Doesn't the fuller quote give context to exactly that,

By contrast with that, the actual infinite is an infinite which is, as it were, complete. The number of items in the collection is not growing toward infinity; it is infinite! It is complete and static and involves an actually infinite number of things.

This type of infinity is symbolized by the Hebrew letter aleph (ℵ) and is used in set theory. In set theory, mathematicians talk about sets like the set of natural numbers which have an actually infinite number of members in the set. The collection is not growing toward infinity as a limit. It is infinity. There are an actually infinite number of natural numbers in this set.

What are you referring to exactly? I don't see anywhere in which Craig uses Hilbert's Hotel differently from Hilbert. And they both come to the same conclusion. As Hilbert states, "The overall result is then: The infinite is nowhere realized. Neither is it present in nature nor is it admissible as a foundation of our rational thinking – a remarkable harmony between being and thinking."

Craig is not the one who alleges that absurdities result from Hilbert's Hotel. That is, he did not come up with this idea that Hilbert's Hotel results in absurdities. You're acting like he came up with all of this all by himself, and is making a fool of himself for having discovered these absurdities, but as I've already pointed out, he's simply mimicking others who've come to similar conclusions. In the end, I can't help but feel that your issue with Craig is less to do with hismisuseof math, and more that you disagree with those mathematicians that he relies on. There are mathematicians, good mathematicians, that do not believe that actual infinities can exist in the real world. I imagine even they would not accept that the use of hyperreals could resolve absurdities with infinities in real life.

I'm no mathematician though, and I know enough to know when I'm out of my depth. So I suppose there's something really obvious here that I'm missing. Have you emailed Dr. Craig your solution to Hilbert's Hotel using hyperreals? He answers emails all the time on both his website and podcast. I couldn't find anyplace where Craig discusses hyperreals to solve Hilbert's Hotel Paradox, but I did find this interesting discussion between the skeptic mathematician Jeffrey Shallit, and a poster named Wade. Wade makes a number of very interesting points that you might want to look over.

I realize you are disputing that Hilbert's Hotel is a paradox. Do you dispute that other mathematicians refer to it as a paradox? It appears as though even The Concise Oxford Dictionary of Mathematics (2009) considers it a paradox.