Question About Hilbert's Hotel

[Tomdstone]

Tom
What do you mean by an interval of time, if not a duration with a finite length??
Yppop

The time intervals that I had given above may become shorter in length by 1/2 but they do not disappear. For any point in the interval 0,1), there will be an interval in the scheme given, that contains it.

 

[Tomdstone]

Tom,
What do you mean by “instants”. If you mean a “point” in time then you are correct to say there are an infinite number of instances of time in 1 hour, 1 minute, 1 second, etc.
If you are talking about the rational numbers, then there are intervals because there is always a “next” number, but an “instant” can be defined as a “point” in time and since it must be associated with a rational number we are then talking about discrete space.

On the other hand, if you are talking about continuous space, there is no “next” point since there is no “next” real number. There can be no interval or probably no instant in continuous space. The correct observation to be gotten from Zeno’s paradoxes is: if there is motion, space cannot be continuous and if space is continuous there can be no motion. If Zeno’s observation is correct what conclusion can you derive?

Yppop

When dealing with continuous space I believe we are dealing with the spiritual element of reality. Only God is actually infinite and that is why the infinitude of continuous space is an analog of the spiritual. We are immersed in the Mind of God. It is amazing that God has laid out this conundrum of infinity for us to contemplate.
Yppop

An instant is a point in time. And I believe that time is continuous and not discrete in the sense that between any two distinct points in time, a, b with a<b, there will always be one which is in between: There exists c with a<c<b.

 

[rossum]

That’s my point. Saying all the even plus odd numbers equal all the even numbers denies that there are different types of infinity

No mathematician says that. The cardinal number of the set of integers is the same as the cardinal number of the set of even integers and is also the same as the cardinal number of the set of odd integers, the set of prime numbers, the set of multiples of 42 and a great many other countably infinite sets.

The cardinal number of the set of real numbers is larger, however, as Cantor showed.

rossum

 

[JapaneseKappa]

Tom
What do you mean by an interval of time, if not a duration with a finite length??
Yppop

Cute, but this definition has a certain lack of mathematical imagination.

Yes, there are an infinite number of intervals with finite length. However, when I say this, I do not mean that in the additive sense. Rather, I mean that there is an interval which goes from 0 to 0.1, an interval which goes from 0 to 0.2, an interval which goes from 0 to 0.3, and so on. There are an infinite number of such intervals that start at 0 and end at some number 0<n<1.

 

[thinkandmull]

No mathematician says that. The cardinal number of the set of integers is the same as the cardinal number of the set of even integers and is also the same as the cardinal number of the set of odd integers, the set of prime numbers, the set of multiples of 42 and a great many other countably infinite sets.

The cardinal number of the set of real numbers is larger, however, as Cantor showed.

rossum

I believe mathematicians have been lead astray for a non-philosophical Cantor. First, nothing is countably infinite. That’s the first huge error. Second, you are thinking, because of the first error, or infinity as only a line with one step after the other. Its a lot more than that

 

[thinkandmull]

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.

Who’s Alex Pruss?

 

[rossum]

I believe mathematicians have been lead astray for a non-philosophical Cantor.

You may believe anything you want. If you wish to convince mathematicians, then you are going to have to produce a mathematical argument to show that Cantor was wrong. There is indeed some argument, still, about Cantor’s continuum hypothesis, but that has no bearing on the countable infinity of the integers.

First, nothing is countably infinite.

Where is your proof? If the integers are not countably infinite, then they are a finite set. If so, then what is the largest integer?

Second, you are thinking, because of the first error, or infinity as only a line with one step after the other. Its a lot more than that

I am well aware of the various infinities: Aleph-0, Aleph-r, Aleph-1, Aleph-2 etc. The continuum hypothesis relates to whether Aleph-r = Aleph-1 or not. As with so much of mathematics, infinity is not a simple concept.

rossum

 

[thinkandmull]

The reason is the 1-10 is great than 1 3 5 7 9. We can push these both to infinity, but the first will still be larger.

 

[rossum]

The reason is the 1-10 is great than 1 3 5 7 9. We can push these both to infinity, but the first will still be larger.

That statement is meaningless. I suggest that you restate it, saying what you actually meant to say.

For instance your “1-10” might be a subtraction, a range (inclusive or exclusive at either end) or perhaps something else. Whatever it it, it is not a set since if it was a set you should have written it as {1 … 10}. Cantor deals with the cardinal numbers of sets. That is how the Aleph numbers are defined: Aleph-0 = C{Z}. Yes, I did teach maths for five years.

rossum

 

[thinkandmull]

1 thru ten is greater than 1 3 5 7 9. We can push these both to infinity, but the first series will still be larger.

 

[Tomdstone]

1 thru ten is greater than 1 3 5 7 9. We can push these both to infinity, but the first series will still be larger.

