a definite problem? Is “Who let the dogs out” a definite problem? Is “Given $f\colon\Bbb{R\to R}$, is $f$ continuously differentiable” a definite problem?

All these questions need more information to be properly answered. They all depend on much much more than just the information given in the question. If the Continuum Hypothesis has to have an answer, then it is just a witness that $\ZFC$ is not sufficiently strong to describe the “true” universe of mathematics and sets, whatever it may be.

But why should everything have a definite answer? Gödel’s incompleteness theorem tells us that if we want to keep working in a setting that (1) has an algorithmic proof verification method; (2) can describe sufficient amount of arithmetic; (3) is not inconsistent, then it is necessarily the case that our setting is incomplete. Why is that such a bad thing? It just goes to show that (1) proving things is important; and (2) much like the physical reality, we cannot know everything, and we will not know everything.

Humans are generally scared of not knowing. And it makes sense that you want to know. But why should we expect to be able to know? Amongst the sonic and light spectra, we are mostly blind and mostly deaf. Why should we be surprised that in the mathematical spectrum we will also be lacking? We shouldn’t be. If anything, that is consistent with our reality. And if someone wants to argue in favor of realism, then “not knowing” is far more realistic than “knowing”.

So what about the continuum hypothesis? Well. I don’t know. As I said, I am not a Platonist. I do not feel the need to believe in a yes/no answer to every “Is it …” question I ask. But I do think that the continuum hypothesis is a very concrete, very definite problem. Its solution being independent of $\ZFC$ shows one of two things: (1) As far as Platonist viewpoint go, $\ZFC$ is insufficient to describe the true nature of mathematics; and (2) maybe Platonism isn’t the way the go.

I wrote a comment on Michael Harris’ blog some time ago. I was confused as to why someone who is a Platonist is not working in a “maximally consistent theory”. If you believe that something is true, it makes no sense to work in a theory where it is not true. Especially if you argue in its favor, and even more if you know that you cannot prove it otherwise. Sure, this runs into problems since you will probably grow into the realm of a theory which is not recursively enumerable, so proofs cannot be checked with an algorithm anymore. But if you believe this is all true, then why does it matter?

And while I do understand that Platonist or not, you want to work with axioms that other people accept as well, which gives us an “incomplete intersection of assumptions” (since the union is almost always inconsistent). But this is more the reason to either work towards adding more axioms to the accepted foundations of mathematics, or prove more theorems of the form “If such and such holds, then so on and so forth”. It’s not a reason to claim that a question which cannot be answered naively is not a good question.