I got balls!!

[Goliath]

This thread is split off from this one in which I pose the really nifty "infinite shoebox" problem.

Wade, you got the solution right. You've won....ummm....the joy that comes from knowing that you've won.

Originally Posted by squian

I thought the problem could be solved as the limit of 9/t as t approaches 0. In which case my answer was not 0, it was infinity. I'm still pondering why that did not come out right.

One of the reasons why it didn't come out right was that as t->0 from the right, the balls that you've counted as being in the box before begin to disappear.

Let me rephrase the solution another way. Suppose for a moment that there were some (possibly infinitely many) balls left in the box at midnight. So we take a ball out of the shoebox that's left over after all of this hoop-la. The question to ask is what is the number on the ball?

It can't be 1, since I took ball 1 out at a minute to midnight.

It can't be 2, since I took ball 2 out at 1/2 minute to midnight.

It can't be 3, since I took ball 3 out at 1/3 minute to midnght.

.
.
.
etc. For any natural number n, n can't be the number on the ball, since I took ball n out at 1/n minutes to midnight. Therefore no such ball can exist, and there are none left at midnight.

Is infinity a non-zero number?

Nope, it isn't even a real number.

Lastly, if we do not number the balls and pull a random one out of the box each time, does it change the answer?

Ah, a very good question. The answer is yes. Let's look at an example.

Suppose instead that we pull out ball 2 at 1 minute to midnight, ball 4 at 1/2 minute to midnght, ball 6 at 1/3 minute to midnight, etc...so that ball 2n is removed at 1/n minutes to midnight. Then there are an infinite number of balls left at midnight--namely, all the odd numbered balls.

In fact, if you choose a random sequence of positive integers for balls to pull out, you can potentially leave any number of balls in the box (in our previous example, the sequence that we removed was {2,4,6,8,....}).

I hope that clears things up.

[squian]

Originally Posted by Goliath

One of the reasons why it didn't come out right was that as t->0 from the right, the balls that you've counted as being in the box before begin to disappear.

OK, I realized that f(0) may not equal the limit of f(x) as x->0. And I can see there is no numbered ball that I can pull; hence, the question about the unmarked balls.

Originally Posted by Goliath

In fact, if you choose a random sequence of positive integers for balls to pull out, you can potentially leave any number of balls in the box (in our previous example, the sequence that we removed was {2,4,6,8,....}).

But if the sequence is random, then the set of removed is the same as the set added, just in a different order. Alternatively, the odds of a particular ball remaining in the box is (9/10)(18/19)(27/28)... which does go to 0.

The counter-intuitive part that keeps me from accepting the answer is that you must add a ball to the box before you can remove it. So it seems to me that at any point in time after we start adding balls to the box, there should be some balls in the box from the last iteration.

The problem is that midnight is not one of the iterations so there is no meaningful way to talk about the last iteration. So all of a sudden we are right back where we started. Infinity is downright weird.

I'm going to stop storing things in shoeboxes just in case.

[Goliath]

Originally Posted by squian

But if the sequence is random, then the set of removed is the same as the set added, just in a different order.

Well, not quite. In the example above where I only took out the evens, I took out a subset of the total set of balls, but an infinite subset (so if by "the same" you meant "the same size", then you're correct).

Alternatively, the odds of a particular ball remaining in the box is (9/10)(18/19)(27/28)... which does go to 0.

Yep, the same probability as randomly choosing a sequence of natural numbers that misses a particular number.

The counter-intuitive part that keeps me from accepting the answer is that you must add a ball to the box before you can remove it. So it seems to me that at any point in time after we start adding balls to the box, there should be some balls in the box from the last iteration.

Ah, but that's just it: there is no last iteration.

Infinity is downright weird.

Yep. We've taken infinitely many balls out, and we've put infinitely many in. This is one illustration of why infinity-infinity is not well defined, because by changing which sequence of balls you take out, you can make infinity-infinity equal to anything you want!

