HELP: restore my faith in Python

[Michael Hudson]

neelk at brick.cswv.com (Neel Krishnaswami) writes:

> Mikael Olofsson <mikael at isy.liu.se> wrote:
> > 
> > On 05-Mar-00 Moshe Zadka wrote:
> >  >  On Sat, 4 Mar 2000, Tim Peters wrote:
> >  > > Your fear of the binary point is much deeper <wink>.
> >  >  
> >  >  Actually, no: both scare me equally. For what I do, 2 and 10 are just as
> >  >  bad -- it's the finite numbers of bits I'm fighting against. And I can't
> >  >  even take the rational way out: I have to deal with square roots and
> >  >  trigonometry.
> > 
> > Ahh, it's time to include extensions of the rational field. Let's get
> > more abstract. Abstraction is beautiful! And we need to represent 
> > transcendental numbers consistently aswell.
> 
> Is this possible? I've forgotten most of my analysis, but I was under
> the impression that you could divide the continuum as follows:
> 
>   o Rationals -- eg 2/3. Countably infinite.
>   o Algebraic irrationals -- eg sqrt(2). Countably infinite.
>   o Transcendental numbers -- everything else. Uncountably infinite.

Well, you could have algebraic extensions of the rationals, i.e. the
rationals extended by the roots of some ploynomial or other; these
certainly don't have to be contained in the reals (eg. Q[i]), though
they will be contained in the complex numbers (if the roots are
rational).

> So presumably you could write numeric classes that properly (if you
> don't care about speed or memory usage) handle rationals and algebraic
> irrationals, but the full transcendentals are just plain impossible. 
> Maybe some interesting subset of the transcendentals are possible; 
> if you throw in e and pi most people will be happy most of the time? 
> Though on reflection this is getting pretty close to a full symbolic 
> computation package.

Food for thought: the computable numbers are countable.

Proof left as an excercise for the reader (it's not hard!).

(Is it even meaningful to ask whether they are [recursively]
enumerable?)

Cheers,
M.

-- 
very few people approach me in real life and insist on proving they are
drooling idiots.                         -- Erik Naggum, comp.lang.lisp