FOM: Precision and Turing-computability

[friedman@math.ohio-state.edu]

Reply to raymond.

> ... Lang was, I believe, talking about numbers which
> actually come up in computing things, or in theorems of classical
> analysis (pi, e, log(x), etc).  Lang has argued some odd things in the
> past, but I doubt he would argue that classical analysis involved only
> computable numbers.  Classical analysis of course requires a complete
> metric space, and without noncomputable numbers, we cannot have the
> metric closure of the rational numbers (the reals).
> 
There is a very interesting issue here, one that has come up (in various forms) 
on the FOM e-mail list before.

It would be interesting to have a clear example of a real number, defined in 
some very elementary and understandable and simple and - yes - beautiful, way, 
beautiful from the point of view of a classical analyst, that is actually a 
nonrecursive real number. This is an open ended project: the more beautiful the 
better. 

I did some work on this kind of issue with regard to a nonrecursive set of 
natural numbers. There is a joint paper in the works with an analyst, which 
improves nicely on my earlier postings on the FOM on this topic. I will 
eventually give a pointer to the final version. 

But this earlier work is not really in the style of classical analysis, which 
is what is required here.  

People can be very interested in such a project even though they fully accept 
complete separable metric spaces as the basis for analysis. 



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