Something That Frustrates Us about 121C (Voice for the Laymen)

[ryan]

Take two sets A and B for example. A = {-3,-2,-1,0,1,2,3} and B = {0,1,2,3}. We clearly have a 121C match from 0 to 3 but not for all of the elements in both sets. Similarly, every natural number in set A matches to every natural number in set B.

Now take the two sets C and D. C = {...,-3,-2,-1,0,1,2,3, ...} and D = {0,1,2,3, ...}. Just like in the other example, each natural number in set D will match to each natural number in C. And it would seeeeeeeeem that I just did the exact same thing as I did with A and B, except with larger sets, yet you say it's not the same thing.


HOW IS C AND D ONE-TO-ONE BUT THE OTHER ISN'T !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ...

 

[beero1000]

A set is infinite if and only if it is in bijective correspondence with some of its proper subsets.

 

[Juma]

Take two sets A and B for example. A = {-3,-2,-1,0,1,2,3} and B = {0,1,2,3}. We clearly have a 121C match from 0 to 3 but not for all of the elements in both sets. Similarly, every natural number in set A matches to every natural number in set B.

Now take the two sets C and D. C = {...,-3,-2,-1,0,1,2,3, ...} and D = {0,1,2,3, ...}. Just like in the other example, each natural number in set D will match to each natural number in C. And it would seeeeeeeeem that I just did the exact same thing as I did with A and B, except with larger sets, yet you say it's not the same thing.


HOW IS C AND D ONE-TO-ONE BUT THE OTHER ISN'T

Because D is infinite.

 

[ryan]

A set is infinite if and only if it is in bijective correspondence with some of its proper subsets.

It just seems like if I match all of the natural numbers from set D to set C, I would have the rest of the negative integers left unmatched in C.

Give me any number in D, and I will match it to the same number in C. Eventually all naturals numbers in D should be matched to all natural numbers in C, no?

 

[beero1000]

A set is infinite if and only if it is in bijective correspondence with some of its proper subsets.

It just seems like if I match all of the natural numbers from set D to set C, I would have the rest of the negative integers left unmatched in C.

Give me any number in D, and I will match it to the same number in C. Eventually all naturals numbers in D should be matched to all natural numbers in C, no?

The set of integers is infinite. A set is infinite if and only if it is in bijective correspondence with some of its proper subsets. Therefore, the set of integers is in bijective correspondence with some of its proper subsets.

 

[Bomb#20]

A set is infinite if and only if it is in bijective correspondence with some of its proper subsets.

You're not helping. You're assuming way too much background.

It just seems like if I match all of the natural numbers from set D to set C, I would have the rest of the negative integers left unmatched in C.

Give me any number in D, and I will match it to the same number in C. Eventually all naturals numbers in D should be matched to all natural numbers in C, no?

The point you're missing, that the other people are taking for granted that you already understand, is that you don't have to match a number in D to the same number in C. You can match any member of one set to any member of the other set that you please. In your original example,

A = {-3,-2,-1,0,1,2,3} and B = {0,1,2,3}

there's no rule that says {-3,-2,-1} are the unmatched set members that are left over after you set up your 121C. If you want you can match them up like this:

A B
-3 0
-2 1
-1
0
1 2
2 3
3

and have {-1, 0, 3} left over. B still matches some of the elements in A but not all. With finite sets this doesn't matter -- which mapping you use never makes a difference as to whether you succeed or not. But with infinite sets this point is crucial. Sometimes you succeed and sometimes you fail, depending on which elements in C you match with which elements in D. The two sets have a 121C if any method at all lets you succeed. The two sets have lots of correspondences -- any way you line them up is a correspondence -- but if one of those correspondences is a 121C then the sets have a 121C. (It's the same as how a zoo has lots of animals, and most of them aren't penguins, but if any of them at all is a penguin then the zoo has a penguin.)

So why do C and D have a 121C? Because here it is:

{...,-3,-2,-1,0,1,2,3, ...} and D = {0,1,2,3, ...}
C D
...
-3 5
-2 3
-1 1
0 0
1 2
2 4
3 6
...

