[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?


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