Infinity Question

Factual Questions
1

this actually isn’t even a question more of a “huh, thats weird, hes absolutely right” moment

a while back I took a class about infinity.

say god got bored and made a bunch of chips with numbers written on them 1 to infinity (he can do that… hes god… he created an infinite amount of chips… )

and yes you can do an infinite amount of things in a finite amount of time… xeno’s paradox and all. (just take an hour on the first one, half an hour on the second one, 1/4th an hour on the 3rd one… as the number you do goes to infinity the time it takes becomes exactly 2 hours)

so god takes his chips… and makes a big dish for them… and puts them in a pile next to them…

he plans to sort them…

his plan is to take two out of his pile and put them both in the dish… then look in the dish and take out the biggest one.

so he does that for every chip (takeing two hours) and he takes out the biggest one

turn one puts in 1 and 2… takes out 2…
second turn puts in 3 and 4 takes out 4
third turn puts in 5 and 6 takes out 6

ect ect ect…

when hes done… he looks at his dish and says “wow, I have every single odd number in my dish” (since 1,3,5, ect never were the highest in the dish)

now jesus comes in and says “dad! thats not fair, takeing out the biggest one… its unfair”

so god comes up with a new way to sort them for the next time he plays

he takes out the smallest number each time.

so he puts in 1 and 2 and takes out 1 (so there is one in the dish)
next he puts in 3 and 4 and takes out 2 (two in the dish now)
next 5 and 6 and takes out 3 (3 chips left in the dish)
7 and 8 takes out 4 (4 left in the dish)

while hes doing this… being all smart he says “every time I do it, there is one more chip left in the dish than last time” being even more smart he knows that after doing that an infinite number of times… there will be an infinite number of chips in it…

so he calls over jesus “hey jesus! look at this! there will be an infinite number of chips in this dish when I am done!”

jesus says “oh yeah, name one?”

god says… well I bet 1 billion is in there…
jesus points out… “nope you took that out the billion and first turn same for a trillion or a jillion”
question is… how does that work? its odviously correct, but my brain isn’t quite sure what that means…
I mean the whole infinity fractions thing is the base of inergal calculus so thats right…

derr… the solution is that no chips remain in the dish after its done… but derr… derr… my head hurts…

2

Yep. Infinity’s a weird number. Chips are put in twice as fast as they’re taken out, so the number of chips in the dish is always increasing; but every chip is eventually taken out, so in the end none are left. If we were watching God do this over the course of a minute by doing the first swap at 1/2 minutes, the second at (1/2 + 1/4) minutes, the third at (1/2 + 1/4 + 1/8) minutes, etc, the rate at which the number of chips was increasing would appear to accelerate exponentially, until suddenly, at t = 1 minute, they’d all disappear.

3

Here’s my uneducated take on it (great spatial skills, no time for class):

Let’s not conceive of infinity. You’re not supposed to actually picture it, I’m told one can get brain cancer from doing so. Infinity is beyond conception. However, what I noted about your pattern is that you’re basically keeping the numbers going up by one, while the range of integers in your chip bowl is always an uninterrupted sequence between x and 2x (x being the chip that gets taken out “next”)… since infinity is inconceivably huge, no number that God could name could be half of infinity (therefore waiting to be taken out rather than already out). Clever Tyrell. No chips in the bowl… at least none that would fit inside our universe. Infinitely big chips. Nifty.
Personally, I think if he did it in a minute, they wouldn’t all disappear, because the bowl is overflowing with half of an infinite number of chips… and even silly people know that half of infinity is still infinity. SO… the universe explodes or something. oops.

4

Yep. It’s kinda weird, ain’t it?

5

You can’t.

Since it is infinity, there is no biggest number.

6

That is not Xeno’s Paradox.

And you can’t infinite amount of things in a finite amount of time. For the last thing, you have to do it in 1/infinity, which bombs out.

7

Well you know, infinity is not a number so it isn’t surprising that the rules of arithmetic, which deals with numbers, don’t apply.

8

I think this is explained in
http://www.mathpages.com/rr/s3-07/3-07.html
http://mathforum.org/isaac/problems/zeno1.html
one of these links explains how an infinite number of events can be done in a finite amount of time.
Granted, it’s just theory and based on ancient physics, but to say it’s not Xeno’s paradox…well.

yes, you can>no, you can’t >yeah>no>…:wink:

I think considering the lack of technology and even a basic understanding of quantum mechanics he did allright.

9

Maybe that’s how God created the supernatural numbers.

