RE: The First Cause? Prime Mover Argument
May 12, 2015 at 1:47 am
(This post was last modified: May 12, 2015 at 2:13 am by Hatshepsut.)
Divergence to plus infinity in the sequence {1, 2, 3, ... } does not require any "potential" or "actual" infinity to be produced. It requires only that for any natural number m, there exists a natural number greater than m. Mathematics doesn't have to worry about the distinction between potential and actual infinities, a topic you won't see in any math textbooks. The set of natural numbers, N, is defined by stating (1) that every natural number has a unique successor distinct from itself, (2) that there exists a natural number called 1 which is not the successor of any number, and (3) that if a set S contains 1 along with the successors of every number in S, then S is the set N. An arbitrary set is infinite if it contains a subset whose elements can be placed in one-to-one correspondence with the elements of N.
There's a cardinality thing where N represents a "countable infinity." Sets of higher cardinality exist, such as the set of all functions from N into N, which we can show is equivalent to the set of real numbers. Yet nothing is said about whether there are, or can be, any collections of real-world objects that can be placed in one-to-one correspondence with N. So, the mathematical set N remains small enough to fit conveniently in a brain regardless.
There's a cardinality thing where N represents a "countable infinity." Sets of higher cardinality exist, such as the set of all functions from N into N, which we can show is equivalent to the set of real numbers. Yet nothing is said about whether there are, or can be, any collections of real-world objects that can be placed in one-to-one correspondence with N. So, the mathematical set N remains small enough to fit conveniently in a brain regardless.
