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.

V=L

[r6.livejournal.com]

More generally you are assuming the V=L (http://en.wikipedia.org/wiki/Axiom_of_constructibility) axiom (schema?). This implies the generalized continuum hypothesis.

Re: V=L

[gustavolacerda.livejournal.com]

Great!

I like this part:

A constructible set is any set that can be outputted by a transfinite computation.

I just wish I knew what I means for a computation to output a set. What sorts of encodings are allowed?

P.S.: WTF?? The Wikipedia article references Joel Hamkins. What a small, small world.

[brianfey.livejournal.com]

hey smart guy! I wonder sometimes...
What would happen if we had a chat. Likely you would dismiss me as a dumbfuck.
I used to be smart... but then I wandered. Somme smart people get pissed off at my statements which include the contradictions of things as they seem to be.

In any case... I am impressed with your smartness.
But I do wonder where is goes really.
I wonder that these lofty thoughts lead to when it comes to day to day living. Perhaps they are some key, or perhaps a distraction.

Life is long. Perhaps we will blather about such things around a campfire or wiki someday.

I hope so.
And even though I don't understand many of your posts... thanks for posting them... it is a wonderful world all those bloggers make.

Irrelevence

[r6.livejournal.com]

I wonder that these lofty thoughts lead to when it comes to day to day living. Perhaps they are some key, or perhaps a distraction.

They are just distractions. Answers to questions about the axiom of choice and the generalized continuum hypothesis basically do not affect day to day living. This is because if &phi is an arithmetic statement (i.e. a theorem that could perhaps affect day to day living) then if ZFC + GHC ⊢ φ then ZF ⊢ φ.

But distractions can be fun to play with.

Re: Irrelevence

[gustavolacerda.livejournal.com]

GHC -> GCH?

Isn't this also the case with *all* controversial axioms?

Re: Irrelevence

[r6.livejournal.com]

Yes I meant GCH.

I believe large cardinal axioms allow you to prove more arithmetic statements, but maybe this means that they are not controversial.

Re: Irrelevence

[rdore.livejournal.com]

Well large cardinal statements will allow you to prove stuff like Con(ZFC), which can be coded as an arithmetical statement.

[spoonless.livejournal.com]

Interesting that the Wikipedia page says Godel took the independence result as evidence that the CH was false. I don't understand that reasoning, especially since he seemed to be involved with this "axiom of constructibility" which can prove that it's true.

I'm a Platonist in that I think mathematics is about describing real stuff, not just playing around with formal axioms. However, I don't think the mathematical world is separate from the physical world... it's just a lot larger and includes things we haven't necessarily encountered (yet) but could, in some sense. I think there are often valid empirical reasons to reject or accept certain axioms. I also think axioms shouldn't be the basis of mathematics, but I'm not sure what should replace them.

Lots of Sets

[r6.livejournal.com]

I imagine, but am not certain, that Gödel wanted set theory to contain as many sets as possible. So any sets that that aren’t explicitly disallowed ought to exist.

Re: Lots of Sets

[spoonless.livejournal.com]

Interesting, I wonder if anyone has proposed a sort of inverse-Occam's razor for mathematics... always choose the axioms which give rise to the most structure, unless there is a reason why said structure is inconsistant. I kind of like that. It goes along nicely with a principle of physics known as the totalitarian principle: "everything which is not forbidden is compulsary".

Re: Lots of Sets

[henriknordmark.livejournal.com]

Yes, this has been proposed.
It's called the Maximize Principle.

Penelope Maddy was the first person to explicitly state this and advocate for it.

I am not entirely sure whether she is still an advocate for this principle after her conversion from being a realist to being a naturalist.

[lingboy.livejournal.com]

rdore's response was basically the same as mine. If you restrict yourself to just constructible sets (in the set theoretic notion of constructible), then you get L, and you get CH. If you don't assume that V=L, it may be the case that with AC you build a set which is interesting, whose cardinality you discover is aleph_1. But then how big is that really? Is that the same size as the reals? You would know that there exists an injection from the set into the reals. As for the other direction, you wouldn't be able to prove or disprove that such an injection from the reals to the set exists, let alone be able to construct one. So if you don't assume V=L, you can still have constructible sets S such that |Naturals}< |S| ≤ |Reals|, and you simply would not be able to prove whether that ≤ should be an < or an =.

by the way, this is krasnoludek.