[zelah]

>I's much easier to consider the infinite strings of digits like "0.765653625367523765..." or "0.5265362556..." or "0.000073468763478..." and also the one with repetitions like "0.0006767000000..." or "0.0072257822222222...". This is essentially a copy of the real number, but in this copy "0.2999999999999..." is different from "0.300000000000000..." This trick makes much easier to prove that the diagonal ´+1 in each one digit is not in the list. Then it's possible to fix the details and use the real numbers instead of the infinite strings of digits.

What am I missing?

"0.765653625367523765..." could be assigned the transfinite natural number beginning "1765653625367523765..."

"0.000073468763478..." could be assigned the transfinite natural number beginning "1000073468763478..."

Transfinite natural numbers must exist otherwise you do not have an infinite set.

[zardo]

>Transfinite natural numbers must exist otherwise you do not have an infinite set

Transfinite

This word does not mean what you think it means. I'm not entirely sure what you think it means, but it's definitely not what it means to everyone else.

[relate]

Suppose I have the set F = { all integers x where x can be written with a finite number of digits }

Are you saying the set F is finite?

Or are you saying that the set F contains transfinite numbers?

[pitaj]

The difference is that `0.7656536...` is smaller than 0.7656537, whereas a similar number with every digit moved left of the decimal point has no bound.

[Cyph0n]

Yes they could, but note that these real numbers are all between 0 and 1. So the "infiniteness" of R is "higher" than that of N.