Even if theism is a failure, it's still superior to atheism

[polymath257]

Hibert's Hotel Assumes 1 guest in 1 room. Why can't the rooms have infinite capacity ?.....

Countable or uncountable?

William Craig likes to use the Hilbert Hotel in an argument against the possibility of infinite regress. He only manages to show he has no real understanding of what is going on.

The problem is that there are many ways to describe the 'size' of a set.

We can say that a set A is smaller than a set B if A is a subset of B.

We can use Cantor's idea and say that A is smaller than B is there is an injective map from A into B and none the other direction.

If the sets are subsets of three dimensional space, we can use volume to describe their size. For subsets of the natural numbers, we can use density.

So, for example, every countably infinite subset of three dimensional space has a volume of zero. And two different volumes will still be the 'same size' according to cardinality. So it is possible to be 'large' in one sense and small in a different sense. it is even possible for an uncountable set to have zero volume.

A set is infinite in terms of cardinality precisely when there is a proper subset that is in one-to-one correspondence with the original set.

Also, the notion of 'infinite' has traditionally had several different versions.

So, we can talk about cardinality. We can talk about ordinality.

We can talk about boundedness or unboundedness.

We can talk about limits (which are close to the classical idea of potential infinity).

But it is possible, for example, for an unbounded set to have finite volume. Limits have almost nothing to do with cardinality. Etc.

The reason there is no much confusion is often that people expect all the different notions to give the same answers and they simply do not  always do so.