Quentin Anciaux wrote:

>Hi, thank you for your answer.
>But then I have another question, N is usually said to contains positive
>integer number from 0 to +infinity... but then it seems it should contains
>infinite length integer number... but then you enter the problem I've 
>so N shouldn't contains infinite length positive integer number. But if 
>aren't natural number then as the set seems uncountable they are in fact
>real number, but real number have a decimal point no ? so how N is
>restricted to only finite length number (the set is also infinite) without
>infinite length number ?

The ordinary definitions of the natural numbers or the real numbers do not 
include infinite numbers, but in at least some versions of nonstandard 
analysis (which as I understand it is basically a way of allowing 
'infinitesimal' quantities like the dx in dx/dy to be treated as genuine 
numbers) you can have such infinite numbers (I believe they're the 
reciprocal of infinitesimals). I know the system of the "hyperreals" 
contains them, see http://mathforum.org/dr.math/faq/analysis_hyperreals.html 
for some more info. I'm not sure if infinite hyperreal numbers have the sort 
of "decimal expansion" that you suggest though, skimming that page it seems 
that infinite hyperreals are identified with the limits of sequences that 
sum to infinity, like 1+2+3+4+..., but different sequences can sometimes 
correspond to the same hyperreal number, you need some complicated set 
theory analysis to decide which series are equivalent. Since the hyperreals 
contain all the reals, the set must be uncountable...I don't know if it 
would be possible to just consider the set of infinite hyperreal "integers" 
or not, and if so whether this set would have the same cardinality as the 


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