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Vaughan Pratt <pratt@xxxxxxxxxxxxxxx> writes:
> While I'm really enjoying this great thread, at the same time I'm
> really bothered that no one has mentioned what to me is by far the
> biggest problem with Cantor's theory, namely its claim to
> universality.
This is certainly the most thoughtful post in this thread. It is a
symptom of Usenet, I guess, that it has generated only one other
response that I see.
The first part of the post (regarding V_4, etc.) seems to me to be the
usual structuralist argument that set theory somehow misses the
essence by forcing one to pick representations. Something like
Benacerraf's point in "What numbers cannot be", right?
The second part is new to me. It seems to be something like this: 2^A
is naturally structured. It is a Boolean algebra. Thus, 2^- should
be viewed as a functor from Set to BoolAlg. But then 2^(2^-) only
makes sense if the outer 2^- is a functor with domain BoolAlg, and
hence should take Boolean algebras B to the set of *Boolean*
homomorphisms of B to 2. Here, I guess, you naturally regard 2 as the
Boolean algebra (0 < 1).
You then deduce that for finite sets A, 2^(2^A) "should" equal A. For
the infinite case, you notice that 2^A is complete and so *there* you
interpret 2^(2^A) as the set of *complete* Boolean homomorphisms from
2^A to 2. Again, you get the same result: 2^(2^A) = A, or at least it
should[1].
Then you mention the role of forgetful functors and this and that
(including Stone duality, which always hurt my poor addled brain).
But the niggling point I don't get is the different cases for the
finite and infinite sets A. When A is finite, you view 2^A as a
functor from Set to BoolAlg and otherwise to CompBoolAlg, but this ad
hoc change of codomain seems to strongly undercut the naturalness of
your analysis, doesn't it? Without it, you lose the identity, but
with it, the whole "should" seems much weaker: it *should* only if
2^(2^A) retains those features of 2^A which come from A's being
finite/infinite[2]. But why those features and not other features of
A? Or why *any* features of A per se? Why not just BoolAlg?
Anyway, I hope I haven't missed the point entirely. Thanks for any
comments.
Footnotes:
[1] I don't know the proper notation for "should =", but I suppose
that it is O(x = y) (where O is the deontic circle).
[2] Is there a term like "parity" which applies to the
finite/infinite dichotomy?
--
"I am a force of Nature. Time is a friend of mine, and We talk about
things, here and there. And sometimes We muse a bit [...] and then We
watch them go... in the meantime, Time and I, We play with some of
them, at least for a little while." --- JSH and His pal, Time.
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