The what? I’d provide a link, but I can’t find any good sites out there. Basically, supernatural numbers can be thought of as “infinitely big numbers.” Real numbers can be expressed in the form xx.xxxx… (e.g. pi = 3.141592…), and supernatural numbers can be expressed in the form …xxxx.xx (you can’t write them as a numeral). You could think of infinitesimals like dx as the inverses of supernatural numbers. The best discussion I’ve seen of supernatural numbers (and a great book that looks at all sorts of questions like this) is Gödel, Escher, Bach: An Eternal Golden Braid by Douglas Hofstadter.

God does have to have an “infinite number” of chips in the dish, as is obvious (more or less) if you ignore the numbers on the chips and just count the number of chips. A sorta rigorous way of thinking about this is that, if you name any number n, then I can find some time before the 2 hr. limit when there are more than n chips in the dish, and, since the # of chips in the dish is always increasing (or at least non-decreasing), there will be more than n chips in the dish at any later time. Since the # of chips grows greater than any number n as you approach the 2-hr mark, it must be approaching infinity.

10

urban ranger… well its related to xeno… alot of calculus only works because you can add an infinite number of things and get a whole number.

11

No, no, a thousand times no! Calculus works because as you add more and more terms, the sum gets closer to a specific value. There is no adding of an infinite number of terms.

Sorry if I sound harsh, but this is probably one of the most confusing facts about math, and I’ve seen a few heated arguments arise from it.

12

I think the trick is in the definition of infinity in different contexts. It can be used as either countable infinity or not countable. By limiting it to a set (ie:a space of time with a beginning and an end) you have changed the context to where it can be counted. This way you can actually “measure” infinity in a finite setting.

So, which set is largest…the (infinite) set of even integers or the I-set of odd integers or the set that contains all integers?

13

I don’t think that this is quite right. It is one thing to say that,
[ul]For every number n, there is a time t such that more than n chips are in the dish at time t.[/ul] This is true. But to say that there are ever infinitely many chips in the dish (ie, that “God does have to have an “infinite number” of chips in the dish”) is to say that
[ul]There is a time t such that, for every number n, more than n chips are in the dish at time t.[/ul] This is not true.

14

or small
infinity is not a dimension
a concept to defy dimensionally related concepts:)

15

Uh-oh. A lot of us are in trouble, then.

16

They’re all the same size. The problem deals only with countable infinities.

17

During the time period when 0 (< or =) t < 2 hr., you’re right, there are never “an infinite number of chips” in the dish. However, as the time approaches the 2-hr mark, the number of chips in the dish approaches infinity (i.e. increases without bound). And at the time t=2 hours, how many chips would you say are in the dish? I would say that there are infinitely many, since, for any n, there are more than n chips in the dish at t=2 hr.

18

The OP’s second example is cute, but it seems to me that there is a problem in trying to equate the cardinality of an infinite set {1, 2, 3,…} with the sum of an infinite series 1/2, 1/4, 1/8, … Like the Energizer Bunny, the infinite set just keeps on going. When you say that the sum of the series exists, you mean that the total can be made arbitrarily close to a value by including enough terms. The problem with trying to equate this to how many disks are left in the bowl is that there is no last one that gets put in the bowl - the number is always increasing.

As t-keela mentions, there are different “sizes” of infinity. The first infinity is called countable because a countably infinite set is one which can be brought into a 1-to-1 corespondence with the set of positive integers {1, 2, 3, …} Thus the set of all +ve odd integers, the set of all +ve even integers and the set of all +ve integers are each countably infinite, and thus the same “size”. Any odd number can be expressed as 2n - 1, any even number as 2n for n = 1, 2, 3, …
thus there is a 1-to-1 corespondence between the three sets.

IIRC correctly, the next “size” of infinity is represented by the set of real numbers. It is possible to show that one cannot establish a 1-to-1 corespondence between the set of real numbers x such that 0 < x < 1 and the set of +ve integers, thus there are uncountably many real numbers in this interval. Apparently there are even higher kinds of infinity, but at that point my brain was full so I asked to be excused from class.

19

Well . . . The cardinality of the reals is the next cardinal number, if you accept Cantor’ Continuum Hypothesis.The Continuum Hypothesis is independent of the other axioms of set theory. Since it is independent it can be assumed true or assumed false. Neither will result in a contradiction (that wasn’t already there). The set of reals is strictly larger than the set of integers with or without CH. If you assume CH is false, then you are assuming there are sets of reals that is uncountable, but still smaller than the set of reals.

For any set A, the power set of A (the set of all subsets of A) is larger than A. The Generalized Continuum Hypothesis says that if A is infinite, the next larger “size” set is the power set of A. GCH, like CH is independent of the other axioms. It can be assumed true or assumed false.

20

Huh? You took a class in it, and you have to ask this question? Didn’t they teach you everything?