[FOM] Intuitionist choice sequences
Daniel Méhkeri
dmehkeri at yahoo.ca
Sun Jul 6 18:34:32 EDT 2008
The original Brouwerian idea of "choice sequence" has to do with choices
an ideal mathematician makes over the course of time. One thinks of
this ideal fellow as a regular fellow with unlimited spare time and
scratch paper (or hard drive space) who commits to keep producing values
on request.
But there is no constraint on the value produced - it could be
arbitrarily large. This seems to go beyond even this idealisation. Even
an ideal mathematician can only produce a finite amount of information
in a given amount of time. If there really is a commitment to
"eventually" return a value, and this commitment is to be innocent of
the "fraud" attributed to non-constructive proofs, then it seems that
it would have to include an honest bound on time, which therefore seems
to imply a bound on value.
So shouldn't choice sequences be restricted to a finite number of
choices at each stage? (It seems without loss of generality we can
call it two choices.) Would this have any effect on intuitionist
analysis? It seems that it should - it changes the topology of
the universal spread from Baire space to Cantor space. Has this
question been treated anywhere?
-DM
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