The fuller quote doesn't correct the problem to which I pointed. The symbol ℵ does not represent a "type of infinity," as Craig claims. Rather, it represents the cardinality of infinite sets. The cardinality of an infinite set isn't a "type of infinity." The cardinality of an infinite set is a description of an algorithm which one might use in listing the elements of that set.
Originally Posted by Adrift
I'm referring particularly to when Craig says this:
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."
Suppose all the people in the odd-numbered rooms check out – 1, 3, 5, 7, and so forth. How many guests are left? Well, all the even-numbered guests. An infinite number of guests are still left in the hotel even though an equal number has already checked out and left the hotel. But now let’s suppose instead that all of the guests in the rooms 3, 4, 5, 6, 7, out to infinity checked out. How many guests are left now? If there is a room #0, just three are left. Yet, the same number of guests checked out this time as when all of the odd-numbered guests left. You subtract identical quantities from identical quantities and you get non-identical results, which is absurd.
I have highlighted the particularly offending portion. It is not the case that the same number of guests checked out in each of those cases. Each case describes an infinite number of guests checking out, to be sure, but not the same infinite number. This is not a case of subtracting identical quantities from identical quantities to receive non-identical results, as Craig alleges.
Whether Craig came up with them himself or is simply repeating something he's heard others claim is irrelevant. The fact of the matter is that he presents a misunderstanding of the mathematics in order to support his claim that actual infinites do not exist.
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.
Nope, it's due to his misuse of math. If Craig were making legitimate mathematical arguments, I might disagree with him, but I wouldn't accuse him of being ignorant of mathematics. Our disagreement, in that case, would likely be a philosophical one, and I would cite those reasons for my disagreement. I have done exactly that, in fact, when I've discussed Dr. Norman Wildberger's view on the subject. Dr. Wildberger is a mathematics professor at the University of New South Wales. He understands the mathematics under discussion, but objects to some of the axioms underlying that mathematics.
In the end, I can't help but feel that your issue with Craig is less to do with his misuse
of math, and more that you disagree with those mathematicians that he relies on.
However, Craig doesn't understand the mathematics. He makes claims about the mathematics which are incorrect in order to support his claims about actual infinites.
Those mathematicians would argue that the axioms underlying the Hyperreals do not apply to the real world. They would not argue that Hilbert's Hotel on the Hyperreal number system results in mathematical absurdities.
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 have not. Even if Dr. Craig were to answer such a question, it would likely be a one-off reply, on his part. Given the complexity of the matter, and given Dr. Craig's numerous misunderstandings of the mathematics with which he is somewhat aware, I have no confidence that such a one-sided correspondence would be at all useful, on my part.
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?
Wade doesn't seem to have any understanding of the Hyperreals, either. When the user Gareth McCaughan mentions them, Wade misunderstands and thinks the gentleman is referring to Transfinite arithmetic.
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.
Beyond that, the discussion definitely illustrates some of the problems I have with Craig's discussion of Hilbert's Hotel. Wade mistakes the fact that subtraction is not defined on Transfinite arithmetic for implying that it is impossible to perform subtraction on infinite numbers. Then, after noting that subtraction is not defined, he tries to draw conclusions about that subtraction, which is preposterous. That'd be like bringing up an Onomblattive in a conversation, noting that you have no definition for what an "Onomblattive" actually is, then proceeding to claim that the Onomblattive therefore proves your point.
I certainly don't! People often use the phrase "paradox" colloquially to refer to something which is counter-intuitive or which can easily be misunderstood as self-contradictory. For example, the Twin Paradox of Special Relativity isn't actually a paradox, but because a simple misunderstanding can make it appear to be paradoxical, the name has stuck despite the problem having been resolved. Similarly, even some people who don't actually consider Hilbert's Hotel to be self-contradictory have referred to it as a "paradox," in this colloquial sense.
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.
If there are mathematicians who do still consider the thought experiment to be an actual paradox, I will be happy to disagree with them.