Asaf Karagila
I don't have much choice...

Is the Continuum Hypothesis a definite problem?

There are 4 comments on this post.

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.

There are 4 comments on this post.

By Alon Amit
(Aug 09 2015, 13:19)

Do you feel that sentences in the first order language of arithmetic have a truth value?

By Asaf Karagila
(Aug 09 2015, 13:30 In reply to Alon Amit)

That depends on what you mean a truth value. The statement \(1 1=0\) is a perfectly valid atomic sentence in the language of arithmetic. But is it true or false? That depends on the structure you interpret it in. If you interpret it in a model of Peano arithmetic, then the answer is of course negative. But if you interpret it in modular arithmetic, then it can have a positive answer.

Does the continuum hypothesis have a truth value? Again, depends where. Remember that the language of set theory is just a language with a single binary relation symbol. You can interpret the continuum hypothesis in \((\mathbb{Q},\leq)\) or \((\mathbb{Z},\equiv_{42})\) just fine, and ask if it is true or false there. It's just an extremely weird thing to do. :-)

By Alon Amit
(Aug 09 2015, 18:43)

Yes, sure. I mean when interpreted as a statement about the natural numbers.

(I'm deliberately not asking about models of PA, rather just the natural numbers. As in "the standard model" of PA, or as in Tarski's definition of truth.)

By Asaf Karagila
(Aug 09 2015, 19:01 In reply to Alon Amit)

Well. That's a tricky question to answer, and I again I will have to give the "agnostic evasive" answer. Yes, every statement has a truth value, but it may depend on additional information.

For example "\(\ZF\) is consistent" is a statement about the natural numbers which may or may not be true in the standard model, depending on your universe of sets. I don't believe there is a concrete single universe to which we need to align our notion of "truth". Our experience should give us some sort of guidance, but only so far. So after a while you try to prove that something is inconsistent and if you couldn't do it, then maybe it's safe to assume that it's a reasonable candidate for a truth (rather than "the" truth) of mathematics.

I am not arguing in favor of "truth is what people believe is truth", since I don't think that gods lived on the Olympus some couple of millennia ago. But "truth" is something that depends on context, as a fickle and meaningless speck of existence, your truth might end up being false in the global scheme of things.

So is there a particular truth value? I don't believe in that, although I understand why many people find it satisfying philosophically to believe that there is such a thing, and their work is "true" in that deep sense.

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