Tag Archives: CH

Re: Paper and slides on indefiniteness of CH: My final mail to the Thread

Dear Sol,

My participation in this interesting discussion is now at its end, as almost anything I say at this point would just be a repeat of what I’ve already said. I don’t regret having triggered this Great Debate on July 31, in response to your interesting paper, as I have learned enormously from it. Yet at the same time I wish to offer you an apology for the more than 500 subsequent e-mails, which surely far exceeds what you expected or wanted.

Before signing off I’d like to leave you with an abridged summary of my views and also give appropriate thanks to Pen, Geoffrey, Peter, Hugh, Neil and others for their insightful comments. Of course I am happy to discuss matters further with anyone, and indeed there is one question that I am keen to ask Geoffrey and Pen, but I’ll do that “privately” as I do think that this huge e-mail list is no longer the appropriate forum. My guess is that the vast majority of recipients of these messages are quite bored with the whole discussion but too polite to ask me to remove their name from the list.

All the best, and many thanks, Sy

Continue reading

Re: Paper and slides on indefiniteness of CH

I have two titles to this note. You get to pick the title that you want.

REFUTATION OF THE CONTINUUM HYPOTHESIS THE PITFALLS OF CITING “INTRINSIC MAXIMALITY’

Note that the most fundamental and simple nontrivial equivalence relation on the set theoretic universe is that of “being in one-one correspondence”.

Also very fundamental and simple is the equivalence relation EQ on infinite sets of reals “being in one-to-one correspondence”.

Note that it is consistent with ZFC that this fundamental simple EQ has

i. exactly two equivalence classes. ii. infinitely many equivalence classes.

THEREFORE, by the “intrinsic maximality of the set theoretic universe”, ii holds. THEREFORE, we have refuted the continuum hypothesis (smile).

NOTE: A lesson that can be drawn here is just how important it is to avoid cavalier quoting of “intrinsic maximality of the set theoretic universe”.

In fact, if we factor, we are looking at a set for which it is consistent with ZFC that it has, on the one hand, exactly two elements, and on the other hand, is infinite. So by “intrinsic maximality of the set theoretic universe”, it must be infinite (smile).

GENERAL PRINCIPLE. Let EQ be a simple equivalence relation. Suppose ZFC + “EQ has infinitely many equivalence classes” is consistent. Then EQ actually has infinitely many equivalence classes.

Here is the legitimate foundational program.

  1. Set up an elementary language that is based on only some of the most set theoretically fundamental notions.
  2. Determine which “simple” definitions define equivalence relations. Show that this is robust, in that here truth is the same as provability in ZFC and in ZC.
  3. Determine what is consistent with ZFC about the number of equivalence classes of items in 2.
  4. Now apply the general principle, and show that the resulting statements are (even collectively?) consistent with ZFC. Perhaps the general principle will be seen to be equivalent over ZFC to “the continuum is greater than \aleph_\omega” or perhaps some versions of not GCH?
  5. Rework 1-4 with ever stronger elementary languages and ever less “simple” definitions, until one hits a brick wall.

The immediate problem is to get a good prototype for this elementary language. We want it to be not ad hoc, and so should be in tune with the most basic set theoretic material.

While doing this real time foundations, it now appears, provisionally, that we are best off using “there is a function from x onto y” and not just “there is a bijection from x onto y”. The former is more flexible than the latter, and still very very basic for elementary set theory.

ELST = elementary set theory. We have

  1. Equality, and union operator (set of all elements of elements).
  2. There is a function from x onto y. Written x\geq y.
  3. Convenient to have variables range only over infinite sets.

Something interesting has arisen. This language supports even more naturally the 3-ary relation

T(x,y,z) if and only if

i. The union of y and the union of z are both x.
ii. y \geq z\geq x and z \geq y \geq x.
iii. (Implicitly, x,y,z are infinite).

Two observations.

  1. We have defined T as a conjunction of a small number of atomic formulas in x,y,z, with no nesting of the union operator.
  2. For all x, T_x is an equivalence relation.

Thus there is great simplicity here. We can provisionally concentrate on just cases of 1.2, even perhaps with a limit on the number of atomic formulas. We can also relax the “no nesting”.

So we have a parameterized equivalence relation. We should look at a modified General Principle.

