the Continuum Hypothesis is true

[gusl]

I'm normally not very interested in controversies about foundational axioms for mathematics, since they tend to concern infinities that are too abstract to mean anything to me. Would any scientists care if ZF were all they could work with?
Furthermore, I'm not a Platonist, and I don't like to talk about things in mathematics as being "true" (long ago, I added another argument to this predicate: "theory T is true" became "theory T is good model of domain D", whether D can refer to things in the physical world or to intuitions).

But yesterday, my boss, who is not an academic, let alone a logician, led me to the following insight: in a certain sense, CH is *true*. The independence result implies that one cannot construct a set S such that |Naturals| < |S| < |Reals| (if you could, this would be a proof that ~CH follows from ZFC. I wish I understood how to define this constructibility in terms of axioms).

Any sets that you stick in there are "made up" and cannot be exhibited concretely (not that the Reals is very concrete either, but adding the limits of all Cauchy sequences seems like a rather natural way to construct it).

My boss was thinking about the problem in concrete terms, something which I hadn't done in a long time.

Comments

[rdore.livejournal.com]

What you're describing is Godel's constructible universe, L. Even if you start from just ZF, in L, you get ZFC, CH, and much more. The main objection to using L is that it is too "thin". It has so few sets, that you get unusual pathologies such as very "simple" well orderings of the real numbers. (And as such, very simple unmeasurable sets, etc). One of the central questions of current set theory is how to pick something which is L like, but not as pathologically restrictive. I'm being a bit vague since the technical details quickly get quite messy.

[gustavolacerda.livejournal.com]

Does L have more axioms than ZFC, or does it depend on some sort of constructivist interpretation of things, like "that which cannot be constructed does not exist"?

[rdore.livejournal.com]

Yes, you can make V=L an axiom:

Take L₀ = empty set.
Then you just induct (transfinitely):
L_a+1 is the stuff definable from L_a union {L_a}
And at limit stages you just union everything up.

Then your axiom is just:
for all sets x, there is some level L_g which contains x.

This is all done in standard first order logic. The constructability comes from the notion that every set can be built up in a definable way, it has nothing to do woith intuitionistic logic.

[gustavolacerda.livejournal.com]

Can you show me an example of a set that cannot be constructed in this way?

[rdore.livejournal.com]

I'm not sure what you mean by "show". In some sense, I can't even "show" you the natural numbers. Also, how natural of things I can show you is sensitive to what universe you start in.

[gustavolacerda.livejournal.com]

It has so few sets, that you get unusual pathologies such as very "simple" well orderings of the real numbers. (And as such, very simple unmeasurable sets, etc)

I see nothing wrong with things being simple.

[rdore.livejournal.com]

Well say I take an open subset of Rⁿ.
Then all I do is project it to lower dimensions and take complements.
In L, it is possible to get a nonmeasurable set this way.

There's plenty of reasons why L is strange like this in pathological ways, which is why no one who studies the subject believes that V=L is the "right" axiom (whatever that means).

Really the problem is not with being interested in constructible sets. The problem is that when you assume that everything must be constructible, because then it allows you to do weird logicy things on top of that constructibility.

[gustavolacerda.livejournal.com]

Are we talking Lebesgue-measurability? I would like to see this construction somewhere. Does it have a name? Or are you referring to a non-constructively-proven existence theorem?

This reminds me of the Banach-Tarski "paradox", which is counterintuitive at first sight, until you see how the construction is actually made.