It is true that the set {1,2,3,4,5,6,7,8,9,10} contains ten elements which is more than {1,3,5,7,9} which contains 5 elements. But {1,2,3,4,5,6,7,8,9,10} contains ten elements as does{1,3,5,7,9,11,13,15,17,19}. they both contain ten elements. As you push them toward infinity, they will both contain the same number of elements.

 

[Tomdstone]

1-An actual infinite is impossible.

An actual infinite is possible because there are an infinite number of points in the interval [0,1]. Each one of these points can correspond to a point in time.

 

[thinkandmull]

It is true that the set {1,2,3,4,5,6,7,8,9,10} contains ten elements which is more than {1,3,5,7,9} which contains 5 elements. But {1,2,3,4,5,6,7,8,9,10} contains ten elements as does{1,3,5,7,9,11,13,15,17,19}. they both contain ten elements. As you push them toward infinity, they will both contain the same number of elements.

Nop. By adding 11 13 15 17 and 19 you are adding 11 thru 20 as well to the first set. The first set will always be larger. This philosophical truth is buried today because everyone like Cantor, although he was a product of the analytical age. In those Bertrand Russell days, they minimalized what the mind could know. Cutting the mind off from philosophy has bled over into math in this case

 

[Tomdstone]

Nop.

{1,2,3,4,5,6,7,8,9,10} contains ten elements.
And the set {1,3,5,7,9,11,13,15,17,19} contains ten elements.
It is a simple counting of ten objects.

 

[thinkandmull]

You are not saying that {1,2,3,4,5,6,7,8,9,10} equals {1,3,5,7,9,11,13,15,17,19], but that it equals 1 thru 20

If you want to be an analytical thinker that’s fine

 

[Rhubarb]

Nop. By adding 11 13 15 17 and 19 you are adding 11 thru 20 as well to the first set. The first set will always be larger. This philosophical truth is buried today because everyone like Cantor, although he was a product of the analytical age. In those Bertrand Russell days, they minimalized what the mind could know. Cutting the mind off from philosophy has bled over into math in this case

Wait, what? The elements of the set 1 through 10 is indeed larger than the set of the odd numbers between 1 and 10. But both of these are finite sets. They DO have limits, those two sets. The set of all whole numbers is limitless. The set of all odd whole numbers is limitless. They are both countably infinite. You’re letting your pre-theoretical intuitions muddle the conceptual fact of the matter. “Infinity” isn’t a number. You can’t have “infinity minus one.”

A foot, or an hour, can be infinitely divided. (At least as far as I’ve found a good argument for) This doesn’t mean there’s ‘infinity points’ between 0 and 12 inches. It means that the space between 0 and 12 inches can be divided smaller and smaller without limit.

 

[Tomdstone]

You are not saying that {1,2,3,4,5,6,7,8,9,10} equals {1,3,5,7,9,11,13,15,17,19], but that it equals 1 thru 20

If you want to be an analytical thinker that’s fine

Here’s what I said:
{1,2,3,4,5,6,7,8,9,10} contains ten elements.
And the set {1,3,5,7,9,11,13,15,17,19} contains ten elements.
It is a simple counting of ten objects.

 

[thinkandmull]

If a segment can be infinitely divided, then the points are all there, whether you think of them or separate them or not. If you pass over a bridge with 7 planks, you pass over the seven planks.

Now the set of all odd numbers is a smaller infinity than the set of all odd plus even numbers. In denying this, I knew you would end up denying there can be a greater infinity, and behold, you have: “The set of all whole numbers is limitless. The set of all odd whole numbers is limitless. They are both countably infinite.”

 

[JapaneseKappa]

A foot, or an hour, can be infinitely divided. (At least as far as I’ve found a good argument for) This doesn’t mean there’s ‘infinity points’ between 0 and 12 inches. It means that the space between 0 and 12 inches can be divided smaller and smaller without limit.

Question: how many ways could you cut the space you described into two (not necessarily equal) parts?

For example: I could cut the space at 1 inch, giving me two parts:
one part goes from 0 to 1 inch.
the other part goes from 1 to 12 inches.

How many other ways are there?

 

[Rhubarb]

I haven’t denied there are greater types of infinity. How did you pull that out? I said the set of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} has more elements than the set {1, 3, 5, 7, 9}, naturally. But these sets are finite. They HAVE limits. We set the limits when we defined the set - whole numbers between 1 and 10 for the former and whole odd numbers between 1 and 10. The set of ALL odd whole numbers is not smaller than the set of ALL whole numbers. They are both limitless.

I don’t know what you’re driving at, about planks. Sure, you can cross over one bridge. Or one-seventh of a bridge seven times. Or one fourteenth of a bridge fourteen times, etc. The SUM of certain infinite series (such as the interval of a bridge) can converge to a finite limit. That doesn’t mean the set that makes up the series isn’t without limit.