Question

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Answer 1

Show S is finite:
Every element of S is expressible in no more than 50 characters.
Therefore the total possible amount of elements in
\[(27)(27)......(27)\] 50 times
\[=27^{50}\]
Any other element is we try to make is already in these combinations.
So S is finite.

Now consider the natural numbers. There is an infinite amount. If we take away a finite amount, we still have an infinite amount. In particular, \(\infty - 27^{50} = \infty\) so T has an infinite amount of elements.

Finally, remember T is the elements NOT in S, so the elements in T can't be expressed in 50 characters or less. But the smallest natural number, whatever it is, in T is 'the smallest natural number in T'. But now we've described it in less than 50 characters! So it is in S. Paradox.