[FOM] reply to shipman: countably uncomfortable
pratt at cs.stanford.edu
Thu Jan 31 22:11:55 EST 2008
Gabriel Stolzenberg wrote:
> 1. I was referring to the uncomfortable feeling a mathematician
> may get when she realizes that the system in which we prove that
> the reals are uncountable is countable. Yes, we all know the
> things that can be said to reassure her but I can't think of any
> that would be likely to work.
Things viewed in a mirror have some aspects reversed, but modulo that
difference we tend to view the object and its reflection as otherwise
the same in appearance.
Thinking of duality as a kind of mirror which merely reverses the
direction of the morphisms, Pontryagin duality (the duality of locally
compact abelian groups where the morphisms are the continuous group
homomorphisms) reflects (dualizes) the additive group Z of integers as
the additive group T of reals mod 1, that is, the circle group,
equivalently the multiplicative group of complex numbers z satisfying
|z| = 1, furnished locally with the usual topology on the reals.
So we have an object, the integers, that we usually think of as
countable, becoming an uncountable object when viewed in the mirror of
Pontryagin duality. Yet the two groups are "the same" in that they
transform in the same way when we make the correspondence between the
morphisms to Z with the morphisms from T and vice versa. Put more
concretely, the elements of any locally compact abelian group G
(identifiable with the morphisms from Z to G) become the group
characters of the dual G' of G (identifiable with the morphisms from G'
to the circle group T) and vice versa.
This seems to me a more intimate relation between a countable object and
an uncountable one than the one you mention above between a universe and
its theory, except when the theory captures the universe so precisely as
to actually be its dual, which probably isn't the case in the example
you're thinking of.
Hence Pontryagin duality should make one feel even more uncomfortable.
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