[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,
>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.


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