bystander wrote:...

∞ is not an integer, so your statement "for all numbers x, where x is a member of the set of integers, there is no x such that the successor of x is ∞" is true by default, because ∞ is not a part of the set. That does not mean there exists a greatest integer.

I believe I should have written 'cardinal' rather than 'integer' although the difference seems to be slight, cardinals are specifically for talking about the number of elements in a set, as in the discussion above (so the number of Plank volumes in the observable universe would be a cardinal number). I have looked at the definition of

**cardinal** number, and they are transfinite. I have one last question on this topic before I give it a rest and get back to the lectures. I can't find the answer elsewhere on the Internet. I don't know whether you can answer this for me or whether I need to put this on a math forum, but please let me know either way:

is it also true that there is

no x, where x is a finite cardinal number, such that the successor of x is Aleph Null?

Unlike ∞, Aleph Null is a member of the set of cardinals, so the above is not true by default, although I believe it is true. That is probably what I should have put originally. Thanks again for your time.