On Tuesday, June 11, 2013 12:00:58 AM UTC-7, muec...@rz.fh-augsburg.de wrote: > Matheology § 285 > > > > In this article, I argue that it is impossible to complete infinitely many tasks in a finite time. A key premise in my argument is that the only way to get to 0 tasks remaining is from 1 task remaining, when tasks are done 1-by-1. I suggest that the only way to deny this premise is by begging the question, that is, by assuming that supertasks are possible. > > > 1 By definition, completing infinitely many tasks requires getting the number of tasks remaining down to 0. > > 2 If tasks are done 1-by-1, then the only way to get to 0 tasks is from 1 task, because if more than 1 task remains, then performing a task does not leave 0 tasks. (This reasoning holds in both the finite and infinite cases.) > > 3 When infinitely many tasks are attempted 1-by-1, there is no point at which 1 task remains. > > 4 Then from 2 and 3, there is no point at which 0 tasks remain. > > 5 Then from 1 and 4, it is not possible to complete infinitely many tasks. > > > > JEREMY GWIAZDAArticle first published online: 4 MAR 2012 > > Pacific Philosophical Quarterly > > Volume 93, Issue 1, pages 1?7, March 2012 > > http://onlinelibrary.wiley.com/doi/10.1111/j.1468-0114.2011.01412.x/full > > > > Therefore it is not possible to enumerate all rational numbers (always infinitely many remain) by all natural numbers (always infinitely many remain) or to traverse the lines of a Cantor list (always infinitely many remain). >
You are right an infinite task cannot be completed 1-by-1. At each step you still have more to complete than have been completed.
However, in set theory, if we have an axiom of infinity then we can define a function ( with finite number of symbols ) to do the task.
This works very well enumerating the set of rationals, or to find the diagonal of a countably infinite list of real numbers.
Mathematics is an ideal theory that doesn't not necessarily correspond directly with reality. Despite the lack of correspondence, Science finds Mathematics useful in its endeavors. Even the Infinite!
Supertasks are not physically possible. You can't increase the efficiency of a machine, or operation, an infinite number of times ( in reality ).
Supertasks are a nice heuristic theory used to explain the Infinite to people who lack the Mathematical intuition of the Infinite.