Question About Hilbert's Hotel

[Rhubarb]

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?

Infinitely, I suppose? I’m not a mathematician. But that seems right? Any arbitrary place you decide to cut, that distance can be halved and the cut can be placed there instead.

I mean, unless we discover that space is made of discrete units. But that’d be weird.

 

[JapaneseKappa]

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.”

Prove it. Use a mathematical proof. It really isn’t hard or a particularly long proof. It’s a pretty textbook problem that you’d find in any intro to real analysis course.

 

[Tomdstone]

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?

Uncountably infinite.

 

[Rhubarb]

Uncountably infinite.

Oh, yeah. That’s what I meant to say. Because we’re using real numbers.

 

[rossum]

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.

I’m afraid you do not know enough mathematics to discuss this sensibly. You appear to be saying, C{1 …10} > C{1, 3, 5, 7, 9} which is correct, but you are not expressing it clearly.

Unfortunately for your argument, if you “push these both to infinity” then the cardinal numbers of the two sets will be the same. The contents of the two sets can be put into a one-to-one correspondence so the two sets have the same number of elements.

I suggest that you drop this particular subject until you have learned more about it. You are making too many avoidable errors at the moment.

rossum

 

[Rhubarb]

Perhaps this table will help. For every new element we put on the left, we can put one on the right, when we take both sets without limit. (To infinity) While at first glance it might seem like the former should be twice the latter - after all, it has odd AND even numbers - when we continue without limit we see that the latter set always has an element to line up with an element in the former. They have the same number of elements. Both sets have 13 elements showing - and they can both go on without limit.

All whole numbers | All odd whole numbers

1 → 1
2 → 3
3 → 5
4 → 7
5 → 9
6 → 11
7 → 13
8 → 15
9 → 17
10 → 19
11 → 21
12 → 23
13 → 25
. → .
. → .
. → .

 

[Usagi]

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.

I know that seems right to you, but the “one-to-one correspondence” trick puts that entirely to rest, at least in my mind.

Look, I will start counting the odd numbers.

1, 1.
3, 2.
5, 3.
7, 4.
9, 5.
11, 6.
13, 7.
15, 8.
17, 9.
19, 10…

Notice that counting the odd numbers is the same thing as putting them into a one-to-one correspondence with the whole numbers. That’s what counting anything is; if I want to know how many apples I have, I assign each one a whole number in order until I reach the last one (“one apple, two, three, four, five … Okay, there are five.”)

Of course, in this case there is no “last” odd number, so we cannot literally finish counting them. But the count we have so far shows a pattern (the odd number n corresponds to the whole number 2n-1). Is there any mathematical reason to believe that pattern will change? Can we somehow “run out” of odd numbers and still have whole numbers left over? I certainly don’t think so. That’s why we refer to an infinity like that of the odd numbers as “countable” (because the set of them can provably be placed in one-to-one correspondence with the set of whole numbers), even though we can’t literally count them all. It’s also how we know that the two sets have the same cardinality (are the same “size” infinity) even though both are endless.

Yes, there are five more whole numbers between 1 and 10 than there are odd numbers. But when we’re talking about infinite sets, that makes no difference. I could remove the first five members from the set of whole numbers, starting my count with 6, and that wouldn’t actually get me any closer to the “end,” as there is not one. That awesome last verse of “Amazing Grace” is being not merely poetic but mathematically accurate when it claims that living ten thousand years of infinite time will not shorten the time remaining even by a day. You could, indeed, put “total number of days in infinite time” into one-to-one correspondence with “total number of days in infinite time, less 3,650,000” and the two sets would prove to be of the same cardinality.

Usagi

 

[Pink_Elephants]

Who is the person Craig mentions at the beginning?

I believe he is referring to Al-Ghazali. I did find an interesting exchange between Richard Swinburne and William Lane Craig on this subject.

It appears that some philosophers just like to argue.

 

[Pink_Elephants]

There are an infinite number of split seconds in a one hour period. There is a scheme to show what they are.

Would an example of this be illustrated in the Simpsons episode Itchy & Scratchy Land where in the “Scratchtasia” segment Scratchy literally chops Itchy into millions of little Itchys? With every chop he just creates more and more Itchys LOL

Triflelfirt, thank you for explaining Craig’s argument. I have a much better understanding of it now.

 

[Rhubarb]

I believe he is referring to Al-Ghazali. I did find an interesting exchange between Richard Swinburne and William Lane Craig on this subject.

It appears that some philosophers just like to argue.

Argue is what philosophers are paid to do. It’s their job.

 

[thinkandmull]

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.

First paragraph: Sure they are both limitless, but that is your proof? There are larger and smaller infinities. Infinity plus one is larger than just infinity. You are giving up logic in order to guard a simple error made in “established math”

Second: you said there are not infinity of points between do A and B. That’s false. There is no such thing as a potential infinity without there being an actual infinity which needed to be divided.
[/quote]

 

[thinkandmull]

I’m afraid you do not know enough mathematics to discuss this sensibly. You appear to be saying, C{1 …10} > C{1, 3, 5, 7, 9} which is correct, but you are not expressing it clearly.

