Paradoxes of the Supertask
In set theory, an infinite set is not considered to be created by some mathematical process such as “adding one element” that is then carried out “an infinite number of times”. Instead, a particular infinite set (such as the set of all natural numbers) is said to already exist, “by fiat”, as an assumption or an axiom. Given this infinite set, other infinite sets are then proven to exist as well, as a logical consequence. But it is still a natural philosophical question to contemplate some physical action that actually completes after an infinite number of discrete steps; and the interpretation of this question using set theory gives rise to the paradoxes of the supertask.
The diary of Tristram Shandy
Tristram Shandy, the hero of a novel by Laurence Sterne, writes his autobiography so conscientiously that it takes him one year to lay down the events of one day. If he is mortal he can never terminate; but if he lived forever then no part of his diary would remain unwritten, for to each day of his life a year devoted to that day’s description would correspond.