GENERAL PRINCIPLE. Let T be a 3-ary parameterized equivalence relation. Suppose ZFC + “EQ has infinitely many equivalence classes” is consistent. Then EQ actually has infinitely many equivalence classes.

In this way, we should be getting the robustness referred to above, and also the failure of GCH at every infinite cardinal.

I’ll stop here with this provisional beginning…

Harvey

Re: Paper and slides on indefiniteness of CH

Dear Pen,

On Fri, 24 Oct 2014, Penelope Maddy wrote:

Dear Sy,

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?

Well, I’m not sure it is clear that there will never be a theory whose virtues swamp the rest.

What evidence do you see for the existence of such a theory? All the evidence points to the contrary: The number of different valuable directions in set theory just keeps multiplying.

What I could imagine is that a particular truth value of CH will be required for an optimal foundation for mathematics (Type 2 evidence), but that is just wild speculation at this point. If that occurred, then maybe it would tip the balance between axioms which are valuable for the mathematical development of set theory (Type 1 evidence) yet give different verdicts on CH.

I am however inclined to think that Type 3 evidence (the HP) will not have as much influence as Type 2 evidence, simply because people regard the foundations of mathematics as more important than what can be derived from the maximality of the set concept.

 

Is CH one of the leading open questions of set theory?

No! The main reason is that, as Sol has pointed out, it is not a mathematical problem but a logical one. The leading open questions of set theory are mathematical.

I didn’t realize that you’d been convinced by Sol’s arguments here.  My impression was that you thought it  might be possible to resolve CH mathematically:

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!

That’s why I posed the question to you as I did.

You misunderstood me. I didn’t need Sol to convince me that CH is a logical but not mathematical problem. The HP is a programme based on logic, so any conclusion about CH via the HP would be a logical solution, not a mathematical one.

With apologies to all, I want to say that I find this focus on CH to be exaggerated. I think it is hopeless to come to any kind of resolution of this problem, whereas I think there may be a much better chance with other axioms of set theory such as PD and large cardinals.

PD has a decent chance of winning the blessing of all 3 forms of evidence: It may be that you need it for the best set theory (as mathematics), if mathematicians ever start worrying about the higher projective levels they may appreciate having the Lebesgue measurability of the projective sets, and as Hugh and I mentioned, PD may be a consequence of maximality according to the HP. (Hugh if you thought that I was claiming otherwise then you got confused; where did I say that?) Of course it is too soon to come to any definitive conclusion about PD, but there is a fighting chance for its truth.

I am less optimistic about large cardinals. This past week I was at AIM (American Institute of Mathematics) and based on the work we did I am willing to conjecture that the following is consistent:

(*) Every uncountable cardinal is inaccessible in HOD (the hereditarily ordinal definable sets).

(Cummings, Golshani and I already got the consistency of a weaker version of this: alpha^+ of HOD is less than alpha^+ for all infinite cardinals alpha.)

Now (*) is clearly a maximality principle, but the mathematical evidence is that it contradicts the existence of large cardinals. Indeed, the consistency proofs of these “V is fatter than HOD” principles break when you try to accomodate large cardinals, and indeed Hugh has plausible conjectures which imply that such obstacles are unsurmountable.

So where this is pointing is that maximality denies large cardinal existence. This happened before with the IMH but that got fixed by marrying the IMH to a vertical maximality principle; I don’t see how large cardinals are going to escape from this latest dilemma.

All the best,
Sy

Re: Paper and slides on indefiniteness of CH

Dear Sy,

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?

Well, I’m not sure it is clear that there will never be a theory whose virtues swamp the rest.

 is CH one of the leading open questions of set theory?

No! The main reason is that, as Sol has pointed out, it is not a mathematical problem but a logical one. The leading open questions of set theory are mathematical.

I didn’t realize that you’d been convinced by Sol’s arguments here.  My impression was that you thought it  might be possible to resolve CH mathematically:

 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!

That’s why I posed the question to you as I did.

All best,
Pen

Re: Paper and slides on indefiniteness of CH

Dear Sy,

This doesn’t really bear on any of the debates we’ve been having, but …

It seems to me disingenuous to suggest that resolving CH, and devising a full account of sets of reals more generally, is not one of the goals of set theory — indeed a contemporary goal with strong roots in the history of the subject.

