Saturday, October 31, 2009

Is "Set" Vague?

The (normal) intuitive conception of the hierarchy of sets is roughly this:

The hierarchy starts with the empty set, at the bottom. Then, above every collection of sets at one stage, there's a successor stage containing all possible collections made entirely from elements produced at that stage. And, above every such infinite run of successor stages, there's a limit stage, which has no predecessor, but contains all possible sub-collections whose elements have already been generated at some stage below.

But how far up does this hierarchy of sets go? Is there a fact of the matter, or does our conception not determine this?

The conception/our intuitions about sets doesn't directly tell us when to stop. For any stages we suppose we are looking at, it always seems to make sense to think of new collections that contain only sets generated by that point (e.g. the collection of all things so far generated). Of the sets generated by any collection of stages we can ask:
- Does the proposed next stage/limit stage of these stages really make sense? Are there really such collections?
- If so, are the collections generated at this stage still sets?

A textbook will tell you that at some point the things generated by the process above DO make sense, but DON'T count as sets. So, for example, there is a collection of all sets, but (on pain of paradox) this is not itself a set, but only a class.

a) This just pushes the philosophical question back to classes: is there a point at which there stop being classes? Are there something else (classes2) which have the same relation to sets as sets do to classes? [One of my advisors calls this the "neopolitan" view of set theory]

b) We don't have any idea of WHEN the things generated by the process above are supposed to stop counting as sets.

Note that the issue with b) is not just that we don't know whether sets of a certain size exist. There are lots of things about math we don't know, and (imo) could never know. Rather, the uneasy feeling is that our conception doesn't "determine" an answer to this question in the following much stronger sense:

There could be two collections of mathematical objects with different structures, each of which equally well satisfies our intuitive conception of set.

For, consider the hierarchy of classes (note: all sets are classes). There might be two different ways of painting the hierarchy to say at what point the items in it stop counting as sets. Our intuitive conception just seems to generate the hierarchy of classes, not to say when things in it stop being sets!

In contrast, in the case of the numbers, I might not know whether there are infinitely many twin primes, but any two objects satisfying the intuitive, second order, characterization of the numbers would have to have the same structure (and hence make all the same statements of arithmetic true).

Thus, our intuitive conception of set seems to be hopelessly vague about where the sets end. Hence, even if you are a realist about mathematical objects, we seem forced to understand set theory as making claims about features shared by everything that satisfies the intuitive conception of set, rather than as making claims about a unique object.


1. If you buy the reasoning in the main body of this post, does it give an advantage to modal fictionalism? e.g. the modal fictionalist might say: "You already need to agree with us that doing set theory is a matter of reasoning about what objects satisfying the intuitive conception of set would have to be like. What does incurring extra commitment to the actual existence of mathematical objects (as opposed to their mere possibility) do for you?".

2. An alternative would be to reject the textbook view, and say EVERYTHING generated by the process above is a set. Hence, you couldn't talk about a class of sets. Would this be a problem?

3. [look up] Is it possible that all initial segments of the hierarchy of classes that reach up to a certain point are isomorphic? (I mean, the mere existence of one-to-one, membership preserving, function that's into but not onto -the identity- doesn't immediately guarantee that some *other*, more clever, function that IS an isomorphism)
[maybe you can prove this is not poss by using the fact that one initial segment would have extra ordinals, and this iso could be used to define an iso between ordinals of different sizes which is impossible]

4. Is there some weirdness about the idea that collections in general (whether they be sets, or classes) eventually give out - so there's no collection of all collections.

We could say there are sets, classes, classes2, classes 3 and so forth. This lets us say there's a class of all sets, and a class2 of all classes etc. But as far as collections in general we must admit that there's no collection of all collections, on pain of contradiction via Russell's paradox.

Well, I don't personally find this that problematic. It's a surprising fact about collections maybe, but mathematics often yields surprising results.

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