>>11526029S* contains 0.9, 0.09, 0.009, etc.
S contains 0.9, 0.99, 0.999, etc
all elements of S* are bijected through the set of all numbers N (which has ? elements) via , and all elements of S are bijected through N via .
Gonna drop the 0 index here, trust me i'm not changing the math. Index from 1.
So borrowing from the axiom of infinity, N has ? elements. The size of the set of N is ?. Bijected with S and S* linearly through the sum and lim functions means S and S* are also ? size'd
if is true, lets test k=4
S*[1]+S*[2]+S*[3]+S*[4] =
>0.9 + 0.09 + 0.009 + 0.0009 = 0.9999S[4] =
>0.9999K, seems true.
So S* has ? elements inherited from the size of N it's bijected with. If all S* elements are added together, S* = 0.999... where infinite size of the set has been translated to infinite length in the decimal. Since sequential S* elements summed to n from the first index also equals the element of S at it's own n'th index, then
>S*[1]+S*[2]+S*[3]+S*[...]-> = 0.999... (with infinite 9's) =>S[...]-> = 0.999... (with infinite 9's)So 1 element in S is the contained sum of all S* elements.
this isn't really surprising is it? The set's sizes were infinite to begin with.
I guess the surprising part is the sum.
Figuring to sum all the elements of N,
All elements of N are finite, so summing them all should be a finite number too, even if the process never ended.
All elements of S* are finite, so they too should be a finite number when summed.
And yet this infinite length 0.999... number has finite value???
so, infinity isn't endlessly counting beyond the reals.
Infinity IS endlessly counting the reals.
And exists as a small non-zero real number for every real number n.
0.999... != 1
