Posted: Sep 10, 2015 2:24 pm
Newmark wrote:crank wrote:Calilasseia wrote:Actually, I can write down an expression that constitutes the definition of a real number. Viz:
where for all possible values of i, ai is an element of the set {0,1,2,3,4,5,6,7,8,9}.
If ai=0 for all i>N, where N is some suitable finite integer, then you have a rational number. But not all rational numbers fall into this category. 1/7 being a classic example. The decimal expansion of 1/7 continues indefinitely:
(1/7) = 0.124857124857124857124857124857124857124857124857124857124857...
But in this and other cases, the coefficients ai are cyclically periodic for all i>N, where N is some suitable finite integer.
Oh, and see also, Dedekind Cuts.
That's the definition of a rational, not a real, isn't it? Most reals require an infinite string of integers, incompressible, you have to stipulate each one to define it.
No, Cali gave a correct definition of a (quite generic) real number above, which indeed can be viewed an infinite string of integers. It is not the definition of a rational, which severely makes me doubt your ability to identify "bad math". However, this definition does not cover all reals r, only those for which 0 <= r <= 10, but there should be sufficiently many non-computable ones in that range for Cali's point to be valid...
[Edit: Ninja'd by Cali...]
As I said in my response to cali, I was trying to read more into the definition, I didn't realize it was so basic. As for understanding bad math, any interval will have the same number, the same infinities, as any other, unless there's weird constructions that makes that not true. Any interval, like 0-1, or -infinity to infinity, 6-43.5, have a 1 to 1 mapping. Discussing that some range should have sufficiently many non-computables doesn't really make sense.