Tag Archives: Defending the Axioms

Re: Paper and slides on indefiniteness of CH

Dear Hugh,

On Fri, 24 Oct 2014, W Hugh Woodin wrote:

Dear Sy,

You wrote to Pen:

But to turn to your second comment above: We already know why CH doesn’t have a determinate truth value, it is because there are and always will be axioms which generate good set theory which imply CH and others which imply not-CH. Isn’t this clear when one looks at what’s been going on in set theory? (Confession: I have to credit this e-mail discussion for helping me reach that conclusion; recall that I started by telling Sol that the HP might give a definitive refutation of CH! You told me that it’s OK to change my mind as long as I admit it, and I admit it now!)

ZF + AD will always generate “good set theory”…   Probably also V=L…

This seems like a rather dubious basis for the indeterminateness of a problem.

I guess we have something else to put on our list of items we simply have to agree we disagree about.

What theory of truth do you have? I.e. what do you consider evidence for the truth of set-theoretic statements? I read “Defending the Axioms” and am convinced by Pen’s Thin Realism when it comes to such evidence coming either from set theory as a branch of mathematics or as a foundation of mathematics. On this basis, CH cannot be established unless a definitive case is made that it is necessary for a “good set theory” or for a “good foundation for mathematics”. It is quite clear that there never will be a case that we need CH (or not-CH) for “good set theory”. I’m less sure about its necessity for a “good foundation”; we haven’t looked at that yet.

We need ZF for good set theory and we need AC for a good foundation. That’s why we can say that the axioms of ZFC are true.

On the other hand if you only regard evidence derived from the maximality of V as worthy of consideration then you should get the negation of CH. But so what? Why should that be the only legitimate relevant evidence regarding the truth value of CH? That’s why I no longer claim that the HP will solve the continuum problem (something I claimed at the start of this thread, my apologies). But nor will anything like Ultimate L, for the reasons above.

I can agree to disagree provided you tell me on what basis you conclude that statements of set theory are true.

Best,
Sy

Re: Paper and slides on indefiniteness of CH

Dear Pen and Sol,

Pen: I finally got, and have read most of, Defending the Axioms. It is a beautiful book, which I thoroughly enjoyed! I am glad that you succeeded in demonstrating that some form of (thin) ontology is entirely possible without succumbing to all of the disadvantages of Platonism. (For me the big problem with Platonism is that it doesn’t allow us to learn anything about V via its comparison with other possible universes; for that we need potentialism). Also I’m happy that you agree that intrinsic considerations are perfectly viable without Platonism.

You won’t be surprised to hear that I am sympathetic to a form of Thin Realism, and in particular with what you express as follows:

The key here is that the second-philosophical Thin Realist begins from her confidence in the authority of set-theoretic methods when it comes to determining what’s true and false about sets…

Great!

But it seems to me that a much more thorough discussion of the meaning of “set-theoretic methods” is needed. We don’t want statements to be both true and false, so we need a consistent set of inferences from such methods. You do entertain both extrinsic and intrinsic sources for such inferences. Have I got it right that the source is of no real importance because the axioms that are generated are thrown into one pot (ignoring their source) and after 100 years or so we’ll know what the best set theory was and ignore what source or sources it had? Don’t you find that a bit harsh for the concept of “intrinsic”? I appreciate your AC example (if the math is better, change the concept!) but as I currently understand “intrinsic” we’re talking about the beloved MIC! Would you really trash the MIC for “good set theory”? (I guess I know your answer!)

The latter is not just a theoretical possibility, as in some sense it already happened: The original form of Maximality expressed by the IMH contradicts large cardinal existence, even if you soften it with the form of Reflection that Bill and Peter have been talking about.

But a more serious worry is: What do we do if the axioms dessirable for “good set theory” conflict with those which are best for the foundations of math outside set theory? Large cardinals and determinacy are largely irrelevant to mathematics outside of set theory (so far!) so how do you know that they don’t obstruct other axioms that are good for the foundations of math outside of set theory? Are you prepared to give up on large cardinals and determinacy if that happens? (Of course when I say “you” I mean your friend, the Thin Realist.)

Finally I address Sol. The original topic under discussion was the indefiniteness of CH. Here is my prediction:

(Type 1) Axioms for good set theory are wide and varied, some imply CH and others imply not-CH. There is nothing that we will ever be able to do about that.
(Type 2) We don’t know yet what the best axioms for the foundation of math outside of set theory will say about CH. No prediction here, and the answer will take a lot of research that has not yet even begun (I hope my grant proposal gets approved!).
(Type 3) Not-CH is derivable from the maximal iterative conception. This hasn’t been shown yet (the set theory is very hard) but has a good chance of being shown in my lifetime (I am 61).

So it doesn’t look good for resolving the continuum problem (long face). But I see no argument that CH is in principle undecidable, it is just a case of bad luck.

Many thanks to you both,
Sy