Date: Dec 9, 2012 5:06 AM
Author: mueckenh@rz.fh-augsburg.de
Subject: Matheology § 175

Matheology § 175

Until then, no one envisioned the possibility that infinities come in
different sizes, and moreover, mathematicians had no use for ?actual
infinity.? The arguments using infinity, including the Differential
Calculus of Newton and Leibniz, do not require the use of infinite
sets.
T. Jech: "Set Theory", Stanford Encyclopedia of Philosophy (2002)
http://plato.stanford.edu/entries/set-theory/

There are only countably many names.
An uncountable set of names cannot be well-ordered - because it does
not exist.
A set of numbers cannot be well-ordered unless all the numbers have
names.
This seems to contradict Cantor's diagonal argument - but only if
infinite set are complete.

Conclusion: Infinities do not come come in different sizes. In fact
mathematicians have never had use for actual infinity because they
could not. All they could is to believe that they had use for actual
infinity, i. e., for numbers that have no names and cannot be used.
That's called mathelogy.

Regards, WM