 

[ryan]

It just seems like if I match all of the natural numbers from set D to set C, I would have the rest of the negative integers left unmatched in C.

Give me any number in D, and I will match it to the same number in C. Eventually all naturals numbers in D should be matched to all natural numbers in C, no?

The set of integers is infinite. A set is infinite if and only if it is in bijective correspondence with some of its proper subsets. Therefore, the set of integers is in bijective correspondence with some of its proper subsets.

Okay, I understand. But it's the 121C that I am questioning, and I am also not sold on the idea that the set of integers is infinite.

Please see my response to Bomb#20.

 

[ryan]

You're not helping. You're assuming way too much background.

It just seems like if I match all of the natural numbers from set D to set C, I would have the rest of the negative integers left unmatched in C.

Give me any number in D, and I will match it to the same number in C. Eventually all naturals numbers in D should be matched to all natural numbers in C, no?

The point you're missing, that the other people are taking for granted that you already understand, is that you don't have to match a number in D to the same number in C. You can match any member of one set to any member of the other set that you please. In your original example,

A = {-3,-2,-1,0,1,2,3} and B = {0,1,2,3}

there's no rule that says {-3,-2,-1} are the unmatched set members that are left over after you set up your 121C. If you want you can match them up like this:

A B
-3 0
-2 1
-1
0
1 2
2 3
3

and have {-1, 0, 3} left over. B still matches some of the elements in A but not all. With finite sets this doesn't matter -- which mapping you use never makes a difference as to whether you succeed or not. But with infinite sets this point is crucial. Sometimes you succeed and sometimes you fail, depending on which elements in C you match with which elements in D. The two sets have a 121C if any method at all lets you succeed. The two sets have lots of correspondences -- any way you line them up is a correspondence -- but if one of those correspondences is a 121C then the sets have a 121C. (It's the same as how a zoo has lots of animals, and most of them aren't penguins, but if any of them at all is a penguin then the zoo has a penguin.)

So why do C and D have a 121C? Because here it is:

{...,-3,-2,-1,0,1,2,3, ...} and D = {0,1,2,3, ...}
C D
...
-3 5
-2 3
-1 1
0 0
1 2
2 4
3 6
...

So now I think you are saying that uniqueness of digits doesn't matter, right? If you use this argument, then we can visit the uncountable argument, which seems to use uniqueness to identify numbers that are left out of the infinite row of numbers.

Or, for example, why couldn't I just do this with the sets E = {1,2,3...} and F = {1,2,3...}:

E|__|F
1-not-1
2-not-1
3-not-1
.
.
.

 

[Juma]

A set is infinite if and only if it is in bijective correspondence with some of its proper subsets.

It just seems like if I match all of the natural numbers from set D to set C, I would have the rest of the negative integers left unmatched in C.

Yes. But you only showed that your mapping wasnt a bijection.

Not every mapping is a bijection.

For to set to have the same size it is enough that there is one mapping that is bijective.

 

[ryan]

It just seems like if I match all of the natural numbers from set D to set C, I would have the rest of the negative integers left unmatched in C.

Yes. But you only showed that your mapping wasnt a bijection.

Not every mapping is a bijection.

For to set to have the same size it is enough that there is one mapping that is bijective.

Is that the function that maps every natural number n (from 1 to infinity) to all integers in the order: 0, 1, -1, 2, -2, 3, -3, 4, -4 ... ?

So why does order matter, but it doesn't matter when matching {1,2,3} to {4,5,6}?

 

[Juma]

Yes. But you only showed that your mapping wasnt a bijection.

Not every mapping is a bijection.

For to set to have the same size it is enough that there is one mapping that is bijective.

Is that the function that maps every natural number n (from 1 to infinity) to all integers in the order: 0, 1, -1, 2, -2, 3, -3, 4, -4 ... ?

That is one of many others.

So why does order matter, but it doesn't matter when matching {1,2,3} to {4,5,6}?

Because inifinte sets contain infinite subsets.

 

[beero1000]

You're not helping. You're assuming way too much background.

It just seems like if I match all of the natural numbers from set D to set C, I would have the rest of the negative integers left unmatched in C.

