Tag Archives: Goals of set theory

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,

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.)

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. 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).

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. 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).

All best,
Pen

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

Dear Sy,

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

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.
All best,
Pen