Subject: categories: unbounded choice

From: Peter Freyd <[email-masked]>
Date: Thu, 30 Mar 2000 07:57:56 -0500 (EST)

Max and Mike both mentioned that there are addition-preserving
self-maps on the reals that are unbounded on every non-trivial
interval. But such examples require the axiom of choice. In a
different context I happened to mention that just two months ago:

  Do we need the axiom of choice? If, instead, we add the axiom of 
  measurability, then the counterexamples disappear: let  f  be a 
  measurable midpoint-preserving map from  R  to  R;  we can easily
  specialize to the case that  f0 = 0, hence  f  is a measurable group
  endomorphism; for any real  a  consider the induced group 
  homomorphism from  R/aZ  to  R/(fa)Z;  any measurable group
  character is continuous and that's enough to force  f  to be
  continuous. (28 Jan)

The example that Michael Wendt and I described (well, almost the same
example) we didn't need choice but we did need -- as all good
toponomers know -- the law of excluded middle for the equality
predicate on the reals (in the category of spatial sheaves over a
perfect space it is the case that all real-valued maps from a closed
interval are, indeed, bounded.)