Good luck selling that to the ST public. This is interesting to you, me and many others in this thread, but very few set-theorists think it’s worth spending much time on it, let’s not deceive ourselves. They are focused on “real set theory”, mathematical developments, and don’t take these philosophical discussions very seriously. … Resolving CH was certainly never my goal; I got into the HP to better understand large cardinals and internal consistency, with no particular focus on CH. … It would be interesting to ask other set-theorists (not Hugh or I) what the goals of set theory are; I think you might be very surprised by what you hear, and also surprised by your failure to hear “solve CH”.

The goal I mentioned was resolving CH as part of a full theory of sets of reals more generally. I said ‘resolving’ to leave open the possibility that the ‘resolution’ will be a understanding of why CH doesn’t have a determinate truth value, after all (e.g., a multiverse resolution).

It’s not a matter of how many people are actively engaged in the project: there might be lots of perfectly good reasons why most set theorists aren’t (because there are other exciting new projects and goals, because CH has been around for a long time and looks extremely hard to crack, etc.). I would ask you this: is CH one of the leading open questions of set theory? Is it the sort of thing that would draw great acclaim if someone were to come up with a widely persuasive ‘resolution’?

All best,
Pen

Re: Paper and slides on indefiniteness of CH

Dear Pen,

It seems to me disingenuous to suggest that resolving CH, and devising a full account of sets of reals more generally, is not one of the goals of set theory — indeed a contemporary goal with strong roots in the history of the subject.

Good luck selling that to the ST public. This is interesting to you, me and many others in this thread, but very few set-theorists think it’s worth spending much time on it, let’s not deceive ourselves. They are focused on “real set theory”, mathematical developments, and don’t take these philosophical discussions very seriously.

Surely doing serious set-theoretic mathematics with the hope of resolving CH isn’t a mere ‘philosophical discussion’!

I agree, and I did not mean to imply that the discussion was only philosophical. But my belief is that there are at most 3 or 4 set-theorists actually engaged in the attempt to resolve CH. Resolving CH was certainly never my goal; I got into the HP to better understand large cardinals and internal consistency, with no particular focus on CH. But as this thread began with Sol’s paper on CH, I have been naturally talking about what the HP could offer to that problem. (In any case you already know my views on CH: There will never be a Type 1 solution, we don’t know if there will be a Type 2 solution and I expect a Type 3 refutation.) But if CH motivates Hugh to do good set theory then that is valuable. The motivation fo the HP is much broader than the continuum problem.

In any case, for the record, only the foundational goal figured in my case for the methodological principles of maximize and unify. The goal of resolving CH was included to illustrate that I wasn’t at all claiming that this is the only goal of set theory. Your further examples will serve that purpose just as well:

The goals I’m aware of that ST-ists seem to really care about are much more mathematical and specific, such as a thorough understanding of what can be done with the forcing method.

It would be interesting to ask other set-theorists (not Hugh or I) what the goals of set theory are; I think you might be very surprised by what you hear, and also surprised by your failure to hear “solve CH”.

Best,
Sy

Re: Paper and slides on indefiniteness of CH

Dear Sy,

I think it’s fair to say that contemporary set theory also has the goal of resolving CH somehow.

No, Type 1 considerations (ST as a branch of math) are not concerned with resolving CH, that is just something that a handful of set-theorists talk about. The rest are busy developing set theory, independent of philosophical concerns. Both Hugh and I do lots of ST for the sake of the development of ST, without thinking about this philosophical stuff. Philosophers naturally only see a small fraction of what is going on in ST, for the simple reason that 90% of what’s going on does not appear to have much philosophical significance (e.g. forcing axioms).

It seems to me disingenuous to suggest that resolving CH, and devising a full account of sets of reals more generally, is not one of the goals of set theory — indeed a contemporary goal with strong roots in the history of the subject.

Good luck selling that to the ST public. This is interesting to you, me and many others in this thread, but very few set-theorists think it’s worth spending much time on it, let’s not deceive ourselves. They are focused on “real set theory”, mathematical developments, and don’t take these philosophical discussions very seriously.

Surely doing serious set-theoretic mathematics with the hope of resolving CH isn’t a mere ‘philosophical discussion’!

In any case, for the record, only the foundational goal figured in my case for the methodological principles of maximize and unify. The goal of resolving CH was included to illustrate that I wasn’t at all claiming that this is the only goal of set theory.  Your further examples will serve that purpose just as well:

The goals I’m aware of that ST-ists seem to really care about are much more mathematical and specific, such as a thorough understanding of what can be done with the forcing method.

