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And later still this same structure was used to show even deeper truths.
Gödel demonstrated, using the same sort of self reference as Russel, that any system of math that can express basic arithmetic either allows you to prove some false statements to be true or is unable to prove some true statements (Gödel's Incompleteness Theorems). This result is considered by many to be one of the crowning achievements of modern mathematics.

And this same structure has been used to show even deeper truths.
Gödel demonstrated, using the same sort of self reference as Russel, that any system of math that can express basic arithmetic either allows you to prove some false statements to be true or is unable to prove some true statements (Gödel's Incompleteness Theorems). This result is considered by many to be one of the crowning achievements of modern mathematics.

However using the same structure above (albeit complicated enough I won't get into the details) it was shown that any algorithm must either sometimes produce the wrong answer or fail to produce an answer at all. (This is called the Entscheidungsproblem.)

And this is the exact same in the case of the other problems I presented. Russel's proposition is a nonsense proposition when "set" is taken to mean "Frege set", (and the other ones two but I haven't discussed them in detail). The point is to take a premise which may seem fine and to tease out a contradiction.

However using the same structure above (albeit complicated enough I won't get into the details) it was shown that any algorithm must either sometimes produce the wrong answer or fail to produce an answer at all. This is called the Entscheidungsproblem and it is exactly equivalent to a much more famous problem the Halting problem.

And this is the exact same in the case of the other problems I presented. Russel's proposition is a nonsense proposition when "set" is taken to mean "Frege set". And similarly the proof that the halting problem is incomputable involves constructing a contradictory machine based on a hypothetical solution to the problem.

The point is to take a premise which may seem fine and to tease out a contradiction where it is clear for all to see.

The only way for the box to be correct is for it to provide no answer at all. Any box must reject some sorts of answerable questions, if it wishes to always produce correct answers. If it is fine with producing wrong answers then it can accept every question, no problem, but such a box is probably not very useful.

The only way for the box to not be incorrect is for it to provide no answer at all. And in this case the answer is then "No" because it didn't produce a negative answer.

Any box must reject some sorts of answerable questions, if it wishes to always produce correct answers. If it is fine with producing wrong answers then it can accept every question, no problem, but such a box is probably not very useful.