Craig Feinstein <firstname.lastname@example.org> wrote in news:email@example.com:
> 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? >
By means of limits. Infinity minus infinity is an indeterminate form, and no said that the rules of finite arithmetic apply to non-finite things. We invented limits to deal with non-finite things.