irrational number continuum

0.3+ .03+ .003 etc if carried on "to" "infinty" would have the precise
limit 1/3.
although the LIMIT is well definited, since "infinity" is not an
actuial number, the summing itself never has any definite value.

an "INFINITE NUMBER" of 3's after the decimal point, is no more a
number then an ""INFINITE NUMBER" of 3's before the decimal point;
both are perpetually undefined.

only the sum's LIMIT is definite. (being "the smallest number which
the sum shall never reach")

The number 2 exists and is definite, but on what basis do we accept
that a number, r, who's square is 2 exists?

if a^2 < 2 and b^2>2, is that sufficient justification for accepting
that there must be an r; such that a<r<b, where r^2=2?

why do we say r exists and is irrational rather than saying:

"2 doesn't have a square root, but there's an infinity of numbers
who's squares will differ from 2 by as little as you need for any
practical purpose"

why do we assume a continuum between 0 and 1?



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