Question About Hilbert's Hotel

[Pink_Elephants]

To any mathematically inclined person, I’ve recently come across Hilbert’s Hotel in a talk given by Christian philosopher William Lane Craig:

youtube.com/watch?v=RjCrUEwzCxc

Try as I might I have no idea what he is talking about. Math was easily my worst subject in school. How exactly does Hilbert’s Hotel have to do with the Universe having a cause? What are the objections of atheists on this point?

Also, does a person really need to understand all of the details of a particular argument for God? This argument seems highly technical and not to mention boring. I’m almost at the point of throwing up my hands and leaving this argument to those who are mathematically inclined.

Thanks

 

[Mary_Estelle]

Don’t worry about it. We are told that math is infinite and in the Bible what we believe and accept, God is infinite. Even the great mathmatictions can not agree. It is an abstract.

 

[Tomdstone]

To any mathematically inclined person, I’ve recently come across Hilbert’s Hotel

You have a hotel with a countably infinite number of rooms, all of which are occupied. A new guest arrives and wants a room, but all are occupied. So the hotel manager moves the guest c in room 1 to room 2, the guest currently in room 2 to room 3, and so on, moving every guest from his current room n to room n+1. After this, room 1 is empty and the new guest can be moved into that room. So even though all the rooms have been occupied, it is still possible to accomodate a new guest, or a finite number of guests, or even a countably infinite number of guests in the hotel.

 

[agnes_therese]

“Countably infinite” – see, this is why I stick with history instead of math.

 

[Flopfoot]

Countably infinite means that if you choose any given number in the infinitely sized set, you can count to it. So the set of whole numbers is countably infinite, but the set of real numbers is uncountably infinite (you can count to 100000000, but you can’t count to pi).

 

[belorg]

To any mathematically inclined person, I’ve recently come across Hilbert’s Hotel in a talk given by Christian philosopher William Lane Craig:

youtube.com/watch?v=RjCrUEwzCxc

Try as I might I have no idea what he is talking about. Math was easily my worst subject in school. How exactly does Hilbert’s Hotel have to do with the Universe having a cause? What are the objections of atheists on this point?

Also, does a person really need to understand all of the details of a particular argument for God? This argument seems highly technical and not to mention boring. I’m almost at the point of throwing up my hands and leaving this argument to those who are mathematically inclined.

Thanks

Very simple answer: Craig is wrong. Hilbert’s Hotel in no way shows that the universe is finite, and that’s because there is no contradiction in Hilbert’s hotel, just as there is no contradiction in saying that there are as many even numbers as there are natural numbers.

 

[Rhubarb]

It sounds to me that Craig is taking issue with the properties of infinity - that you can put an infinite number of guests into an infinite number of full hotel rooms. This isn’t absurd - this is how infinite sets work. It seems super counter-intuitive, but it’s just the way infinity works.

That aside, I’m not sure that an infinite number of events, in itself, is absurd.

 

[Linusthe2nd]

To any mathematically inclined person, I’ve recently come across Hilbert’s Hotel in a talk given by Christian philosopher William Lane Craig:

youtube.com/watch?v=RjCrUEwzCxc

Try as I might I have no idea what he is talking about. Math was easily my worst subject in school. How exactly does Hilbert’s Hotel have to do with the Universe having a cause? What are the objections of atheists on this point?

Also, does a person really need to understand all of the details of a particular argument for God? This argument seems highly technical and not to mention boring. I’m almost at the point of throwing up my hands and leaving this argument to those who are mathematically inclined.

Thanks

Don’t worry about it, just read the Bible, look at the beauty and order of the universe. That is all the proof anyone needs. But if you still have a hankering for intellectual proofs give this a try: strangenotions.com/god-exists/

Linus2nd

 

[thinkandmull]

Very simple answer: Craig is wrong. Hilbert’s Hotel in no way shows that the universe is finite, and that’s because there is no contradiction in Hilbert’s hotel, just as there is no contradiction in saying that there are as many even numbers as there are natural numbers.

All the odd numbers line up with all the odd (or even), not both with the odd and even. 1 3 5 line up with 2 4 6, not 1 2 3 4 5 6. Math people often make this mistake

 

[Tomdstone]

All the odd numbers line up with all the odd (or even), not both with the odd and even. 1 3 5 line up with 2 4 6, not 1 2 3 4 5 6. Math people often make this mistake

No. 13579 etc. is in 1-1 correspondence with 1234567 etc so the two sets do “line up”.

 

[thinkandmull]

Not so. All numbers are a bigger infinity than all the odd numbers, although its hard to see when you are going to infinity by single steps

 

[thinkandmull]

To any mathematically inclined person, I’ve recently come across Hilbert’s Hotel in a talk given by Christian philosopher William Lane Craig:

youtube.com/watch?v=RjCrUEwzCxc

Try as I might I have no idea what he is talking about. Math was easily my worst subject in school. How exactly does Hilbert’s Hotel have to do with the Universe having a cause? What are the objections of atheists on this point?

Also, does a person really need to understand all of the details of a particular argument for God? This argument seems highly technical and not to mention boring. I’m almost at the point of throwing up my hands and leaving this argument to those who are mathematically inclined.

Thanks

Who is the person Craig mentions at the beginning?

 

[Tomdstone]

Not so. All numbers are a bigger infinity than all the odd numbers, although its hard to see when you are going to infinity by single steps

Both sets are “countably infinite.” So the number of odd integers is the same as the number of all integers. Have you heard of Georg Cantor?

