Actual infinities.

[Jehanne]

Here it is:



https://en.wikipedia.org/wiki/Zermelo%E2...f_infinity
https://en.wikipedia.org/wiki/Axiom_of_infinity

What I am asking, is not to define what infinity is, nor to prove it as an abstract theory.  But when you are talking about two points, attributing infinity between those two points, what are you describing here?  Infinity of what?

It cannot be related to distance or a physical thing with dimensions, this would lead to a contradiction.  I don't think you can define what is infinite, without making in then finite.  You can potentially make smaller and smaller fractions, but each one will represent a finite amount, and will never get you an infinite result.

The set of Natural Numbers is an (countably) infinite set, an actual infinite of numbers, even if we cannot "count" them.  So, too, are the set of numbers between any 2 numbers of the real number line, which would be an uncountable infinite set.  I realize that no one can "count" those numbers, just as no one can enumerate the number of past events, if, in fact, the Cosmos is without a beginning.  I am just claiming that "actual infinities" of physical things may exist, just as they do in transfinite arithmetic.  If space is infinitely divisible, and, time, too, then those are actual infinities of things; if not, then they are finite.  You (and, Craig) seem to be conflating the act of enumerating a set with the intrinsic membership of that set!  Again, what is the cardinality of the set of "future praises" in Heaven?!  Don't say that it is a "potential infinite"; I am not asking you to count it!  I am asking you what its cardinality is?!  Now, if you are going to insist on sets being "potentially infinite", then please define what the cardinality of a "potentially infinite set" is!  And, then, answer the question as to if some potentially infinite sets are bigger than others.