All best,

Pen

Re: Paper and slides on indefiniteness of CH

Dear Pen,

On Sun, 19 Oct 2014, Penelope Maddy wrote:

Dear Sy,

For present purposes, what matters is that set theory has, as one of its goals, the kind of thing Zermelo identifies. This is part of the goal of providing the sort of foundation that Claudio and I were talking about (a kind of certification and a shared arena).

I interpreted the Zermelo quote to mean that ST’s task is to provide a useful foundation for mathematics through a mathematical clarification of ‘number’, ‘order’ and ‘function’, Is that correct? This goal is then Type 2, i.e. concerned with ST’s role as a foundation for mathematics.

Yes, in your classification (if I’m remembering it correctly), this would be a Type 2 goal, that is, a goal having to do with the relations of set theory to the rest of mathematics. (My recollection is that a Type 1 goal is a goal within set theory itself, as a branch of mathematics, and Type 3 is the goal of spelling out the concept of set, regardless of its relations to mathematics of either sort, as a matter of pure philosophy.)

Well, I never talked about “goals” really, but simply about evidence for the truth of set-theoretic assertions. Yes, Type 1 refers ST as a branch of math and Type 2 refers to ST as a foundation for math, with their corresponding forms of evidence. Type 3 concerns evidence for set-theoretic assertions resulting from an analysis of the set-concept, so it is not just a matter of pure philosophy, it is the generation of mathematical assertions which are evidenced through such an analysis.

I don’t see that it’s being Type 2 in any way disqualifies it as a goal of set theory, with attendant methodological consequences.

I agree! I was just trying to understand your use of the word “foundation” in your message.

It’s true that set theory has been so successful in this role and is now so entrenched that it’s become nearly invisible, and neither set theorists nor mathematicians generally give it much thought anymore, but it was explicit early on and it remains in force today (as that recent quotation from Voevodsky indicates).

Good point.

I think it’s fair to say that contemporary set theory also has the goal of resolving CH somehow.

No, Type 1 considerations (ST as a branch of math) are not concerned with resolving CH, that is just something that a handful of set-theorists talk about. The rest are busy developing set theory, independent of philosophical concerns. Both Hugh and I do lots of ST for the sake of the development of ST, without thinking about this philosophical stuff. Philosophers naturally only see a small fraction of what is going on in ST, for the simple reason that 90% of what’s going on does not appear to have much philosophical significance (e.g. forcing axioms).

It seems to me disingenuous to suggest that resolving CH, and devising a full account of sets of reals more generally, is not one of the goals of set theory — indeed a contemporary goal with strong roots in the history of the subject.

Good luck selling that to the ST public. This is interesting to you, me and many others in this thread, but very few set-theorists think it’s worth spending much time on it, let’s not deceive ourselves. They are focused on “real set theory”, mathematical developments, and don’t take these philosophical discussions very seriously.

To say this is in no sense to deny that you and Hugh and other set theorists have many other goals besides. (Incidentally, I don’t see why you think forcing axioms are of no interest to philosophers, but let that pass.)

There are others.

Such as? I think that just as the judgments about “good” or “deep” ST must be left to the set-theorists, perhaps with a little help from the philosophers, so must judgments about “the goals of set theory”.

I haven’t attempted to list other goals because, as a philosopher, I’m not well-placed to do so (as you point out).

The goals I’m aware of that ST-ists seem to really care about are much more mathematical and specific, such as a thorough understanding of what can be done with the forcing method.

Going back to:

“I realize this is frustrating for you, but what you’ve said so far about the ‘reduction to ctms’ hasn’t yet produced understanding (for me) or conviction (for Hugh or Harvey).
I’m still stuck, as above, on figuring out where the ctms live, what makes everything ‘ideal’, and so on. They have other concerns.”

Pen: If you have questions about this, please pose them.

Harvey: I repeat the argument below (for the 5th time? Harvey, please read these e-mails!). As a mathematician you should have no trouble following it.

It is what I wrote to Pen on 13.October:

Maybe this will help: If a first-order sentence holds in all countable transitive models of ZFC then it holds in all transitive models of ZFC. That is LS (Löwenheim-Skolem). Now we want the same for “maximal” models (if a first-order sentence holds in all “maximal” countable transitive models of ZFC then it holds in all “maximal” transitive models of ZFC). Now this is not obvious because “maximality” in its various forms is not first-order. And this kind of transfer from countable to arbitrary doesn’t work for second-order sentences. But the point of my mail to Geoffrey is that “maximality” as treated in the HP is only slightly worse than first-order, it is first-order in a “slight lengthening” of the model, and this is good enough to apply LS again.

