A Critique of William Lane Craig's Use of Hilbert's Hotel

This critique was written by Principia Atheologica based on exapologist's critique of Craig's Tristram Shandy paradox, which supposedly demonstrates the impossibility of an actual infinite.

John said...

Can you change your stylesheet so that links are more contrasted with regular text? Maybe it's just because I'm viewing on an LCD monitor, but I could barely tell what I was supposed to click on in this post.

John W. Loftus said...

We'll see what we can do.

TKD said...

Thanks for the link John! It's an honor to have caught your attention.

kiwi said...

The point of the Hilbert's Hotel (HH) paradox is to show that even if all rooms of a hotel with an infinite amount of rooms are occupied, you can still welcome an infinite amount of guests. (And indeed you can).

But you're arguing the hotel wasn't really full in the first place? ... What? If you claim the hotel isn't really full in the first place, then we're not longer talking about HH paradox. The paradox starts with the claim that the hotel IS really full.

I don't see any reason to accept your definition of full anyway. A hotel is full if all rooms are occupied. Changing the definition of full is like I said no longer the HH paradox.

Having said that, there cannot be a hotel with an infinite number of rooms; that's a chimera. So I think it's shocking that a philosopher would use the paradox to make a conclusion about reality.
It's unclear with the blog entry how Craig comes to his conclusion using the paradox, but I cannot see how he can be successful.

Steven Carr said...

Dealing with infinite numbers is always tricky.

Let us take a simpler version of the paradox.

Suppose we match all the integers up with the even numbers.

1-2, 2-4,3-6,4-8 etc.

Then all the integers are 'full' ie they have been matched up with even numbers.

How could we then match up a new even number which shows up? There is no room for it, as all the numbers are 'full'

We simply follow Craig and shuffle all the nummbers along 1. 2-2, 3-4, 4-6,5-8 etc.

And then match the new even number with the 1, which is no longer 'full.'

But this is absurd. No new even numbers are going to show up.

So we can match an Aleph-null set with another Aleph-null set.

But Craig has invented a scenario where new members of one infinite set are going to 'show up'.

Craig's point is that even if we match up one set of infinite things, with another set of infinite things, new members of the second set may 'show up', and they can not be matched, as the first set is 'full'.

But mathematicians already know there are different orders of infinity.

There is not just Aleph-null.

However, people form an Aleph-null set.

If Hilbert's Hotel is infinite and it is full, no new guests are going to show up.

Every single person must already be in the hotel.

The only way you can make an infinite hotel full is by putting everybody in it.

So new new guests are going to show up.

Just as if you match up the integers 1,2,3,4..., with the even numbers 2,4,6,8.... , no new even numbers are going to 'show up', demanding a room for the night.

Steven Carr said...

'But you're arguing the hotel wasn't really full in the first place? ... What? If you claim the hotel isn't really full in the first place, then we're not longer talking about HH paradox'

An infinite hotel cannot ever be full.

You can always make as much space as you like, by mapping the rooms to guests in any way that checks them all into specified room numbers.

kiwi said...

You're missing the point.

Arguing that the hotel is not really full in the first place is like someone watching an illusionnist trick and complaining: "you're not using a real saw!!!".

Just do like Jesus says and become a child again.

The hotel is full, as every room is occupied. But you can still welcome guests! OMG. That's magic.

Steven Carr said...

KIWI
Arguing that the hotel is not really full in the first place is like someone watching an illusionnist trick and complaining: "you're not using a real saw!!!".

CARR
But the infinite hotel isn't full, as you can create a mapping between guests and hotel room numbers that means the hotel is 0.00001% full.

kiwi said...

Yawn.

Gary Charbonneau said...

There is no paradox here. The premise is that there exists a set (the hotel) consisting entirely of an infinite number of elements of a certain type -- full rooms. Therefore, there is nothing either surprising or paradoxical about a potential guest not being able to find, in the hotel, an element of an entirely different type: an empty room. By the very nature of the initial premise, that type of element DOES NOT EXIST in the set (the hotel), because a room cannot be full and empty at the same time.

exapologist said...
This comment has been removed by the author.
exapologist said...

I should note that my criticism of Craig's kalam argument referenced by the current post doesn't address Hilbert's Hotel, which he uses to argue against the possibility of the existence of concrete actual infinites. Rather, my criticism is of Craig's most popular version of his argument against the possibility of traversing actual infinites. You can find my criticism here.

jgphxguy said...

Come on guys...The context of Craig's reference to Hilbert's Hotel was how you can not have an infinite regress of past events, and the fact that the universe BEGAN to exist. He used it again recently at Saddleback(Rick Warren's church)and can be seen on the web. It's airtight. The problem with atheism is the obsession of trying to fathom details of the unfathomable. Even Dawkins concedes that if God exists he is beyond what we could ever imagine, so therefore God can perform feats beyond what we could ever imagine or explain.