Question About Hilbert's Hotel

[Tomdstone]

They are equal in the same number of elements they have, but not equal element-by-element. .

Right. One infinite set can be a proper subset of another, but they will both have the same number of elements.

 

[thinkandmull]

Right. One infinite set can be a proper subset of another, but they will both have the same number of elements.

That is mathematically and philosophically absurd and you have not proven it nor refuted my reasoning. It doesn’t matter how many people jumped over this issue and accepted hasty reasoning by Cantor or others

 

[thinkandmull]

In modern times we are told that a finite area can be bounded by an infinite length. Why not relativism then?

 

[Rhubarb]

That is mathematically and philosophically absurd and you have not proven it nor refuted my reasoning. It doesn’t matter how many people jumped over this issue and accepted hasty reasoning by Cantor or others

You want… a proof that two sets of the same size has the same number of elements? Um.

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10} has the same number of elements as {1, 3, 5, 7, 9, 11, 13, 15, 17, 20} this is self-evident. (I hope you’ll agree)

Now, consider the sets:
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10… N+1} and
{1, 3, 5, 7, 9, 11, 13, 15, 17, 19…2N-1} (or whatever the function that spits out the odd-numbers are - it’s been awhile)

Now, chose some random number for N. Any number you like. Let’s start with N=198,476,501. Both sets will have 198,476,501 elements. There is no number for N that will make the sets differ in the number of elements they have. They will always be in a one-to-one correspondence with each other. If you can find a counting number that will make the sets have a different number of elements (like, say, one set would have 10 elements while the other has 15) then you’ve shown an absurdity. The reasoning isn’t hasty. It’s been around for hundreds of years. These concepts are both analytically shown (in various proofs by various mathematicians, and in how they are used every day all over the globe in math for centuries) as well as in actual applied math and science.

 

[thinkandmull]

That logic can be used to show that any two infinities are the same, therefore there are no such thing as larger and smaller infinities in that system

 

[thinkandmull]

1 2 3 4 5 6 7 8 9 10 has more units then 1 3 5 7 9. We can take both to infinity, but does that make the second set greater than the first? That is the nature of this disagreement, and I think too many people have no thought about what infinity means when it goes on forever “out there”. They are stuck in the idea of single steps (“one to one correspondence”)

 

[rossum]

That logic can be used to show that any two infinities are the same, therefore there are no such thing as larger and smaller infinities in that system

No. All countable infinities are the same; there are larger infinities which are not the same – the uncountable infinities. Cantor’s diagonal argument shows that the real numbers cannot be put into a one-to-one correspondence with the integers. Since the integers form a countable infinity, then the real numbers form a larger infinity. There are a lot of larger infinities, see the Aleph numbers for the details.

As I said previously, you appear to have insufficient knowledge of this area of mathematics. You need to lean more to get the most out of this discussion.

rossum

 

[thinkandmull]

You keep saying that you have the mathematical knowledge on this, but you haven’t stated the argument yet. Uncountable infinities would still have units. Take ten of the units, line them up with 1 through ten, and then say " both sets go to infinity so they are equal". Why doesn’t that argument work? Its the one you used in your previous post.

I am not the one saying there aren’t larger infinities. I am saying that part of all natural numbers, the odd numbers, are not equal to the whole. So far I haven’t ben shown otherwise

 

[Rhubarb]

I gave you the demonstration of what you want in post 107. Many people have explained to you why this works.

{1, 2, 3, 4, 5, 6, 7, 8, 9…N+1}

{1, 3, 5, 7, 9, 11, 13, 15, 17, 19…2N-1}

There is no number, N, that would make the two sets have a different number of elements. These work because when you take an infinite set, you’re not talking “from 1 to 10” and saying it goes to infinity. We’re showing the first ten elements to establish the pattern. The sets I’ve shown, the whole numbers and odd whole numbers, aren’t equal - clearly. They have different elements. But they are the same size - they have the same number of elements. Because no matter how many elements they have, they can both add on one more. Count the elements of the sets as I wrote them - they are the same. 10 and always one more, and 10 and always one more. Chose any number for N. Each set will have N elements, and always one more.

We’re not filling up a balloon here. We’re adding elements to a set. Each element counts as one element, even if it’s ten digits across.

 

[Usagi]

thinkandmull, you are accusing us of trying to reason from finitude to infinitude with the one-to-one corespondence method, when in fact you are the one trying to apply the same rules governing finite sets to infinite sets.

Yes, there exist infinite sets that are provably larger than other infinite sets. But that difference is not made by adding a finite number of elements/units. It does not matter that the natural numbers take longer to get to 10 than the odd numbers. Infinity plus one is NOT greater than just infinity. Infinity plus a like infinity (your example of even plus odd) does not make a larger infinity. Infinite sets do not work like that.

If you want to prove us wrong, show us a point where the one-to-one correspondence breaks down. Yes, we have to stop and say “and it goes on like that” at some point because we cannot literally keep going forever. But a pattern has been established, and just from the way numbers work we cannot see any reason that it will change as you extend the correspondence. If we’re wrong, show us.