Give me any number in D, and I will match it to the same number in C. Eventually all naturals numbers in D should be matched to all natural numbers in C, no?

The point you're missing, that the other people are taking for granted that you already understand, is that you don't have to match a number in D to the same number in C. You can match any member of one set to any member of the other set that you please. In your original example,

A = {-3,-2,-1,0,1,2,3} and B = {0,1,2,3}

there's no rule that says {-3,-2,-1} are the unmatched set members that are left over after you set up your 121C. If you want you can match them up like this:

A B
-3 0
-2 1
-1
0
1 2
2 3
3

and have {-1, 0, 3} left over. B still matches some of the elements in A but not all. With finite sets this doesn't matter -- which mapping you use never makes a difference as to whether you succeed or not. But with infinite sets this point is crucial. Sometimes you succeed and sometimes you fail, depending on which elements in C you match with which elements in D. The two sets have a 121C if any method at all lets you succeed. The two sets have lots of correspondences -- any way you line them up is a correspondence -- but if one of those correspondences is a 121C then the sets have a 121C. (It's the same as how a zoo has lots of animals, and most of them aren't penguins, but if any of them at all is a penguin then the zoo has a penguin.)

So why do C and D have a 121C? Because here it is:

{...,-3,-2,-1,0,1,2,3, ...} and D = {0,1,2,3, ...}
C D
...
-3 5
-2 3
-1 1
0 0
1 2
2 4
3 6
...

Eh, I've tried that already, for essentially this very same point of confusion. See http://talkfreethought.org/showthread.php?7793-Regarding-Cantor-s-Diagonal-Argument. It didn't work.

I hope you get through, but now I'm trying this.

The set of integers is infinite. A set is infinite if and only if it is in bijective correspondence with some of its proper subsets. Therefore, the set of integers is in bijective correspondence with some of its proper subsets.

Okay, I understand. But it's the 121C that I am questioning, and I am also not sold on the idea that the set of integers is infinite.

Please see my response to Bomb#20.

The set of integers is in bijective correspondence with some of its proper subsets. A set is infinite if and only if it is in bijective correspondence with some of its proper subsets. Therefore, the set of integers is infinite.

 

[ryan]

Is that the function that maps every natural number n (from 1 to infinity) to all integers in the order: 0, 1, -1, 2, -2, 3, -3, 4, -4 ... ?

That is one of many others.

So why does order matter, but it doesn't matter when matching {1,2,3} to {4,5,6}?

Because inifinte sets contain infinite subsets.

Okay, but that only begs the question of why order matters with an infinite number of subsets.

 

[EricK]

That is one of many others.

So why does order matter, but it doesn't matter when matching {1,2,3} to {4,5,6}?

Because inifinte sets contain infinite subsets.

Okay, but that only begs the question of why order matters with an infinite number of subsets.

I'll try to explain in as non-mathematical way as possible:

When you have a finite set and you remove one element, the resulting set is smaller. If you repeat this often enough, you will eventually get to the empty set. The number of times you have to repeat this is the size of the set. And it doesn't matter in which order you remove elements, you will still end up with the empty set after the same number of steps.

If two finite sets have the same number of elements, then forming a 121C between them is equivalent to removing entries from each set one at a time until they are both empty. If they both end up empty at the same time then they were the same size. If one runs out before the other then it was a smaller set. But because it doesn't matter in which order you remove entries from either set, you can form 121C's between them any way you like.

Infinite sets behave differently. If you remove one element from an infinite set, you are still left with an infinite set [Simple proof: Assume not. If the resulting set is finite of size n, then adding back the element you removed will result in a set of size n+1. i.e. a finite set]. If you remove an infinite number of elements at once, you might be left with an infinite set, or you might be left with a finite set [eg From {1,2,3,4...} remove all the even numbers, and you are left with the infinite set {1,3,5...}; but remove the set of numbers bigger than 3 and you are left with {1,2,3}]. So when you are forming 121C's between two infinite sets you can't just take one entry at a time from each set willy-nilly and continue until you both sets are empty, because they will never be empty. So instead you need some way to remove an infinite number of elements in one go from each set. But because removing an infinite number of elements can have very different effects you end up with the problem that it matters precisely how you try to form a 121C as to whether you succeed or not.

