# A stupid question about infinity

1. Feb 21, 2012

### Rensom

Lets say you have an indestructible jar. The jar contains two biscuits.

Now imagine you had an infinite number of these jars, each containing two biscuits. Would you have more biscuits than jars?

2. Feb 21, 2012

### pbuk

No.

For a further explanation of the concepts involved, google 'Hilbert hotel infinity'.

3. Feb 21, 2012

### micromass

Basically, what you can do in your case is to label your jars 1,2,3,4,etc.

Take a cookie in jar n, send the first cookie to jar 2n-1 and send the second cookie to jar 2n.

For example send the cookies in jar 1 to 1 and 2.
Send the cookies in jar 2 to 3 and 4.
Send the cookies in jar 5 to 6.
Etc.

This shows that you can get exactly one cookie in each jar. This implies that there are as much cookies as jars.

Last edited by a moderator: May 5, 2017
4. Feb 22, 2012

### Draqla

So that would mean if we look jar at position n, we would have one cookie in jars from 1 to n, and 3 cookies in jars n+1 to 2n, which would still make number of cookies 2x more than jars if we look 2n jars. And in jars>2n, we would have jar with 2 cookies.

5. Feb 22, 2012

### micromass

No, you are not understanding it. You do the process for ALL jars.

You seem to suggest that we only do it for jars 1 to n. But we don't, we do it for all jars.

Jar 1 will send cookies to Jar 1 and Jar 2
Jar 2 will send cookies to Jar 3 and Jar 4.

Jar 4432 will send cookies to Jar 8863 and Jar 8864.

We do this for ALL jars.

6. Feb 22, 2012

### Char. Limit

This is probably what makes infinity so hard to understand. Yes, there are exactly as many even numbers as there are integers. There are also as many rational numbers as there are integers.

7. Feb 22, 2012

### SteveL27

It's historically interesting to note that Galileo pondered these issues hundreds of years ago. He noted that the square numbers 1, 4, 9, 16, 25, ... were a proper subset of the whole numbers, yet could also be put into 1-1 correspondence with the whole numbers.

8. Feb 22, 2012

### daveyinaz

We've all heard the question: What weighs more a ton of bricks or a ton of feathers?

The answer is they weigh the same. This is the same concept, simplistically, when talking about an infinite number of jars and an infinite number of biscuits.

9. Feb 22, 2012

### micromass

I fail to see how your response has anything to do with this thread.

You do know about countable and uncountable right?

10. Feb 22, 2012

### chiro

I think what he means is that certain infinities are equivalent in the same sense that a tonne of feathers vs a tonne of bricks is also equivalent.

Of course not all infinities are created equal, but his comment at least to me had a point.

11. Feb 24, 2012

### daveyinaz

Just to poke fun....I fail to see how your response has anything to do with this thread.

You do know that the set of jars is countable right? Which means even mentioning uncountable sets is fruitless...

12. Feb 24, 2012

### micromass

That was the entire point of this question. Maybe the set of cookies was uncountable!!!

It is not, as we have shown, but we didn't know this a priori.

13. Feb 24, 2012

### SteveL27

Did you know that a ton of feathers weighs more than a ton of gold? That's because gold is measured in troy weight.

Precious metals such as gold are measured in troy weight. A troy pound is 12 troy ounces, and each troy ounce is 480 grains, making a total of 5760 grains to the pound of gold.

Most materials use pounds and ounces from the avoirdupois system, and such a standard pound is made up of 16 ounces, where each ounce is 437.5 grains, making a total of 7000 grains to the pound of feathers.

All this means that a "pound" of feathers (or bricks, or lead) is heavier than a "pound" of gold.

Just as not all infinities are equal, neither are all tons!

14. Feb 24, 2012

### Norwegian

Why is the set of jars countable?

15. Feb 24, 2012

### HallsofIvy

This all depends on how you are measuring the "number of things" in an infinite set. If you are measuring by cardinality (the most usual way) 2 times any infinite cardinality is that same cardinality. Yes, the original post only said that the set of jars was infinite, not whether it was countable or uncountable (I'm still wondering why it was necessary to postulate that the jars are 'indestructible') but whether countable or uncountable, the cardinalities of the set of jars and the set of cookies is the same.

16. Feb 24, 2012

### pbuk

I have a shelf with a countably infinite number of jars on. The first jar has capacity of 1l, the second 0.5l, the third 0.25l... Where would you put an uncountably infinite number of jars?

17. Feb 24, 2012

### pwsnafu

Clearly the shelf spans an uncountable number of dimensions! :tongue:

18. Feb 24, 2012

### daveyinaz

Mountains out of mole hills with you people, lol

let's see...the jars are countable as they could be considered as the natural numbers for all intents and purposes...anything said to the contrary is just wrong.

19. Feb 24, 2012

### Matt Benesi

There are distinct infinites in the hyperreal numbering system, as well as distinct infinitesimals (distinct from 0).

2H/H = 2 (with H representing an infinite)

20. Feb 24, 2012

### Deveno

the question IS rather ill-posed.

the reason being, we're not told "how infinite" our set of jars is. if we had a very large space to put our jars in, perhaps we have uncountably many, in which case micromass' technique breaks down, we don't have enough labels for the jars, so we can't decide which biscuit should go where.

(caveat: i admit, thinking of an uncountable collection of jars is rather mind-boggling...think of the cost! but, by the same token, thinking of a countably infinite collection of jars is also rather whimsical, as no such collection has ever existed).

infinite things behave rather oddly, so it's best to be clear about what KIND of infinite things you mean. there's a lot of different kinds.

at one point it was thought:

∞+1 = ∞
∞+2 = ∞
....
∞+∞ = 2∞ = ∞
3∞ = ∞
....
(∞)(∞) = ∞
3 = ∞
....
= ∞ <--here, it was shown we have a problem.

that is, ∞↑∞ is suddenly "bigger" than all those "smaller infinities".

in fact, the problem happens in the "up" part:

2↑∞ is already "too big" (who would have thought that "2" would cause so many problems? obviously, if we want to avoid infinite problems, we should stick to numbers like 0 and 1, which at least behave themselves).

it of course, gets worse from there. you can form the number:

∞↑∞ (an infinite "power tower" of infinities raised to an infinite power), which is so big...how do you even start to get there?

it's a lot worse than just being able to "add 1". you can add an infinite number of an infinite number of infinite infinities (and more) to each "new, bigger" infinity you get, and then form sequences of these "huge" infinities, and take the limit! perhaps you can see that after a while, it becomes very difficult to tell "which" infinite collections of the infinitely infinite infinities one is talking about. set-theorists often talk about some "infinity" called κ, as if that clears up everything. i, for one, find that amusing.