You are literally saying that Cantor and every mathematician since has missed a simple truth that, according to you, is so obvious that the alternative is intuitively absurd. Does THAT not seem a little absurd to you? It’s like the people who insist that there are simple and obvious holes in the whole notion of evolution, that laymen can easily grasp but entire generations of people devoting their adult lives to the subject have either missed or deliberately ignored. Yeah, knowledge progresses and someone someday may demonstrate flaws in both those notions, but it’s not going to be untrained folks like you and me preaching “the obvious” on the internet.

rossum or someone, can you give us a quick sketch of the diagonal proof that shows the one-to-one method does NOT work for the real numbers? I’m afraid I only read about all this stuff in one pop-math book back in junior high. It made sense to me after I thought about it for a bit (though, yes, it was counter-intuitive at first and I had the same immediate reaction as thinkandmull), but I can’t reproduce the more involved bits off the top of my head.

Usagi

 

[rossum]

You keep saying that you have the mathematical knowledge on this

I do.

but you haven’t stated the argument yet.

I have referenced it, see my link to the Aleph numbers. Understanding the Aleph numbers is fundamental to the understanding of the mathematical infinities.

Uncountable infinities would still have units.

No they would not. These are real numbers, in the mathematical definition. There is an uncountable infinity of real numbers between 0 and 1.

Take ten of the units …

There are no “units”, these are real numbers, not integers.

… line them up with 1 through ten, and then say " both sets go to infinity so they are equal". Why doesn’t that argument work? Its the one you used in your previous post.

Because Cantor’s diagonal argument always allows you to construct a new real number that is not already in the list. Because there is always a real number that is not already on the list, then the two lists cannot be in a one-to-one correspondence. There are too many real numbers for the correspondence to work.

I am not the one saying there aren’t larger infinities. I am saying that part of all natural numbers, the odd numbers, are not equal to the whole. So far I haven’t ben shown otherwise

You have been shown otherwise many times, but you refuse to accept the mathematical reality. The sets, {Z}, {2Z} and {2Z-1} are all countably infinite sets, where Z is the integers, 2Z is the even integers and 2Z-1 is the odd integers.

rossum

 

[thinkandmull]

You’re not showing any proofs. The “pattern” is that 1 2 4 5 6 7 8 9 10 is greater than 1 3 5 7 9

“There are no ‘units’, these are real numbers…”

uh…

Ok, put ten of those real numbers to 1 2 3 4 5 6 7 8 9 10, than they go to infinity, so they are equal infinities! This fails, therefore argument fails.

I don’t think you really know what you are talking about.

Just as you say most philosophers are wrong, since they are not in line with Catholic thought, so I believe there are huge errors from mathematicians, many of whom were probably atheists and believed in positivism

 

[thinkandmull]

Its not hard to see that an infinity plus one is greater than that infinity. Take a line going from you towards the horizon infinitely. Each inch is a number or unit. Add a foot behind you. You just made the infinity greater. How else can you have a greater infinity? Is one not enough to make it “greater” or something?

 

[Rhubarb]

You’re not showing any proofs. The “pattern” is that 1 2 4 5 6 7 8 9 10 is greater than 1 3 5 7 9

“There are no ‘units’, these are real numbers…”

uh…

Ok, put ten of those real numbers to 1 2 3 4 5 6 7 8 9 10, than they go to infinity, so they are equal infinities! This fails, therefore argument fails.

I don’t think you really know what you are talking about.

Just as you say most philosophers are wrong, since they are not in line with Catholic thought, so I believe there are huge errors from mathematicians, many of whom were probably atheists and believed in positivism

The problem is that you’re going to N=10 for the first set, and N=5 for the second set. You’re bounding them - so of course one is going to have more elements than the other. These are unbounded sets, so if N=10, then you need to go 10 elements in each set. We’re not just going from one to ten. We’re going to N=10. There’s a difference.

Everyone is saying that the set of whole numbers, and the set of whole odd numbers, are both equal infinities. They are both “countable infinities”, which is a technical mathematical term. The set of all whole numbers are all real numbers. But the SET of ALL real numbers aren’t just whole numbers. You understand what they mean when they say ‘real number’, right? The set of all real numbers includes much more than just the counting numbers. And moreover, the set of all real numbers are ‘uncountably infinite’ because they do not line up one-to-one with the elements of a countably infinite set.

I don’t mean to question anyone else here. I really don’t. But you seem very confused about simple math concepts that pre-teens learn in grade-school. I know that conceptions of infinity are counter-intuitive. But our number theory depends on it, and it works.

 

[Rhubarb]

Its not hard to see that an infinity plus one is greater than that infinity. Take a line going from you towards the horizon infinitely. Each inch is a number or unit. Add a foot behind you. You just made the infinity greater. How else can you have a greater infinity? Is one not enough to make it “greater” or something?

