Re: Reality check

[Martin Escardo <mhe@dcs.st-and.ac.uk>]

I can't answer Andrej's questions, but I can make a few
observations. 

 > It would be interesting to simplify this presentation even further by
 > using a signed representation with digits -1 and 1 only, in base B
 > strictly between 1 and 2. For example, the golden ratio base B = (1 +
 > sqrt(5))/2 seems to be very popular among exact real arithmetic
 > people. But it's unclear how to make a finite state automaton for
 > negation of === in this case.

In base Golden Ratio with digits 0 and 1 (proposed by Pietro Di
Gianantonio), the family of identities that generates === is

   ... 100 ... === ... 011 ... 

It corresponds to the fact that the Golden Ratio is the positive
solution of the equation x^2 = x + 1.

    (i.e. 1 x^2 + 0 x^1 + 0 x^0 = 0 x^2 + 1 x^1 + 1 x^0)
          =       =       =       =       =       =

What I have reported about signed-digit binary notation has also been
developed for Golden-Ratio notation by David McGaw in his Honours
project ( http://www.dcs.st-and.ac.uk/~mhe/macgaw.ps.gz ).  You may be
able to get a finite automaton from his algorithm for solving the word
problem. It should be even simpler than the one for signed binary,
because there are fewer cases to consider.

 > [Discussion about intuitionistic versions of Freyd's construction
 > deleted.]
 > This [discussion] leads to the idea that we should think of
 > the closed interval I as being glued like this:
 > 
 >              I
 >       |------------|--R--|
 >                    |--L--|------------|
 >                             I

This is precisely what the Golden-Ratio notation achieves.

The interval I is now [0,phi], where phi is the Golden Ratio.

Then the "digit maps" are l(x)=(x+0)/phi and r(x)=(x+1)/phi.

The intersection of the images of l and r is a closed interval with
non-empty interior, as in your picture. 

The above family of identities is equivalent to the single equation
l o r o r = r o l o l.

One could try to consider algebras with two operations and this
equation in order to get the interval in a more constructive way.