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lavinia
#6
Feb11-12, 12:16 PM
Sci Advisor
P: 1,716
Quote Quote by Jano L. View Post
Hello everybody,

Yesterday I've read that there exist a real number r which cannot be defined by a finite number of words. This result, although quite awesome, is so strange that it lead Poincaré to doubt Cantor's work and state "never consider objects that can't be defined in finite number of words".

In our course on analysis, our professor postulated the existence of a supremum for any bounded sequence of rational numbers and I think the existence of the set of real numbers followed.

I am not entirely satisfied by this approach though, because it postulates supremum, which is not particularly simple notion.

Do you think the set of real numbers can be defined in finite number of words, using only basic concepts? (whole numbers, simple logic, ...)

Or do you think it is impossible and the set has to be postulated by force?

What are your feelings about this?
Which possibility do you prefer?

Jano
The number of finite sets of words is countable. So they can not specifically define each real number since there are uncountably many of them

one can define the reals from the rationals using the idea of Dedekind cuts. This require no other ideas than the euclidean metric on the rationals. I am not sure how ones extends the arithmetic to these without notions of Cauchy sequence. Maybe it is not possible.