More details: Take the IMH as an example. It is expressible in V-logic. And V-logic is first-order over the least admissible (Goedel-) lengthening of V (i.e. we go far enough in the L-hierarchy built over V until we get a model of KP). We apply LS to this admissible lengthening, that’s all.

It is not a hard argument, so I don’t know what all the fuss is here. Of course I’m willing to answer any questions but I don’t want to just keep repeating myself!

Hugh hasn’t complained about the reduction to the Hyperuniverse, but for some reason he feels that this reduction is of no importance and has decided that all that matters is the end-product of that reduction, an analysis of ctm’s. Then he goes further and makes no distinction between the study of maximality criteria for ctm’s and the much bigger study of ctm’s in general! I really don’t understand why Hugh makes these moves. As with Harvey, anything I can say about this would just be a repetition of what I have already said many times.

Pen, can you be of some help here? How are your “negotiating skills”?

Thanks,
Sy

Re: Paper and slides on indefiniteness of CH

Dear Pen,

I agree completely with Pen, but would like a clarification of just one thing: What do you mean by “the goals of set theory”? You have used that phrase before and I think it could very easily be misinterpreted. Do you just mean what you attribute to Zermelo above, or something more?

For present purposes, what matters is that set theory has, as one of its goals, the kind of thing Zermelo identifies.  This is part of the goal of providing the sort of foundation that Claudio and I were talking about (a kind of certification and a shared arena).

I interpreted the Zermelo quote to mean that ST’s task is to provide a useful foundation for mathematics through a mathematical clarification of ‘number’, ‘order’ and ‘function’, Is that correct? This goal is then Type 2, i.e. concerned with ST’s role as a foundation for mathematics.

I think it’s fair to say that contemporary set theory also has the goal of resolving CH somehow.

No, Type 1 considerations (ST as a branch of math) are not concerned with resolving CH, that is just something that a handful of set-theorists talk about. The rest are busy developing set theory, independent of philosophical concerns. Both Hugh and I do lots of ST for the sake of the development of ST, without thinking about this philosophical stuff. Philosophers naturally only see a small fraction of what is going on in ST, for the simple reason that 90% of what’s going on does not appear to have much philosophical significance (e.g. forcing axioms).

There are others.

Such as? I think that just as the judgments about “good” or “deep” ST must be left to the set-theorists, perhaps with a little help from the philosophers, so must judgments about “the goals of set theory”. I suspected that there was a misunderstanding here, and I was right.

It is very hard to formulate convincing “goals” for ST. The field constantly develops in ways that we cannot predict.

You’re probably wondering:  what makes a goal legitimate?  Could we just set up any old goal and justify whatever we want to do that way?   Perhaps my answer is predictable by now:   goals are legitimate insofar as they generate ‘good’ (‘deep’) mathematics.  We have pretty good evidence that the sort of foundational goal in play here has been immensely productive.  Harvey thinks the goal of resolving CH is unlikely to be legitimate in this sense, but others (obviously) disagree.  Time will tell.

I don’t think that the good set theory that comes out of programmes to clarify truth in ST is as good as other forms of good ST, at least not yet. Pen, the ST associated to these philosophical discussions is not central, but somewhat peripheral to the subject! Let’s not lose perspective here.

Best,
Sy

Re: Paper and slides on indefiniteness of CH

Pen,

You wrote to Sy,

“OK!  So we have an answer to the question Peter has been asking:  are you an actualist or a potentialist?  Answer:  a potentialist.  So you aren’t really out to settle CH in the ordinary way people think of that project; you aren’t out to discover new things about V (because there is no V).”

But here, ‘V’ can be replaced by “any standard initial segment of a (not “the”!) cumulative hierarchy of sets with full power sets up to and including rank \omega+2, for CH is determinate (semantically) there. And to accommodate the proof theory of “the ordinary way people think of that project”, one can respect that by replacing “new things about V” with “new things about any standard (well-founded with full power sets) model of T, where T is the relevant extension of ZFC for the result in question. This seems to be at least one way in which a potentialist can respect mathematical practice of higher set theory.