s a countably infinite
number of action
s carried out in a finite
amount of time
. Infinite algorithm
s such as Zeno's motion paradoxes
and Thomson's lamp paradox
fall into this category
, as does the infinite urn problem
The way we get around filling a finite amount of time with an infinite number of actions is that we have each action take half the time as the previous one, i.e., the first one occurs at time 1s, the next at time 1.5s, the next at time 1.75s, then 1.875s, and so on. This sequence converges to 2 s, at which point a countably infinite number of actions have taken place. Alternatively, we could have the first action occur at time 0.3s, the next at 0.33s, then 0.333s, until an infinite number of actions have taken place after 1/3 s. A machine that computes this way (carries out an infinite number of instructions in finite time) is called a supertask machine.
Supertasks behave differently than finite algorithms, so sometimes we get counterintuitive results. Here's a good example (along the same lines as the infinite urn problem):
Let's say you have a million dollars, and the Devil approaches you and offers you a deal. He proposes to give you two dollars for every one dollar you give him, infinitely, in a finite amount of time. Should you accept this deal? Of course there's a catch -- the Devil only takes from you the dollar with the lowest serial number, and will only give you dollars with higher serial number. So how many dollars will you have after you've played this game? The answer is zero. If you think you have any dollar, tell me what the serial number is, and I can tell you during what step the Devil took it from you, and never gave it back (the serial numbers of course are enumerable). So not only do you not gain infinite dollars, you also lose your million.
However, if the Devil instead takes a different dollar from you each time, say one of the two he gave you in the previous step, then you will end up with infinite dollars -- and the Devil will also have taken infinite dollars from you.
This counterintuitive result is possible because of the nature of countable infinity. In particular, proper subsets of a countably infinite set can be infinite. Take the natural numbers for instance. The even numbers are a proper subset of the naturals, but there are countably infinitely many of both. Removing the even numbers from the naturals still leaves a countably infinite set -- this corresponds to the second case above. But if we remove every natural number from the set of natural numbers -- a set which is equally as infinite as the even numbers -- we are left with the empty set. This corresponds to the first case above.