> > The problem of 1 = 0.(x) appears for any possible base of > numeration . > If you're bothered by the representation being 'lossy' , you can > always try continued fractions for the numbers in the interval [0,1] : > Each real number is represented by a (possibly infinite) sequence of > strictly positive integers : > You represent r by [a1,a2,a3 ..... an ...] meaning that > r = 0 + 1/ (a1 + 1 / (a2 + 1/ (a3 + .... ))) > I'm pretty sure you can build up the whole of analysis this way , > though nobody's bothered to do it, so it must be tedious. > That being said, I was never really bothered by the whole 0.(9) = 1 > business , it's just a quirk in notation .
I am aware of continued fractions. And, there is nothing about the particular statement of equality that bothers me.
You seem to be focused on the wrong part of the post. That is fine. I know that most mathematicians are not accustomed to the kind of logic that comes from Frege, Russell, Carnap, Lesniewski, Wittgenstein, Tarski and others. But, in fact, what do most mathematicians intend when they say that mathematics is "logical" but then ignore the presumptions and opinions upon which that is based (in the modern sense)?
That is a rhetorical question. In your arena, there is an entirely different set of people such as Turing, Kolgomorov, Markov, Church, Curry, Kleene, etc.
> What seems far more troublesome is the representation of finite > fields , you always have to 'choose' one of many irreducible > polynomials if you want to work with them . >
I have recently run into that problem. I have been fascinated by a particular presentation of the elements of the Galois field over 2^4 generated by
I doubt that it is unique. My interest, however, is not computational.