FOM: Real analysis in a topos  Colin Mclarty cxm7 at Tue Jan 13
15:24:37 EST 1998  Reply to message from simpson at of Mon, 12
Jan >Has there ever been a decent >foundational scheme based on
functions rather than sets?  I know that >some category
theorists want to claim that topos theory does this, but >that
seems pretty indefensible.  For one thing, there doesn't seem to
>be any way to do real analysis in a topos.         There is a
large literature on real analysis in toposes, with applications
to classical analysis, Brouwer's analysis, and recursively
realizable analysis, among other topics.           The best
single reference would be Springer Lecture Notes in Mathematics
no.753, with many articles on real analysis. This was  published
in 1979.           The definitions are exactly like real
analysis in ZF. The difference is that, because excluded middle
and the axiom of choice  fail in most toposes, the set of
Dedekind reals is in general much larger than the set of Cauchy
reals. On the other hand, in the  realizable case, e