 

[ryan]

The set of integers is in bijective correspondence with some of its proper subsets.

But this statement is what I am trying to understand.

 

[Bomb#20]

So why do C and D have a 121C? Because here it is:

C = {...,-3,-2,-1,0,1,2,3, ...} and D = {0,1,2,3, ...}
C D
...
-3 5
-2 3
-1 1
0 0
1 2
2 4
3 6
...

So now I think you are saying that uniqueness of digits doesn't matter, right?

No, uniqueness absolutely matters. Check the two columns. Every member of C shows up exactly once in the left column, and every member of D shows up exactly once in the right column. (Of course you can have the same number twice, once in the C column and once in the D column; but you can't have it more than once in the same column.)

If you use this argument, then we can visit the uncountable argument, which seems to use uniqueness to identify numbers that are left out of the infinite row of numbers.

Or, for example, why couldn't I just do this with the sets E = {1,2,3...} and F = {1,2,3...}:

E|__|F
1-not-1
2-not-1
3-not-1
.
.
.

I don't think you're quite ready for the uncountable argument; first you need to really grok how "countable" infinite sets behave. Let's keep going on the integers.

(But the quick answer is that with finite sets which mapping you use never makes a difference as to whether you succeed or not, but with infinite sets sometimes you succeed and sometimes you fail. What you're doing with that example is exhibiting one line-up that fails -- it isn't a proper 121C because 1 never shows up in the F column. If these were finite sets, you'd be done -- when you show one failure you've shown that all the line-ups fail. I think that's where your intuition is letting you down: you think you're done because you're used to finite sets. Since these are infinite sets, you're not done. You've shown one example of a line-up that has a missing set member on one side; but you haven't shown that every line-up has a missing set member. (And of course we know there's a line-up with no missing members, since E and F are the same set.) But in Cantor's "diagonal" argument for the reals being an uncountable set, he shows that every line-up has a missing real number.)

Is that the function that maps every natural number n (from 1 to infinity) to all integers in the order: 0, 1, -1, 2, -2, 3, -3, 4, -4 ... ?

So why does order matter, but it doesn't matter when matching {1,2,3} to {4,5,6}?

Order doesn't matter. Order never matters for whether sets can be matched up. But which member of one set is matched with which member of the other set does matter. This is a subtle distinction; one of the things you need to get used to is that there are a lot of subtle distinctions that we can ignore with finite sets but which will bite us in the ass if we ignore them with infinite sets. Lining sets up the way you did it:

0 0
1 1
2 -1
3 2
4 -2
5 3
...

is exactly the same as lining them up like this:

1 1
3 2
5 3
...
0 0
2 -1
4 -2
...

The two line-ups have the numbers in totally different order in the two columns, but which number goes with which is the same in both versions.

If you're asking why which number goes with which matters when you're lining up infinite sets but not with finite sets, it's because of the "Hilbert's Hotel" phenomenon. With an infinite hotel you can free up some space by moving the guests around to different rooms. With a finite hotel moving guests around doesn't help, because if you try to free up room 1 by moving every guest from room X to room X+1 you run out of space when you get to N, the last room, and you end up having to put the guy in room N into room 1, and you haven't freed up any space. Which number goes with which doesn't matter when matching {1,2,3} to {4,5,6} because changing it doesn't help. But with infinite sets it does help, because none of the rooms are the last room so you never get stuck and there's nobody you have to move from his current room into room 1.

 

[Bomb#20]

... and I am also not sold on the idea that the set of integers is infinite.

Why aren't you sold on that?

If the set of integers were finite, then, no matter how many there are, in principle you could compare every one with every other one and identify which is larger, the same way you can compare 1 with 2, compare 1 with 3, and compare 2 with 3, supposing there were only those three integers. So you'd just systematically go through all the possible comparisons until you find the largest integer, the one that's bigger than every other integer. Then you'd have an integer that you couldn't add 1 to, because there wouldn't be any integer for the sum to be. But how could you possibly have an integer that you can't add 1 to?

