That defines a 1-to-1 mapping between the naturals and all the sets of the naturals. So by induction, for every n+1 starting at 1, the number representing the set of all numbers up to and including n+1 (call it the set number) can be gotten by adding 2^(n+1) to the set number that corresponded to n. So clearly, for every set of natural numbers, there is a set number that is also a natural number. So because the set of ALL natural numbers is itself a set of natural numbers, there must be a set number for that too, also contained in n. Clearly, the length of this number is infinite, so there must be an infinitely long number in n.
You've defined a one-to-one correspondence between the natural numbers and all FINITE subsets of the natural numbers. But if you look at the subsets {2, 3, 5, ...} and {10, 20, 30, ...} (the set of primes and the set of multiples of 10), then they would both correspond to divergent series instead of actual natural numbers.
So, in short, your first argument is right.
However, it is still true that 0.9999... = 1.
And now, for some fun: what is the smallest positive integer that can not be described using fewer than one hundred letters?
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