[FOM] Re: Excluded middle & cardinality of the reals
ronzitti at nous.unige.it
Wed Jun 30 16:00:43 EDT 2004
Mark van Atten wrote:
> If one thinks of the decimal expansions of the reals as choice sequences,
>a continuity principle suffices to show that the reals cannot be enumerated.
It can be noticed, however, that many small sets which from a classical point of
view can be enumerated are not enumerable, or only 'weakly enumerable' intuitionistically as a consequence of
the continuity principle (as Brouwer noticed, see for example Brouwer ).
For example the set (more precisely the spread) of all motonone binary sequences, intuitionistically
understood, is weakly enumerable (Brouwer calls this set S_3 in his  if I recall correctly).
This last observation can, perhaps, also be seen as an indication that 'small' but continuum-like sets,
to which the continuity principle applies, cannot be enumerated, so to say, in the same way as the discrete-like sets.
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