On Tuesday, February 12, 2013 4:19:02 PM UTC+1, Craig Feinstein wrote: > Let's say I have a drawer of an infinite number of identical socks at time zero. I take out one of the socks at time one. Then the contents of the drawer at time zero is identical to the contents of the drawer at time one, since all of the socks are identical and there are still an infinite number of them in the drawer at both times. But the contents of the drawer at time zero is also identical to the contents of the drawer at time one plus the sock that was taken out, since they are exactly the same material. So we have the equations: > > > > Contents of drawer at time 0 = Contents of drawer at time 1 > > Contents of drawer at time 0 = (Contents of drawer at time 1) plus (sock taken out of drawer). > > > > Subtracting the equations, we get > > > > Nothing = sock taken out of drawer. > > > > This is false, so infinity cannot exist. > > > > How does modern mathematics resolve this paradox?
Mathematics defines the concept of a 'set'. A set can have an INFINITE number of elements, an example is the set of integers. If you have two disjoint sets, each with a number n1 and n2 of elements, you may want to define the SUM of n1 and n2 as the number of elements of the union set. For finite sets, this gives ordinary addition. For finite sets, you get new, perfectly consistent, albeit possibly unfamiliar rules of summation, like INFINITY + 1 = INFINITY.