“In 1952, the logician Leon Henkin flipped Gödel’s idea around and asked about a sentence in the language of arithmetic that says: ‘This sentence is provable.’ He asked: is this provable or not? The answer is much less obvious than for Gödel’s sentence.”
“Suppose that you and all your future selves believe their later self’s promise to finish your paper. Then you just refer the paper to your future self, who refers it to their future self, and so on indefinitely. All of you believe the paper will eventually get done, but there is no finite time at which the paper gets written. We call this the Procrastination Paradox.” In the example they consider, each future self believes that the paper will get written some nonstandard (i.e. infinite) integer number of iterations in the future.
“Viktor Klee, in a parody of the usual optimistic prophecies people like to make, wrote that: ‘…progress on this question, which has been painfully slow in the past, may be even more painfully slow in the future.’”
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