I just finished reading the thread on Marble Bonanza. Quite an interesting read. Before I state my thoughts, I think I should post my credentials: I'm not an engineer, never studied set theory, my father never factored liscense plates. If only he had. Okay

1. It took a while to find the definition of "non-sequitur" on the internet.

2. Is infinity - infinity = 0? I don't know. It is logical to conclude that if I remove everything I put in, nothing will be left.

3. I think the premise that someone finished an infinite task is difficult to accept, but hey; It's just a PUZZLE. It's not the real world. It's like reading a knight and knave puzzle, then making the argument that there is no such thing as someone who always tells the truth. We use the premise given in the puzzle.

4. I really would like to know if infinity - infinity = 0 ever occurs in math.

Infinity-infinity can be anything, finite or infinite. For example, if you remove all prime numbers from the set of all natural numbers, you are still left with an infinite set. But if you remove the set of all odd prime numbers from the set of all prime numbers, the result is finite.

Comparing infinite sets (or doing set operations with them) requires being very careful. I'd suggest googling for "CANTOR" and "INFINITE" or "INFINITY".

Bruce,

If I had a hotel with inifinite rooms (cardinality aleph null), numbered (one-to-one) with the positive integers, which was 100% occupied, and you came to it and said you want a room for the night, I could honestly respond: sorry, we are 100% full. And we could end right there, and be done with it.

But you could also suggest that we move every person into the room numbered one higher, so 1 to 2, 2 to 3, 406 to 407, etc... And then you could go into room 1. And we could end right there, and be done with it.

Or you could come and say you have an infinite number of friends all of whom are numbered with the positive integers, and they all want to room for the night as well. And I could say: sorry, we are 100% full. And we could end right there and be done with it.

But you could also suggest that we move every person into the room numbered twice their number minus one. So 1 to 1, 2 to 3, 3 to 5, 406 to 811, etc..., freeing up all the even numbered rooms. And then your friends could ALL go into the room that is twice their number: 1 into 2, 2 into 4, 3 into 6, etc....

The point of all of this is that we are potentially confusing cardinality with set membership. And we musn't do that.

We all agree that the set of positive odd integers is equal in size to (that is I can put the elements into a one to one correspondence with the elements of) the set of positive integers. Nevertheless, as shown above, if I take those particular elements (the odd ones) away from the latter set (the evens and odds), I am left with a set (the evens) that is just as large as the original set. And in fact, I can then RENUMBER all the elements (balls) with a number that is half its current value. And I am left with the positive integers again (even and odd).

"Marbles Bonanza" is asking a question about element membership, not size (cardinality) of the sets. It is somewhat analagous to asking the following two questions:
What set remains from the set of all positive integers if I remove:

• the even positive integers?


• all positive integers?

Bruce,

I don't know what it means for infinity to be 100% full. For a hotel, I would define 100% full as "every hotel room is occupied". And certainly, for our hypothetical infinite-room hotel, it could be "full".

Yes, I used the word potentially because I didn't want to limit the audience to just you.

Yes, basically, if the elements in both sets can be mapped one-to-one, they are of equal size. But this doesn't mean they have the same elements, such as in the previous example of odd positive integers and even positive integers. Or we could use all positive integers and even positive integers, where in this case, the latter is a subset of the former.

I hope this has helped.

Comparing usual sets is simple: if I have two boxes and I want to know which has more things, I can start taking one thing out of the first box and one thing out of the second box. If the first box runs empty while the second one still has something in it, I can say the first has less things; if they both are empty at the same time, they must have had the same number of things, and so on.

Cantor's trick was noting that the same method was the only one that could be applied to infinite sets. Given two infinite sets, IF (a big IF) you can find a way to pair each and all of the elements of the first set with an element of the second set (a one-to-one relationship), THEN you say both sets are of the same cardinality. (Saying that they both have the same number is elements is problematic, for that would make infinite a number, and it just doesn't work as a number.)

However, IF, no matter what pairing you do, there still are elements "left over" in a set, then you can say that set has greater cardinality.

For example, which set is greater: the natural numbers {1,2,3,4,...} or the natural even numbers {2,4,6,8....}? At first sight one could say the first one is larger, for it includes all the members of the second set, and furthermore adds the odd numbers. HOWEVER, if we pair the elements thus: 1-2, 2-4, 3-6, 4-8, ... for each number in the first set there is a number in the second one, and vice versa -- we must conclude that both sets have the same cardinality.

It's hard and difficult, but (as with relativity or quantum physics) the problems come because we are dealing with quantities we are unaccostumed to... no one ever drives so fast, or becomes so small, or gets so large sets to work with! ;-)

Hi! That wouldn't work -- in theory, you could pick any mapping you want, so you could say that PI goes with 3/5. (The square root of -2 is an imaginary number, not a real number.)

Cantor's argument can be difficult to understand at first. First, you must prove that there are as many rationals as there are natural numbers. Then, you must prove that there are as many numbers in (0,1) as in (0,∞).

Now, instead of comparing rationals to reals, we can compare naturals to reals in (0,1). Imagine there is one possible mapping between naturals and reals. Then, we could write out a table such as:

1 ---> 0.1549870298471987....
2 ---> 0.9748734292409098....
3 ---> 0.8721312672137712....
4 ---> 0.9818718912312136....
5 ---> 0.1531115123166678....

and so on. (The table is an infinite one.) Now, we'll invent a number such that its first digit is different from the first digit of the first number; its second digit will be different from the second digit of the second number; and so on.

1 ---> 0.1549870298471987....
2 ---> 0.9748734292409098....
3 ---> 0.8721312672137712....
4 ---> 0.9818718912312136....
5 ---> 0.1531115123166678....

For example, we could have produced 0.26163.... We agree that this is a real number. Is it in the table? It isn't in the first row, for its first digit is different. It isn't in the second row either, for its second digit doesn't coincide. And, in general, it isn't in the n-th row, because its n-th digit is different.

SO: the number we built isn't in the table anywhere! However, when we built our mapping, we supposed ALL reals were in the table... so that must have been false, and the mapping cannot exist. QED.

HTH!