[gus_massa]

[I'll try a non technical argument to convince you. It's also not a complete argument, so you must think about this for a while.]

> If you believe n is a natural number then you must also believe that n x 10 is a natural number. One more digit!

If you interpret the natural number in this way, the important property is that they have only a finite amount of "interesting" digits. Almost all their digits are zero.

You can define the set of number that have a finite amount of non zero digits. Let's call them the "Very Boring" numbers.

The set of the "Very Boring" numbers is infinite, but it's the same infinite that the Natural numbers.

You can extend this set to include the periodic numbers, for example 46.2222222222222... and 462.2222222222222... and 4622.2222222222222... ... Let's call them the "Boring" numbers. You still get the same infinity.

You can also 71.3535353535... and 713.5353535353... and 7135.3535353535... and all the periodic numbers. Now you have the "rational" numbers. You still get the same infinity.

The Cantor's diagonal argument fails with Very Boring, Boring and Rational numbers. Because the number you get after taking the diagonal digits and changing them may not be Very Boring, Boring or Rational.

--

A somewhat unrelated technical detail that may be useful:

Most of the times you don't prove that the cardinal of real numbers between 0 and 1 is a bigger infinity than the cardinal of the natural numbers.

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.

[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.