 

[ryan]

That is one of many others.

So why does order matter, but it doesn't matter when matching {1,2,3} to {4,5,6}?

Because inifinte sets contain infinite subsets.

Okay, but that only begs the question of why order matters with an infinite number of subsets.

I'll try to explain in as non-mathematical way as possible:

When you have a finite set and you remove one element, the resulting set is smaller. If you repeat this often enough, you will eventually get to the empty set. The number of times you have to repeat this is the size of the set. And it doesn't matter in which order you remove elements, you will still end up with the empty set after the same number of steps.

If two finite sets have the same number of elements, then forming a 121C between them is equivalent to removing entries from each set one at a time until they are both empty. If they both end up empty at the same time then they were the same size. If one runs out before the other then it was a smaller set. But because it doesn't matter in which order you remove entries from either set, you can form 121C's between them any way you like.

Infinite sets behave differently. If you remove one element from an infinite set, you are still left with an infinite set [Simple proof: Assume not. If the resulting set is finite of size n, then adding back the element you removed will result in a set of size n+1. i.e. a finite set]. If you remove an infinite number of elements at once, you might be left with an infinite set, or you might be left with a finite set [eg From {1,2,3,4...} remove all the even numbers, and you are left with the infinite set {1,3,5...}; but remove the set of numbers bigger than 3 and you are left with {1,2,3}]. So when you are forming 121C's between two infinite sets you can't just take one entry at a time from each set willy-nilly and continue until you both sets are empty, because they will never be empty. So instead you need some way to remove an infinite number of elements in one go from each set. But because removing an infinite number of elements can have very different effects you end up with the problem that it matters precisely how you try to form a 121C as to whether you succeed or not.

It still doesn't seem like a legitimate 121C. If all naturals in the set of only naturals corresponds to all of the naturals in the set of all integers, you seem to be saying that we are left with negative integers with no match, okay. But, then how is that 121C if the correspondence of the set of naturals is totally used up. Aleph 0 has been exhausted by the naturals. Aleph 0 naturals was not enough to match aleph 0 integers; how is that 121C?

 

[Juma]

That is one of many others.

So why does order matter, but it doesn't matter when matching {1,2,3} to {4,5,6}?

Because inifinte sets contain infinite subsets.

Okay, but that only begs the question of why order matters with an infinite number of subsets.

I'll try to explain in as non-mathematical way as possible:

When you have a finite set and you remove one element, the resulting set is smaller. If you repeat this often enough, you will eventually get to the empty set. The number of times you have to repeat this is the size of the set. And it doesn't matter in which order you remove elements, you will still end up with the empty set after the same number of steps.

If two finite sets have the same number of elements, then forming a 121C between them is equivalent to removing entries from each set one at a time until they are both empty. If they both end up empty at the same time then they were the same size. If one runs out before the other then it was a smaller set. But because it doesn't matter in which order you remove entries from either set, you can form 121C's between them any way you like.

Infinite sets behave differently. If you remove one element from an infinite set, you are still left with an infinite set [Simple proof: Assume not. If the resulting set is finite of size n, then adding back the element you removed will result in a set of size n+1. i.e. a finite set]. If you remove an infinite number of elements at once, you might be left with an infinite set, or you might be left with a finite set [eg From {1,2,3,4...} remove all the even numbers, and you are left with the infinite set {1,3,5...}; but remove the set of numbers bigger than 3 and you are left with {1,2,3}]. So when you are forming 121C's between two infinite sets you can't just take one entry at a time from each set willy-nilly and continue until you both sets are empty, because they will never be empty. So instead you need some way to remove an infinite number of elements in one go from each set. But because removing an infinite number of elements can have very different effects you end up with the problem that it matters precisely how you try to form a 121C as to whether you succeed or not.

It still doesn't seem like a legitimate 121C. If all naturals in the set of only naturals corresponds to all of the naturals in the set of all integers, you seem to be saying that we are left with negative integers with no match, okay. But, then how is that 121C if the correspondence of the set of naturals is totally used up. Aleph 0 has been exhausted by the naturals. Aleph 0 naturals was not enough to match aleph 0 integers; how is that 121C?

