[FOM] correction to 'constructive cauchy'
Gabriel Stolzenberg
gstolzen at math.bu.edu
Wed Jan 30 12:56:10 EST 2008
I wish to make a correction to my posting, "constructive cauchy."
Nov 9, 2007. In it, I made the following remark about the convergence
of a sequence of functions on [0,1].
> Furthermore, on a meta-level, the claim that "point-wise = uniform"
> is a consequence of the meta-theorem that an integer-valued function
> of a real variable is locally constant.
In a message to me, Ulrich Kohlenbach gave a natural reading of
this on which it is blatantly wrong. (As I recall, my thinking about
this point was so sketchy that it may be more accurate to say that I
didn't even have an incorrect argument, much less a correct one.) Even
better, he offered a correct version, which I quote:
'The correct "metatheorem" to use to infer "pointwise->uniform" is
the fan principle (one version of which actually is equivalent to
the "pointwise->uniform" statement) or the closure of the system
of reasoning one uses under the fan rule. Here one argues on the
level of representatives which for say x\in [0,1] can be chosen
in the compact Cantor space 2^\NN. So it is not the (uniform)
continuity of a function [0,1]\to\NN but of a function 2^{\NN}\to\NN
which is used.'
Gabriel Stolzenberg
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