On 15 Jun., 18:57, stevendaryl3...@yahoo.com (Daryl McCullough) wrote:
> > >We should not use oracles in mathematics. > > On the contrary! Many real numbers in physics are not computable > to infinite precision (for example, the fine structure constant).
Numbers are computable. The fine structure constant is a name. It has soem 20 letters.
> Yet, we can certainly compute other real numbers *relative* to such > parameters.
There are a countable number of parameters and a countable number of relative numbers.
> We can easily devise an algorithm to compute the square > of the fine-structure constant, for example. This algorithm will > take as an input an approximation to the fine-structure constant, > and will return an approximation to the square of the fine-structure > constant. > > In this sense, the square of any real number is computable relative > to that real number.
The fine structure constant, or what we know about it, is a name. This is physics - and it will be the last to support transfinity.
Every real number that can be communicated belongs to a countable set. This set includes all names in all languages, all dictionaries (they come at some later lines) and all objects relative to those.
0 1 00 01 10 11 000 ...
It is impossible to diagonalise it. It is possible to diagonalise infinite lists only when they have infinite lines. But never anybody has obtained a meaning from an infinite sequence of digits. Beware: This is not an infinite sequence of digits "0.111..." but a finite definition (from which an infinite sequence can be obtained) that cannot be used as a line in an infinite list to be diagonalized.