Infinite sets contains infinite subsets.

That means that you can map the entire set to one of its subsets.

Do you realize what that means?

An example:
You can map all natural number to the the subset containing only the even natural numbers by using the mapping x-> 2x.
In your own wording:
"Aleph 0 has been exhausted by a subset of itself. Aleph 0 naturals was not enough to match aleph 0 even naturals".

Ponder this.

 

[ryan]

So now I think you are saying that uniqueness of digits doesn't matter, right?

No, uniqueness absolutely matters. Check the two columns. Every member of C shows up exactly once in the left column, and every member of D shows up exactly once in the right column. (Of course you can have the same number twice, once in the C column and once in the D column; but you can't have it more than once in the same column.)

If you use this argument, then we can visit the uncountable argument, which seems to use uniqueness to identify numbers that are left out of the infinite row of numbers.

Or, for example, why couldn't I just do this with the sets E = {1,2,3...} and F = {1,2,3...}:

E|__|F
1-not-1
2-not-1
3-not-1
.
.
.

I don't think you're quite ready for the uncountable argument; first you need to really grok how "countable" infinite sets behave. Let's keep going on the integers.

(But the quick answer is that with finite sets which mapping you use never makes a difference as to whether you succeed or not, but with infinite sets sometimes you succeed and sometimes you fail. What you're doing with that example is exhibiting one line-up that fails -- it isn't a proper 121C because 1 never shows up in the F column. If these were finite sets, you'd be done -- when you show one failure you've shown that all the line-ups fail. I think that's where your intuition is letting you down: you think you're done because you're used to finite sets. Since these are infinite sets, you're not done. You've shown one example of a line-up that has a missing set member on one side; but you haven't shown that every line-up has a missing set member. (And of course we know there's a line-up with no missing members, since E and F are the same set.) But in Cantor's "diagonal" argument for the reals being an uncountable set, he shows that every line-up has a missing real number.)

But doesn't the argument only show that the line-up of the naturals is missing real numbers?

Is that the function that maps every natural number n (from 1 to infinity) to all integers in the order: 0, 1, -1, 2, -2, 3, -3, 4, -4 ... ?

So why does order matter, but it doesn't matter when matching {1,2,3} to {4,5,6}?

Order doesn't matter. Order never matters for whether sets can be matched up. But which member of one set is matched with which member of the other set does matter. This is a subtle distinction; one of the things you need to get used to is that there are a lot of subtle distinctions that we can ignore with finite sets but which will bite us in the ass if we ignore them with infinite sets. Lining sets up the way you did it:

0 0
1 1
2 -1
3 2
4 -2
5 3
...

is exactly the same as lining them up like this:

1 1
3 2
5 3
...
0 0
2 -1
4 -2
...

The two line-ups have the numbers in totally different order in the two columns, but which number goes with which is the same in both versions.

If you're asking why which number goes with which matters when you're lining up infinite sets but not with finite sets, it's because of the "Hilbert's Hotel" phenomenon. With an infinite hotel you can free up some space by moving the guests around to different rooms. With a finite hotel moving guests around doesn't help, because if you try to free up room 1 by moving every guest from room X to room X+1 you run out of space when you get to N, the last room, and you end up having to put the guy in room N into room 1, and you haven't freed up any space. Which number goes with which doesn't matter when matching {1,2,3} to {4,5,6} because changing it doesn't help. But with infinite sets it does help, because none of the rooms are the last room so you never get stuck and there's nobody you have to move from his current room into room 1.

But I thought that uniqueness matters. Each hotel room is the same, and each person is the same. In other words, I have no problem with an infinite number of eggs each sitting in an infinite number of nests. But this uniqueness seems to pop up in some ways but go away in other ways. For example, Cantor's diagonal argument. There is an issue with the "kind" of number that doesn't get listed, and aleph 0 won't eat it up. Then there is the kind of number that I don't want in aleph 0, but aleph 0 eats it up anyways.