Solving unsolvable problems

In 1936, Alan Turing wrote a landmark paper [4] in which he showed that a particular well-defined mathematical problem is impossible to solve. In this case, the word “impossible” means that no classical algorithm running for a finite amount of time will ever reach a final, concluding step. To prove his claim, Turing created a precise definition of what it means to be an algorithm and went on to devise a problem that contains its own circular knot.

Consider the following sentence:

Is “no” the correct answer to the question that you’re reading right now?

This sentence is like a double-edged sword:

• If you answer “yes” to the sentence, you’re saying “Yes. It’s true that “no” is the correct answer.” But, if “no” is really the correct answer, your utterance of the word “yes” is incorrect.
• If you answer “no” to...