Originally posted by alofatti
when I talk about the decimal representation I am talking about an element which has this form:
where x is a digit between 0 and 9 and which could extend to the infinite, I am not talking about rational numbers (which are not biyective to the real numbers).
This decimal representation number system has the same cardinal than the real numbers, there is a mapping between them that is both injective and suryective.
The inyection is easy, it is clearly seen that the real numbers are included in this set.
The surjection is somewhat trickier. It is possible to show that this set includes both this sets:
- the set of numbers which do not end with a repeating 9 (which are easily shown to be bijective to the real numbers)
- the set of numbers which do end with a repeating 9 (which are numerable).
Since c (the cardinal of R) + Aleph_O (the cardinal of a numberable set) equals c, then they are both coordinable between themselves.
A complete set is a set which, in an informal way, does not have successions which converge to "ghost" (i.e., inexistent) points. More formally, every Cauchy sequence does have a limit.
I have said in a post before this that the set of decimal representable numbers are not complete, which is not right. First is necesary to define a distance between points to talk about completeness.