The "absurdity" of Hilbert's Hotel

This is a continuation of "A few things wrong about the cosmological argument," an ongoing series.  Today we will discuss William Lane Craig's treatment of Hilbert's Hotel.  I will assume you are familiar with Hilbert's Infinite Hotel; if not, you can read William Lane Craig, me, or any other source on the internet.

William Lane Craig (WLC) thinks Hilbert's Infinite Hotel is absurd.  If Hilbert's Hotel is full, then you can add or remove people and it will still have the same number of guests.  Absurd!

Can anyone sincerely believe that such a hotel could exist in reality? These sorts of absurdities illustrate the impossibility of the existence of an actually infinite number of things.

This is an argument from absurdity, which has the following form:

A implies B.
Both me and my opponents agree that B is obviously false.
Therefore we should agree that A is false.

There is nothing wrong with an argument from absurdity if it is done correctly.  But it is not done correctly.  I can agree that Hilbert's Hotel is absurd.  But not all infinities are necessarily absurd.  We can have infinities without specifically having an infinite hotel.  Hilbert's Hotel is absurd because we will never have the resources, the manpower to create such a hotel.  We don't have the people to fill it, the means to maintain it.

Of course, WLC thinks there is a more abstract property of Hilbert's Hotel which is absurd.  Specifically, he claims it is absurd that you can add or remove things, and still have the same number of things.  I do not think this is absurd.  The argument from absurdity relies on my agreement at this point, but I do not agree.

(An aside: WLC makes a technical error here.  We do not say that there is the "same number" of things, we say that the two sets of things have the "same cardinality".  Neither set has a well-defined "number" of things, since both sets are infinite.  There is a reason we use technical terms like "cardinality", and it is to avoid making mistakes by accidentally applying intuition where it does not apply.  WLC may have simply wanted to avoid technical jargon...)

WLC cites a couple people who objected, like me, that infinities are not absurd.  His basic response is, "It looks pretty absurd to me."  "Nuh uh."  "Yeah too."  We seem to be at an impasse.  If you're keeping track, that means WLC lost, since he is the one presenting the argument, and he has failed to convince.  But let's stop keeping track, and focus on resolving the impasse.

An anecdote: I first learned about infinite sets in high school.  This was back in the day, when the internet was still a novelty to me, and I wasn't smart enough to use a pseudonym.  I used to exchange puzzles with people and argue about mathematics.  In one of these arguments, someone told me to read about infinite set theory.  It was crazy!  Infinite sets blew my mind.  But by the time I got to college, they became intuitive and familiar, like a favorite old joke.  Non-math people think that when math people get together, they make jokes about pi and squares.  In my experience, they make jokes about infinite sets.  And yet, the set of untold jokes about infinite sets remains as big as ever.

I contend that infinities are not "absurd" in the sense of "obviously false".  Rather, infinities are only counter-intuitive (and only at first).  Infinite set theory is well-established in mathematics.  Due to some complications*, it is impossible for me to simply prove that set theory is consistent.  But we think that set theory is consistent for the same reason we think arithmetic is consistent, and I don't see WLC waving his arms incredulously at arithmetic.

*See Godel's second incompleteness theorem.

This is a point that clearly needs a response, so WLC has already responded to it.

Hence, one could grant that in the conceptual realm of mathematics one can, given certain conventions and axioms, speak consistently about infinite sets of numbers, but this in no way implies that an actually infinite number of things is really possible.

WLC distinguishes between "logical" possibility and "real or factual" possibility.  Unfortunately, this places infinite sets in a very odd place.  What kind of absurdity is this, that is too absurd for the real world, but not absurd enough for mathematics?  And for what kind of "real" is it too absurd for?  Lastly, if there are different degrees of absurdity, how do we know which degree it is?

There is no way to answer these questions, because "absurdity" is an intuitive idea.  Either we think something sounds absurd or it doesn't.  Intuition doesn't whisper in our ear, "It is absurd in reality, but not mathematics.  Also, 'reality' is a category that includes past events, but not future events, and it excludes inconvenient counterexamples."  Anyways, my intuition never says anything like that to me.

Put it this way.  If WLC didn't know anything about what mathematicians said, he would have guessed infinite sets were bad math.  And then when he finds out that it's good math, what is the appropriate response?  WLC's response is to say that infinite sets are still absurd, just not in mathematics.  I think the proper response is to revise what we previously thought was absurd.

I must say one last thing, as a physicist.  WLC thinks absurdity is a good argument against a physical theory?  Even when said absurdity is mathematically consistent?  I question his knowledge of modern physics.

"A few things wrong about the cosmological argument"
1. Actual and potential infinities
2. Actual infinities in physics
3. What is real?
4.  The "absurdity" of Hilbert's Hotel
5. Interlude: God is infinite