the word infinite exists to describe what it can’t
not finite…? jillion or whatever:D
abstract is the closest you will get…and then you can half that distance forever (is that a term) and never get there…which cant be because it would be finite.
No discussion of infinity is complete without a reference to Hilbert’s Hotel - a problem that is rather similar to the one outlined by the OP.
OK, so why did that link not work?
this worked
infinitely so:D
??? no it didnt:(
=http://www.wikipedia.org/wiki/Hilbert’s_paradox_of_the_Grand_Hotel]well then
brkt missing?:eek:
=http://www.wikipedia.org/wiki/Hilbert’s_paradox_of_the_Grand_Hotel eternally yours
I’ve started an ATMB thread about this problem; we needn’t fill this one up with an inifinte number of posts on the topic.
Fixed link - Hilbert’s paradox
Can’t remember the book, but I remember infinity from it:
"Far to the north there is a big black rock, one hundred miles high and one hundred miles wide. Every thousand years a little bird flies to the rock to sharpen its beak.
“When all the rock has worn away only one second in a minute of infinity will have passed.”
eternity would be more appropriate
still there is no time measure for eternal things.
http://www.everything2.org/index.pl?node_id=800558
There’s a link to the little bird parable.
“You’ve reached the end of the internet…” hmmm
No, no, no. It’ll only take a finite time for the rock to be worn away. And even if that’s only a second in a minute of some other time (whatever that means), we’re still finite, here.
An illustration is in order, here. Consider Graham’s Number, a very large number which has come up in theoretical mathematics. We’ll need to introduce some new notation here, called arrow notation (I can’t make arrows here, so I’ll use carats instead). For starters, let 3^3 = 3[sup]3[/sup]. Then, let 3^^3 mean 3sup[/sup], 3^^^3 = 3sup[/sup], etc. OK, now, take 3^3. Take that number, and put that many arrows in the next step, 3^^^^^^^^^^^^^^^^^^^^^^^^^^^3. Now take the number that results from that, and put that in the next step. Continue this for a total of 64 steps, and you’ve got Graham’s Number.
If your brain hasn’t yet exploded from the incredible hugeness of this number, you can now ask yourself this: How does this number compare to anything infinite? It’s not “almost infinity”, it’s not even a tiny fraction of infinity. If you take away that many integers, the number of integers left over is not just “not diminished significantly”, it’s not diminished at all.
Cabbage wrote:
The"zero chips left" argument seems to depend on not being able to name a chip left. If you assume the continuum hypothesis is false, there is an order of infinity between the integers and the reals. So there must be a subset of the reals which has more elements than the set of integers, but fewer than the set of reals. In fact, there must be an infinite number of them. Yet it is apparently impossible to specify such a subset (bolding mine, quoted post is two posts before my first post):
So just because you can’t “name” something apparently doesn’t mean it doesn’t exist. Can you reconcile these two arguments?
But the deal with this set of chips in the bowl after two hours is that we proved, not only that it is impossible to name any chip in there, but moreover that each and every chip is demonstrably not in there. By “demonstrably”, I mean that we can name the exact time at which any given chip is removed from the dish, and in each case, without exception, the time is before two hours have passed.
This is different from a counterexample set to CH. We may not be able to name any of the particular elements in such a set. But that is a far cry from being able to show that, given any real number, that real number is not in there. For if we could do such a thing, we would have shown that the set was in fact empty, and therefore not a counterexample to CH.
ZenBeam, that’s a good point; you’re right that my “Name one” “argument”, by itself, doesn’t demonstrate that the box of chips is empty, for the reasons you mention.
However, I can demonstrate that those reasons don’t apply in this situation. First of all, God only puts integers in the box, so we know that after two hours (that’s the timespan we’ve been using, isn’t it?) the chips left in the box are a subset of the integers. Any integer can be described (just write it out), and for any integer that can be describe, I can describe the time it was taken out of the box. Therefore, no integers are in the box, and, since only integers are in the box, the box must be empty.
knock-knock’s flaw in his/her logic is using limits. If f(t) is the number of chips in the box, it’s incorrect to say that the number of chips in the box after two hours is equal to lim f(t) as t goes to two (from the left). In general, you can’t evaluate a function at a particular point by taking a limit at that point–that assumes a continuity that may not be there. We can’t evaluate f(2) as the limit t->2[sup]-[/sup] because we haven’t demonstrated that f is continuous at t=2.
Solving this problem doesn’t have anything to do with limits, or in what order the limits are taken (as was mentioned earlier), the point is that we shouldn’t be dealing with limits in the first place. The above paragraph demonstrates the box is empty, and we’re done.
Cabbage, do you agree that there are infinitely many chips in the first example - when god takes out the highest number in the dish, so that it seems that all the odd integers end up in the dish?
The number of chips in the dish at any time can be given by one piece of information - either it’s some nonnegative integer or it’s “infinity.” Putting 2 chips into the dish means adding 2 to this number, and then taking 1 chip out means subtracting 1 from this number. How can it matter which chip comes out of the dish? You can imagine that god puts two chips in with his right hand, mixes up the chips in the dish, and then pulls one out with his left hand. In one case, this chip happens to be the one with the largest number on it (of the ones remaining in the dish), and in the other, it happens to have the smallest number on it. So what? In one case, this chip had been in the dish for a little while, in the other case, it just went into the dish. So what? We’re just counting chips in & chips out, since that’s all the affects the current number of chips in the dish. So there should be infinitely many chips in the dish after 2 hours when god takes out the smallest one, just as when he takes out the largest one.
Does anyone here know enough about supernatural numbers to say if that could be a reasonable answer to the question of what chips end up in the dish?
Once again, if you believe that there are any chips left in the box, name one.
You’re not just counting chips here. Your argument that “adding two and subtracting one” leads us to infinity is flawed, because that series is divergent. That doesn’t mean that the sum is infinite; that means it doesn’t have a sum.
These are not sequences of integers. They’re sequences of sets, and as such should not be expected to behave as sequences of integers.
And we probably should be dealing with ordinal numbers to solve this, but that’s too complicated, and Cabbage has come up with a simpler proof.
Supernatural numbers don’t enter into play here, for two reasons:[ol][li]We haven’t all agreed that they exist. Supernatural numbers are a model-theoretic construct, and it’s quite (relatively) consistent to assume that there are no such beasts.You’ll never break out of the natural numbers by adding one. And that’s all you’re ever doing–adding one.[/ol][/li]
Yeah, it’s only a two-item list. Tough stuff.