Supertasks

I first heard about this when being taught about Zeno of Elea during a philosophy lecture in university, though at the time I simply knew of it as one of Zeno's paradoxes. A supertask is a task which has an infinite number of steps but is completed in a finite amount of time - something that seems like a certain impossibility. But, we in fact complete a supertask every time we move from one place to another, and this brings me to Zeno's paradox:

If you are to move from A to B you first must arrive at the halfway point between A and B - let's call this point A'. Before you get to A' you must first arrive at the halfway point between A and A' (A''), but before you get to this point you must arrive at the halfway point between A and A'', and so on, ad infinitum. Because a supertask seems to be impossible by definition, and motion was itself a supertask, Zeno came to the unlikely conclusion that motion was impossible. The more plausible conclusion would seem to be that some supertasks are possible.

In 1936 Alan Turing developed the idea of a Turing machine, though at the time he referred to it as an "a-machine" for automatic machine. It wasn't something that he actually physically built, but it was a sort of thought-experiment that described how a machine could hypothetically perform computations. The Turing machine, as far as I can understand it, accurately describes modern day computing algorithms, which I think was a pretty remarkable achievement for Turing in the 1930's. 

I say all this about Turing because there is one supertask which may become possible in the future. Where will this exponential increase in computing power take us? How far can it go? Perhaps computing an infinitely large numerical set in a finite amount of time? In this somewhat difficult interview with maths professor Joel David Hamkins he describes his theory of "infinite-time Turing machines", which are again not physically real but exist as a thought-experiment,

"Nevertheless, there are several groups of researchers working on the fantastical idea that it might be consistent with the laws of physics to have a physical realisation of a supertask computational device."

 So what might an example be of computing an infinite set of numbers in a finite amount of time?

"With an infinite time Turing machine, one could solve the prime pairs conjecture (which
asserts that there are infinitely many prime pairs—pairs of primes, such as [11, 13] or [107,109], which differ by two), for example, and the question of whether there are infinitely many Fermat primes (primes of the form 2^2n +1) and so on."

http://arxiv.org/pdf/math/0212047v1.pdf