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Cantor’s theorem establishes that while the set of natural numbers is infinite yet countable, the real numbers in the interval (0,1) form an uncountable set of strictly greater size.

My question is this: suppose one were to create natural numbers at the right-hand end of an infinite sequence of numerals, then extend that sequence indefinitely to the left—thereby constructing “infinitely long” natural numbers. Could one then apply Cantor’s diagonal argument to this enlarged collection?

I recognize that such constructions violate the rule that each natural number must be a finite numeral. Is this violation alone enough to rule out the proposed idea?

Nils
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  • Hi, welcome to Math SE. By NN, do you mean $\mathbb{N}^\mathbb{N}$, a set of functions (or equivalently sequences)? If so, a diagonalisation argument can show it's uncountable. (If you want a set of two-sided sequences, you can take $\mathbb{N}^\mathbb{Z}$.) – J.G. May 12 '25 at 12:49
  • Sorry I meant Natural numbers (N) – Nils May 12 '25 at 13:06
  • Cantor's diagonal theorem is about infinite sequences. It proves that the real numbers are uncountable by representing real numbers as infinite sequences of digits, then showing that the set of such sequences is uncountable. There's no similar way to connect natural numbers with infinite sequences of digits because natural numbers are never represented by infinite sequences of digits. – MJD May 12 '25 at 13:27
  • @MJD That actually seems to be addressed by the last paragraph in the question---the asker seems to be aware that they are creating a set of number-like things which are not actually natural numbers. – Xander Henderson May 12 '25 at 13:50
  • Yes, I was trying to confirm their surmise. – MJD May 12 '25 at 14:25

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