You are comparing infinite apples to finite oranges. Infinity+1 is no larger or smaller than infinity or infinity-1,000,000,000. There is no ‘greater’ when we’re dealing with infinities of the same type. An addition or subtraction from infinity doesn’t change that it’s infinity.

 

[rossum]

You’re not showing any proofs. The “pattern” is that 1 2 4 5 6 7 8 9 10 is greater than 1 3 5 7 9

“There are no ‘units’, these are real numbers…”

This is where your mathematical knowledge is insufficient. The mathematical definition of a “real number” differentiates it from an “integer”. 2 is an integer; 2.0, 2.000349127913754, 2.33333333… and too many others to count are real numbers.

The interval [2 … 3] contains two members in the integers. The interval [2.0 … 3.0] contains uncountably many real numbers. Integers and reals are very different. Your lack of relevant knowledge and vocabulary means that you do not understand the proofs we are referring to. Your confusion of integers with reals is a basic elementary error. Your continued misuse of “greater” between sets is also an error, though not such a big one. The sets themselves are not greater or lesser; it is the number of elements in the sets that is greater or lesser. Good mathematics requires accurate use of the correct terminology. You are failing to do that at the moment.

I don’t think you really know what you are talking about.

If you do not know the difference between integers and reals, then you need to think very carefully before making such statements.

Just as you say most philosophers are wrong, since they are not in line with Catholic thought, so I believe there are huge errors from mathematicians, many of whom were probably atheists and believed in positivism

Euclid, Pythagoras, Fermat, Gauss, Euler and many others were not atheists. Many of them were Catholic, or Protestant. Al-Khwarizmi was Muslim, and certainly not an atheist. Again your lack of knowledge is causing you to make errors.

rossum

 

[Tomdstone]

That logic can be used to show that any two infinities are the same, therefore there are no such thing as larger and smaller infinities in that system

No. you cannot show that the countable infinity of the integers is the same as the uncountable infinity of the real numbers.

 

[Tomdstone]

rossum or someone, can you give us a quick sketch of the diagonal proof that shows the one-to-one method does NOT work for the real numbers? I’m afraid I only read about all this stuff in one pop-math book back in junior high. It made sense to me after I thought about it for a bit (though, yes, it was counter-intuitive at first and I had the same immediate reaction as thinkandmull), but I can’t reproduce the more involved bits off the top of my head.

Usagi

Take the real numbers between 0 and 1. They can all be written as decimals with infinite expansion. Here for example:
1/2 = 0.500000000000000000000000000000000000000000000000000000000
1/3 = 0.333333333333333333333333333333333333333333333333333333333
pi/4 = 0.785398163…
sqrt(2)/2 = 0.707106781…
Now suppose that you have a 1-1 correspondence between the natural numbers and all reals between 0 and 1.
Then it is shown, that you have made a mistake, or that you cannot do that, because there is always at least one real number which did not make the list.
To construct the number which did not make the list, you look at the numbers one by one down the list and for the first number, you take an integer different from that in the first decimal place.
For the second, number, you take an integer which is different from that in the second decimal place,

For the third number, you take an integer which is different from that in the third decimal place,
Etc. Then you construct the new number by placing the different integers in their respective decimal places, and then by construction this number did not appear in the countable list.

 

[thinkandmull]

How is that process different from saying 1 2 3 4 5 is greater than 1 3 5? Sure, real numbers are an infinity all of their own, exponentially so. I didn’t say they weren’t a greater infinity. The issue is the seeming lack of a demonstration that all odd numbers have a one to one correspondence to all the natural numbers

 

[Rhubarb]

How is that process different from saying 1 2 3 4 5 is greater than 1 3 5? Sure, real numbers are an infinity all of their own, exponentially so. I didn’t say they weren’t a greater infinity. The issue is the seeming lack of a demonstration that all odd numbers have a one to one correspondence to all the natural numbers

I’ve written the demonstration twice. That’s why people are wondering if you are up on your math enough. The problem is that you’re going to N=5 in the first series and N=3 in the second series. To see the pattern you need to go out to N=5 for both sets. N=5 will look like {1, 3, 5, 7, 9} in the set of odd numbers. Then you’ll see the correspondence. For countably infinite sets, there will always be a one-to-one correspondence. Cantor’s proofs showed that there are sets that aren’t in a one-to-one correspondence. That means that one set has more elements in the other. One-to-one correspondence is just making pairs of elements. It’s the trick that helps to establish a theory of number in many conceptions of arithmetic. The following numbers are in a one-to-one correspondence. It’s like having X pennies in your pocket, and the same number in the other pocket. Those pennies can be paired off evenly, so, you have a one-to-one correspondence. Infinite sets of the same type have this correspondence in their elements. For every element in one set, there’s one element in the other set with no left-overs. If you want to show that two sets aren’t of the same type of infinity - that one is larger than the others - you need to show that there’s leftovers. That’s what Cantor did. He demonstrated through his Diagonal Argument that the set of all real numbers will have leftovers.