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

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