I want to answer your original question in this comment to show you the importance of being well defined and the use of sound logic.
JDE: "Consider a hypothetical hotel with a countably infinite number of rooms, all of which are occupied."
JG: "Consider a hypothetical pink cow, meat from it must taste like fish".
IF "pink cow" THEN "FISHY TASTE"
The premise is FALSE, so we can't proceed. I can reify a pink cow by splashing pink paint on it, but you can't *reify* a hypothetical hotel!
Why is reification so important in terms of well defined concepts? Well, it gives us a method of checking the validity of those objects we form from concepts in our minds. It's fine to hypothesize, but the premise must be realistic, that is, it must be possible. :-)
JDE: The nature of countable infinity is such that there cannot be a last room, and that's fundamental to the fact that new guests can be accommodated in an infinite hotel at will, itself an illustration of the fact that we can count endlessly.
Let's analyse what you wrote there. You *assumed* that infinity exists. Then you granted the *attribute* of "countable" to it, without realising that its not infinity which is countable, but *sets*. Then you *hypothesized* about the *nature* of an assumed attribute and *deduced* that there cannot be a last room (hypothesis).
Do you think that the first 3 steps can ever lead to any deduction? Imagine if airplane designers tried to engineer aircraft this way. Do you think the aircraft would ever work? If engineers were like "mathematicians", then we would have been in serious trouble. :-)
JDE: But, if there is no last room, it can never be the case that *all* rooms are occupied, hence the whole argument, for how informal, falls apart since inception.
JG: And now you move into "meta-mathematics" with the new premise: "But, if there is no last room..."
IF (Assumption -> Attribution -> Hypothesis -> Deduction) THEN CONCLUSION
IF "no last room" THEN "not ALL rooms are occupied".
Can you see how ridiculous and absurd you have become? :-) I am not attacking you, just wanting to get you to think for yourself.
JDE: More logical seems to say that, while there can be ideal constructs such as the actual infinities of super-tasks and corresponding "super-numbers", there can be no such thing as the standard countable infinity, as that would be something that at the same time is and is not exhausted, i.e. a self-contradictory notion.
JG: That first sentence is impossible to comprehend. It's meaningless rot. One of the great attributes of a real mathematician is his ability to explain what he thinks and understands. Use of well-defined concepts, clear sentences and exegesis are hallmarks of great minds. Honestly, JDE, anyone who claims they know what you intend to convey in that first sentence is a rotten liar.
JDE: Finite infinity is rather just that, the ever unfinished.
JG: "Finite infinity" ? That is an oxymoron. :-) The first red flag in Cantor's mathobabble is his use of oxymoronic concepts.
These ideas are not mathematics, but false Jewish fables. As another interesting exercise, I challenge you to read up on Cantor's quotes. A man is known by his quotes. :-)