Is a number with infinite digits within the set of natural numbers?

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I have 237474 ..... that number should be found within the set of natural numbers no? because if we say that the set of natural numbers has infinite elements, therefore that number should be found in the set. if that is not the case then what set is it in?

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No, that does not follow. There are infinitely many natural numbers, but the decimal representation of each natural number is only a finitely long string of digits.

An infinite string of digits would be in the set of .... strings of digits.