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
>shown,
>so N shouldn't contains infinite length positive integer number. But if
>they
>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 ?
>
>Thanks,
>Quentin

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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
continuum.
Jesse
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