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

[Jehanne]

Seems to me like you could have two tallest trees of equal height. No way to prove there's just one without measuring.

The cardinality of the set of primes and that of the natural numbers are identical, even though all the even numbers (except for #2) are not prime.

[Angrboda]

The main area that Rotkit is wrong in his claim that we can demonstrate the existence of a tallest tree without observation or detection is that one or the other is required to confirm the existence of at least one tree. If there are zero trees, then there is no tallest one. Another difficulty is if there are an infinite number of trees. In that case, demonstrating the existence of a tallest tree is immeasurably harder.

[BrianSoddingBoru4]

The main area that Rotkit is wrong in his claim that we can demonstrate the existence of a tallest tree without observation or detection is that one or the other is required to confirm the existence of at least one tree.  If there are zero trees, then there is no tallest one.  Another difficulty is if there are an infinite number of trees.  In that case, demonstrating the existence of a tallest tree is immeasurably harder.

Similarly, since he’s discounting observation and detection, we can claim that all existing trees are of equal height.

Boru

[LinuxGal]

Seems to me like you could have two tallest trees of equal height. No way to prove there's just one without measuring.

The cardinality of the set of primes and that of the natural numbers are identical, even though all the even numbers (except for #2) are not prime.

Infinite rooms, booked solid with all the primes, and a convention of all even numbers shows up looking for rooms.  Proprietor David Hilbert says, "No problem!" and sends his bellhop to tell every nth prime to relocate to room 2n so the nth even number can have room 2n-1.

[The Architect Of Fate]

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

[Jehanne]

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

As an abstraction, or, in reality? If, as an abstraction, the rooms would not exist, physically, and would instead be abstract objects; if, physical, the space of the room would not be finite, but, infinite, which would mean that the room could not have six walls, which would denote a finite area.

Is God, which theists denote with infinite attributes, just an abstraction?

[polymath257]

I'm always leery about popular treatments of mathematics, especially treatments of infinity.

What would you say is the core of his argument?

He introduces ZFC and says that, apart from finitists, mathematicians no longer (since Cantor) view the concept of infinity as being equinumerous; as you know, there are an infinite number of infinite sets, one countable and an infinite number that are not.  Applying infinite attributes to a God in a qualitative manner is a cop-out, and doing so in a quantitative manner is meaningless.

One aspect is that there are several, very distinct, notions of infinity. Cardinality is only one and probably the easiest one to describe to a layperson. But there are also ordinals (having to do with the order structure). In ZFC, every cardinal number is an ordinal, but not the reverse.

Also, not only are there an infinite 'number' of different sizes of infinite sets, the collection of all cardinalities is a proper class and not a set (like the class of all sets). But this gets into the paradoxes of naive set theory (as opposed to axiomatic set theory--given by ZFC).

[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.

[Jehanne]

As Dr. James Lindsay wrote in his book, "This shit gets weird".

[The Architect Of Fate]

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

As an abstraction, or, in reality?  If, as an abstraction, the rooms would not exist, physically, and would instead be abstract objects; if, physical, the space of the room would not be finite, but, infinite, which would mean that the room could not have six walls, which would denote a finite area.

Is God, which theists denote with infinite attributes, just an abstraction?

It was a joke  Jeh.......