 

[thinkandmull]

Countable infinity is an absurdity. It is trying to understand the infinite through succession of the finite. The odd numbers equal all the even numbers, not all the integers, regardless of what modernist math thinkers say today. I’ve already demonstrated how you are mistaken from correspondence: 1 3 5 7 9 correspond to 2 4 6 8 10, not 1-10

 

[Tomdstone]

Countable infinity is an absurdity. It is trying to understand the infinite through succession of the finite. The odd numbers equal all the even numbers, not all the integers, regardless of what modernist math thinkers say today. I’ve already demonstrated how you are mistaken from correspondence: 1 3 5 7 9 correspond to 2 4 6 8 10, not 1-10

I don’t believe you are correct. Look up Georg Cantor.

 

[thinkandmull]

Infinity is not as linear as he thought

 

[Tomdstone]

Infinity is not as linear as he thought

There are different types of infinity.

 

[Triflelfirt]

Very simple answer: Craig is wrong. Hilbert’s Hotel in no way shows that the universe is finite, and that’s because there is no contradiction in Hilbert’s hotel, just as there is no contradiction in saying that there are as many even numbers as there are natural numbers.

I don’t think you understand Craig’s argument at all. Hilbert’s Hotel doesn’t have to be “contradictory” to prove Craig’s point. That’s because what Craig is arguing for is ontological finitism, not mathematical finitism.

Now, to answer OP’s question: Hilbert’s Hotel is used as an argument in favor of premise 2 of the Kalam cosmological argument. The Kalam cosmological argument is presented as follows:

1-Everything that begins to exist has a cause
2-The universe began to exist
C-Therefore, the universe has a cause

To defend premise 2, Craig uses 2 different philosophical arguments and 2 scientific arguments. The first philosophical argument can be defined as follows:

1-An actual infinite is impossible
2-An infinite past would imply an actual infinite number of past events
C-Therefore, the past is finite

Hilbert’s Hotel is used as an argument for premise 1 here (premise 2 is fairly obvious; there would be an actual infinite of past events if the universe was past eternal, or there could be an actual infinite of particles or whatever – Alex Pruss seems to have made an argument like that, but I don’t quite remember it) to show that an actual infinite, while being mathematically acceptable, would be absurd in concreto, that is, in our reality. A hotel like Hilbert’s Hotel would not be possible in our world.

There are other possible examples: imagine a library with an infinite number of books. You walk in and take one of the books, but actually the library still has the same number of books it had before you took a book from it! Such paradoxes wouldn’t be a problem in mathematics (at least not for mathematical infinitists; intuitionists would find problems with it) because mathematics has very precise rules that would prevent people from doing absurd operations with the infinite, but there is nothing in the real world that could, for example, prevent me from taking a book from an infinite library. This shows how an actual infinite could not possibly exist in concreto. Thus, following the above syllogism, the past is finite. But that means (according to the argument) that the universe has a cause.

I recommend you to read Craig’s books on the subject, if you’re interested. Pick up Reasonable Faith or On Guard. They’re good books.

 

[Tomdstone]

I don’t think you understand Craig’s argument at all. Hilbert’s Hotel doesn’t have to be “contradictory” to prove Craig’s point. That’s because what Craig is arguing for is ontological finitism, not mathematical finitism.

Now, to answer OP’s question: Hilbert’s Hotel is used as an argument in favor of premise 2 of the Kalam cosmological argument. The Kalam cosmological argument is presented as follows:

1-Everything that begins to exist has a cause
2-The universe began to exist
C-Therefore, the universe has a cause

To defend premise 2, Craig uses 2 different philosophical arguments and 2 scientific arguments. The first philosophical argument can be defined as follows:

1-An actual infinite is impossible
2-An infinite past would imply an actual infinite number of past events
C-Therefore, the past is finite

Hilbert’s Hotel is used as an argument for premise 1 here (premise 2 is fairly obvious; there would be an actual infinite of past events if the universe was past eternal, or there could be an actual infinite of particles or whatever – Alex Pruss seems to have made an argument like that, but I don’t quite remember it) to show that an actual infinite, while being mathematically acceptable, would be absurd in concreto, that is, in our reality. A hotel like Hilbert’s Hotel would not be possible in our world.

There are other possible examples: imagine a library with an infinite number of books. You walk in and take one of the books, but actually the library still has the same number of books it had before you took a book from it! Such paradoxes wouldn’t be a problem in mathematics (at least not for mathematical infinitists; intuitionists would find problems with it) because mathematics has very precise rules that would prevent people from doing absurd operations with the infinite, but there is nothing in the real world that could, for example, prevent me from taking a book from an infinite library. This shows how an actual infinite could not possibly exist in concreto. Thus, following the above syllogism, the past is finite. But that means (according to the argument) that the universe has a cause.

I recommend you to read Craig’s books on the subject, if you’re interested. Pick up Reasonable Faith or On Guard. They’re good books.

An actual infinite is not impossible. Between 1:00 PM today and 2:00 PM today you have an actually infinite number of time intervals.

 

[Triflelfirt]

An actual infinite is not impossible. Between 1:00 PM today and 2:00 PM today you have an actually infinite number of time intervals.

1- No, we don’t have an “actually infinite number of time intervals” between 1:00 PM and 2:00 PM, only a potential infinite number of time intervals. Craig isn’t so stupid as to overlook “objections” like that; he writes about them in detail (for instance, in his article on the Blackwell Companion to Natural Theology), while also dealing with Zeno’s paradoxes. There’s one absolute difference between 2 and 1, and that is 1;

2- Regardless, would you then believe that a library with an infinite number of books (where we could take a book and the library would still have the same number of books it had before we took a book from it) would be possible? I find that a very, very hard bullet to bite. But absurdities like that would have to follow from the possibility of actual infinites in concreto.