FOM: {n: n notin f(n)}
Richard Heck
heck at fas.harvard.edu
Thu Aug 29 14:29:46 EDT 2002
Thanks, first, to Dean, for his kind welcome to the discussion.
>Let me translate my actual position into English.
>
>>>1. Let f be a function, f: N -> S, from objects to sets of objects. Suppose f is "onto"
>>>
>I'm uncomfortable with set-theoretic way of putting things precisely because it commits one to nasty things like infinite sets of objects. I'd translate this as: there is a way ("f") of associating things of a certain kind (say natural numbers) with collections of objects of the same kind. The function is "1-1 and onto" i.e. we can pair up the things and the collections. I'm using the word "kind" and collection" deliberately. By a "kind" of thing I mean roughly any thing or things that satisfy the same predicate, by collection I mean a "finite set" (but I hold that "finite" is otiose because, really, there aren't any other kinds of set).
>
The point of the translation into second-order logic was precisely to
allow for an interpretation of Cantor's argument that is compatible with
this sort of anti-set-theoretic view. Unlike the set-theoretic reading,
which commits one to these entities called sets (though not, for the
purposes of the argument, to all that much about them), the second-order
version of the argument need have no such ontological commitments. If
Boolos is right, it commits us only to what plurals do, and that may be
nothing but that to which ordinary singular reference commits us. That's
very controversial. However, and much more interestingly, because the
argument for Cantor's theorem uses only /predicative/ comprehension, the
second-order variables can, in fact, be interpreted substitutionally. To
put the point differently, assuming that "ways of associating" can be
represented by formulae, that is, that these ways can actually be
described, the second-order argument shows that there can be no way of
associating some objects with all collections of those objects, and it
needs no ontological claims about collections or what have you to make
that point. Even collections can be disposed of. All we need are
predicates, because, again, the second-order variables can be
interpreted substitutionally: One can assume, if one likes, that the
only collections there are can also be described. (One could make that
explicit, by writing the argument out schematically. I won't.)
There is a model-theoretic perspective upon this sort of situation that
is familiar from Skolem, of course. Since, again, the logic is
predicative, there will be natural models in which, say, the domain is
countable, and there are only countably many collections of objects in
that domain. So /really/, one might want to say, from outside the model,
there is a one-one map between the objects in the domain and the
collections of objects in the domain this model thinks there are. But
that function won't be in the model.
Dean is troubled that I have him "proposing a relation R such that, for
each subconcept G of F, there is some x such that Fx, where aR__ just is
G", that is, formally:
(ER)(G)[(x)(Gx --> Fx) --> (Ey)(Fy & (z)(Gz <--> yRz))].
But Dean does seem committed to such a claim. For consider his (1),
above, and now define xRy to hold just in case y is in f(x). Then R, so
defined, satisfies the condition just mentioned. In English, that is to
say, R is: y is a member of the set with which x is associated, by
whatever way of associating we have chosen. I don't see what isn't well
defined.
Dean says that his apparent commitment to this thesis is removed by his
view
> that there are two radically different kinds of predication, as follows:
> Some people were at that bar. Some of them were drunk. Some other
> people at the bar were laughing at them.
> We have "x is a person at the bar", which can be satisfied by any one
> of either of the groups of people at the bar, and we have "x was one
> of them" which can only be satisfied by one of the first group
> mentioned. I'd hold that the first kind of predicate can be true of an
> infinite number of things (and is how we get to the idea of infinity
> in the first place), and that the second kind can be true only of a
> "finite" number of individuals.
I'm afraid I don't understand that, however. First, "x is a person at
the bar" isn't satisfied by groups at all, but by people, at least if
the word "satisfied" is used as it usually is: This predicate is
satisfied by the people at the bar. But that may just be a slip. Dean
probably means that "x is a person at the bar" is satisfied by everyone
at the bar, drunk and otherwise. But the important point is supposed to
concern the predicate "x was one of them". I'm not myself clear to which
occurence of "them" Dean is alluding, as there are two, one in "Some of
them were drunk" and one in "...laughing at them". The first of these is
anaphoric on the quantifier in the first sentence, and is satisfied by
all of the people at the bar: The sentence "Some of them were drunk"
means, roughly, that some of the people at the bar were drunk. The
second occurence is satisfied by only the drunk people. As Chris Menzel
noted in his posting, it's a mystery, at present, how all these
anaphoric relations come to hold, but those do seem to be the relations
that hold.
I am perplexed, though, why there are supposed to be two kinds of
predication here. And I am even more puzzled what of it has to do with
finitude and infinitude. Consider:
Some natural numbers are odd. Some of them are powers of 3. Some
other odd numbers are relatively prime to them.
That has the same logical form as Dean's example, and it's sensible,
even true. And "them" is clearly satisfied by infinitely many numbers,
in both occurences. Note that it's a simple fact of number theory, too.
It's got nothing to do with sets and Cantor. It'd be sad to see us have
to abandon such things in our quest to avoid Cantor's result.
Chris's translation of such a story into broadly set-theoretic terms is
reasonable enough, so I won't try to do better here. As he notes,
though, many of the problems one encounters here have to do with issues
in semantics that it is hard to see matter in this context.
Richard Heck
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