Non-enumerability of R

[Andrej Bauer <andrej...@andrej.com>]

I have been haunted by the question "Is there a surjection from N to
R?" in a constructive setting without choice. Do we know whether the
following is a theorem of Unimath or some version of HoTT:

     "There is no surjection from the natural numbers to the real numbers."

Some remarks:

1. Any reasonable definition of real numbers would be acceptable, as
long as you're not cheating with setoids. Also, since this is not set
theory, the powerset of N and 2^N are not "reals".

2. It is interesting to consider different possible definitions of
"surjection", some of which will probably turn out to have an easy
answer.

3. In the presence of countable choice there is no surjection, by the
usual proof. (But we might have to rethink the argument and place
propositional truncations in the correct spots.)

4. It is known that there can be an injection from R into N, in the
presence of depedent choice (in the realizability topos on the
infinite-time Turing machines).

With kind regards,

Andrej