I'm going to stop storing things in shoeboxes just in case.

LOL.

[Clutch Munny]

This is why one of my favorite entries from the "Mistaking my ignorance for insight" files is a gem from Descartes, talking shite about infinity. In the Third Meditation he gets himself out of a jam by stipulating that the idea of infinity (which he equates both with God and with perfection) is "the most clear and distinct of all ideas". It is "clear and distinct in the highest degree".

He was a mathematical genius, to be sure, but wasn't above telling himself a leetle tiny fib when he had to...

[wade-w]

Originally Posted by Goliath

Nope, it isn't even a real number.

But "it" is a surreal number.

Seriously though, Squian, not all infinities are the same; some infinite sets are "more infinite" than others. Some definitions and results:

• The cardinal number of a finite set is the number of elements the set contains.
• A one to one correspondence between two sets S and R is a mapping from S to R such that each element of S is mapped to exactly one element of R, and vice versa.
• Two sets have the same cardinal number iff there exists a one to one correspondence between them.
• Given a set S, it's Power Set, P(S), is the set of all subsets of S.
• A set S is infinite iff there exists a proper subset of S, S1 such that there exists a one to one correspondence between S and S1.

The first three items in the above list are an abstraction of how we count things. It's so basic we don't think about it, but what we do when we count a deck of cards, for example, is set up a one to one correspondence between the set {1, 2, 3, ... , 52} and the cards in the deck. This is why the Natural Numbers (N) are sometimes called the counting numbers.

So, the cardinal number of a finite set is essentially the size of the set. In the late 19th century, Georg Cantor extended this concept to infinite sets. Any set which can be placed in a one to one correspondence with N is said to be countably infinite, or countable. Many common infinite sets are countably infinite. The Integers, for example. The Rational numbers are countable as well. But then Cantor found that the Reals (R) are not countable! In other words, he was able to prove there is no one to one correspondence between R and N. So in a very real sense N and R are not the same size, even though both are infinite. We call a set that is not countable an uncountably infinite set, or uncountable.

Wait, it gets worse! For any infinite set S, there is no one to one correspondence between S and P(S). This means there are an infinite number of infinities! Cantor assigned N the cardinal number aleph-null, and created a heirarchy of what he called transfinite numbers: aleph-null, aleph-one, aleph-two, etc. He also thought that the cardinal of R was aleph-one. This is called the Continuum Hypothesis (R is sometimes called the Continuum), and is now known to be indeterminate in ZFC set theory. In other words, the Continuum Hypothesis can be neither proven nor disproven.

Cantor went on to develop an arithmetic for these new numbers, called, of course, transfinite arithmetic. And, as the problem Goliath posed illustrates, subtraction is not a well defined operation in transfinite arithmetic.

At the time, all of this was extremely controversial; after all, infinity is not a number! In fact, one mathematician who was vehemently opposed to Cantor's ideas used his considerable influence over the major journals of the time to prevent much of Cantor's work from being published. Eventually, however, it was accepted, and is no longer contested by most.

In the 1960's, a new type of number was constructed called the hyperreals, and a new field of mathematics was born, called Nonstandard Analysis. The hyperreals are an ordered field *R which contain R as a subfield. There are hyperreals that are larger than any real. These are called infinite numbers. The reciprocal of an infinite number is called an infintesimal number, and an infintesimal is smaller than any positive real, but still larger than 0 (these are basically Newton's "fluxions" which were abondoned when the epsilon-delta definition of limits was developed). Also, for any real number x, there are hyperreals that are closer to x than any other real.

Then there are the surreal numbers, which contains *R as a subset.(As an aside, I find it interesting that *R is constructed using sequences of reals, much like the reals are somtimes constructed using Cauchy Sequences of Rationals, while the surreals are built using a method that is very similar to Dedekind Cuts.)

So, in the context of Nonstandard Analysis, yes, in some sense we can consider infinity to be a number, and there are infinitely many of them!