[FOM] Re: Excluded middle & cardinality of the reals
Giuseppina Ronzitti
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,
>
>then
>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 [1925]).
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 [1925] 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.
Best,
GR
----
My home page:
http://www.geocities.com/giuseppina_ronzitti
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