FOM: categorical dis-foundations

Vaughan Pratt pratt at cs.Stanford.EDU
Mon Feb 2 19:37:39 EST 1998

From: kanovei at (Kanovei)
>Can someone present a consistent proof, 
>from the McLarty axioms 1 - 15, of the following theorem: 
>there is a real function which equals 0 on Q and 1 on R - Q 

In set theory the disjointness of the rationals and the irrationals
follows immediately from first principles.  In category theory their
disjointness is not a foregone conclusion by any means (see below),
and I would not be surprised if the above required wellpointedness.
I also would expect that this and a few other Lakatosian monsters would
provide the only occasions where well-pointedness was needed in applying
topos theory to analysis, if such monsters can even be considered part
of analysis.

John Isbell's seminal 1972 paper "Atomless Parts of Spaces" treats as
its main example the structure of the intersection of the rationals
and the irrationals, as a nonempty but atomless part of the real line
("pointless" as locale theorists like to kid).

If you are a set theorist who wants a better idea of just how wide
the conceptual gap is between set theorists and category theorists.
try picturing Q and R-Q as having a nonempty intersection.  Hard,
isn't it?  Isbell is clearly another of those charlatans.  Go read the
paper (Math.Scand. 31, 5-32) to see how you can have a nonempty space
that contains no points.

Vaughan Pratt

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