Unfortunately for your argument, if you “push these both to infinity” then the cardinal numbers of the two sets will be the same. The contents of the two sets can be put into a one-to-one correspondence so the two sets have the same number of elements.

I suggest that you drop this particular subject until you have learned more about it. You are making too many avoidable errors at the moment.

rossum

I am not mistaken here. All the odd numbers have a one to one correspondence to all the even numbers, but there are left over the odd numbers of the second set. You guys are trying to count to infinity. Its not philosophically sound, and its a misuse of math. What else does a larger infinity mean than that there are more units in the larger one?

 

[JapaneseKappa]

I am not mistaken here. All the odd numbers have a one to one correspondence to all the even numbers, but there are left over the odd numbers of the second set. You guys are trying to count to infinity. Its not philosophically sound, and its a misuse of math. What else does a larger infinity mean than that there are more units in the larger one?

Prove it. Use a mathematical proof. It really wouldn’t be a hard or a particularly long proof. It’s a pretty textbook problem that you’d find in any intro to real analysis course.

 

[thinkandmull]

Here’s is what I’ve been wondering about today:

Take cube 4 by 4 by 4. Break it in half, than break that piece in half, and again. (This will go on forever) Line up all the pieces from the largest (half the first cube) to the smallest (which doesn’t exist, because you can always divide it further).

Now from this it seems that geometry, here described, contradicts arithmetic, because you can cross of the dimensions of the sphere, yet you cannot get to the end of its parts lined up. Something finite seems to explode with infinity

 

[thinkandmull]

Prove it. Use a mathematical proof. It really wouldn’t be a hard or a particularly long proof. It’s a pretty textbook problem that you’d find in any intro to real analysis course.

I already have. 1 thru ten is greater than 1 3 5 7 9. For one infinity to be greater than another requires one infinity to have more units than another. Real numbers have an infinite number greater than merely odd numbers

 

[JapaneseKappa]

I already have. 1 thru ten is greater than 1 3 5 7 9. For one infinity to be greater than another requires one infinity to have more units than another. Real numbers have an infinite number greater than merely odd numbers

That is not a real mathematical proof. You have failed to demonstrate why two arbitrary subsets of the infinite sets should have the same properties as the infinite sets themselves.

If you think your statement does constitute a real proof, then you would be proving Rossum’s accusation: “I’m afraid you do not know enough mathematics to discuss this sensibly” correct.

 

[thinkandmull]

You are the one saying an arbitrary subset of an infinite set has the same properties as the infinite set itself. I know enough about infinity to understand this issue, but maybe you don’t.

 

[thinkandmull]

I am starting to believe in idealism.

This is fun, but maybe unrelated: youtube.com/watch?v=WabHm1QWVCA

 

[JapaneseKappa]

You are the one saying an arbitrary subset of an infinite set has the same properties as the infinite set itself. I know enough about infinity to understand this issue, but maybe you don’t.

In order for your loose collection of ideas to be a proof, you are missing several crucial steps.

The original idea you were trying to prove was:

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

We were talking about the relative size of two sets, the odd numbers and the naturals, not the real numbers, as you have (perhaps deceptively) stated in other posts.

Your “proof” of the proposition that the set of evens is a different size from the set of naturals is roughly:

Given some natural n (you chose 10 as an example), the set of naturals that are less than n has more elements than the set of even numbers that are less than n.

You therefore conclude that the set of even naturals has fewer elements than the set of naturals. However, this is a non-sequitur. I can easily demonstrate this by asking the following:

1. Given some natural n, how many naturals are there that are greater than n?
2. Given some natural n, how many even numbers are there that are greater than n?

Surely someone who “knows enough about infinity to understand” will realize that both the sets described in 1. and 2. have an infinite number of elements.

I’m sure that someone who claims to know about infinity would also immediately realize that the number of elements in a set could be expressed as:

given some natural n, the total number of elements in the set is:

the number of elements, x, such that n >= x
+
the number of elements, y, such that y > n

Now, as I’m sure you already know, the first set, 0 < x <= n is finite for any n, for both the even numbers and the naturals. I’m also sure that you know the second set is infinite, for any n. So you must finally realize that in both cases (the naturals and the evens) the total size of the set is not determined by the finite set, n >= x > 0, but rather by the infinite set, y > n

If you have any sort of mathematical competence, you will see the problem here.

 

[thinkandmull]

Again, your “proof assumes that all infinities are equal”. This is not hard to understand. If you have a straight line going forever north from where you are sitting, and another going south, both lines together are longer than just one, although all three are infinite. Most mathematicians don’t think about this like that because they are entrenched in there analytic world