Calilasseia 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?

No. What part of the words "infinite sum" did you fail to derive from the expression I provided?crank wrote:Most reals require an infinite string of integers, incompressible, you have to stipulate each one to define it.

Which is precisely what that expression says if you read it.

Plus, since the expression converges to a finite limit for all choices of coefficients as defined, and indeed converges to a finite limit in the interval [0,10] (courtesy of 100 being a part of the expression), it satisfies all the conditions required to be a definition of a real number. Indeed, setting all the ai to 9, which results in the largest of the numbers thus defined, results in the expression converging to the value:

9[1/(1-[1/10])] = 9[10/9] = 10.

Plus, as I already stated, there are rational numbers whose decimal expansions are infinite. I provided the example of 1/7: indeed, this is true for all rationals of the form 1/p, where p is a prime number greater than 2.

EDIT: 5 is an exception, having the decimal expansion 0.2. All other primes apart from 2 and 5 have infinite decimal expansions.

OK, I was trying to read too much into the definition. Is it anything more than saying the reals have a decimal representation? I thought there must be something more that I'm not getting, because this is trivial. How is this relevant to Chaitin's claim? I must be missing something because this doesn't seem to have anything to do with anything I've said, maybe it pertains to something in Chaitin's work, I don't know, I never claimed to have serious understanding of any of this, in fact just the opposite.