Infinity Biscuit Dilemma: Do You Have More Biscuits or Jars?

[Rensom]

Thread title says it all.

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?

 

[pbuk]

No.

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

 

[micromass]

See our 3-part FAQ about infinity: https://www.physicsforums.com/showthread.php?t=507003

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.

 

[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.

 

[micromass]

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.

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.

 

[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.

 

[SteveL27]

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.

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.

http://en.wikipedia.org/wiki/Galileo's_paradox

 

[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.

 

[micromass]

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.

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

You do know about countable and uncountable right?

 

[chiro]

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

You do know about countable and uncountable right?

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.

 

[daveyinaz]

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

You do know about countable and uncountable right?

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...

 

[micromass]

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...

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.

 

[SteveL27]

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.

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.

http://wiki.answers.com/Q/Which_weighs_more_a_pound_of_gold_or_a_pound_of_feathers

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

 

[Norwegian]

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

Why is the set of jars countable?

 

[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.

 

[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?

 

[pwsnafu]

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?

Clearly the shelf spans an uncountable number of dimensions!

 

[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.

 

[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)

 

[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.

 

[chiro]

Are you familiar with Cantors work Deveno?

His work (and work that has followed from other authors) covers different kinds of 'infinities' in a bit more detail.

But despite that examples like Hilberts Hotel are able to deal with your kind of situation even if the principle has to be adapted slightly in some way to explain it.

 

[Deveno]

Are you familiar with Cantors work Deveno?

His work (and work that has followed from other authors) covers different kinds of 'infinities' in a bit more detail.

But despite that examples like Hilberts Hotel are able to deal with your kind of situation even if the principle has to be adapted slightly in some way to explain it.

i actually have his book on transfinite induction, which is rather a small gem, in my opinion. and certainly modern mathematics has "taken infinity and run with it" (the uncountable nature of the reals is "not so bad" because we can use the metric to form a local countable base, in other words, we can use rational-radius ε-balls to prove the theorems we need).

a careful re-reading of my prior post may indicate i am aware of cantor's proof that |A| < |P(A)| (hint: where did the "2" come from?).

if one agrees that 2^A is, in fact, (set-isomorphic; i.e., bijective to) the "power SET" of A (emphasis on set, not "power") (as is usual for the "standard" interpretation of the ZF universe V), then one gets a sequence of "larger" infinities right there:

A, B = 2^A, C = 2^B,...

as i understand it, it is "unknown" if this forms a complete list...apparently, it's a matter of personal taste as to whether there's anything between A and B (a rather non-rigorous paraphrase of the continuum hypothesis), as neither adding this as an axiom to ZF, or adding the negation of this assertion, is logically inconsistent (it's "true if we want it to be").

this underscores, in my opinion, that sets are not, perhaps, the "ideal" concept for expressing things we wish to be undeniably true. they are just the best we've come up with...we may hit upon a better idea at some point (or: what i feel is more likely, come up with a different primitive concept that has sets as one possible variant).

personally, i believe that ideation is subtler than the symbolism we come up with to express it. that "what is out there" and "what we expereience in here" does not admit of a complete description. period. in order to communicate, we sacrifice some of the essence of what IS. as long as we are only talking about "part of it", we can be reasonably clear. in the restricted part of human experience that is mathematics, this is known to be true. i suspect it's true outside of mathematics, as well.

******

as far as the original poster's question...it's more than likely meant to be viewed in the context of a countable infinity (we have "jars" and "biscuits", discrete objects, not continua), and of course, in that case, micromass is right. the problem should say that, because "hidden assumptions" can lead to faulty reasoning, no?

 

[chiro]

as far as the original poster's question...it's more than likely meant to be viewed in the context of a countable infinity (we have "jars" and "biscuits", discrete objects, not continua), and of course, in that case, micromass is right. the problem should say that, because "hidden assumptions" can lead to faulty reasoning, no?

This is a huge deal no matter what kind of field we are talking about.

It is important that everyone understands this whether they are layman and/or scientists. I have a feeling that a lot of scientists do understand this, but I get the feeling that not everyone does. It's good that any scientist with the right doubt and skepticism that applies it equally to others as well as himself will no doubt incorporate this into their scientific, and no doubt other life endeavors.

The best we can do IMO is to not only acknowledge our assumptions and constraints but to then go on and specify the limitations that are inherent with this.

By doing this, I foresee that any person doing investigate work and presenting an argument will be able to see the holes easily and end up developing a way to eliminate them bit by bit if this information is made explicit by everyone that puts forward an argument.

The thing is that ironically, most of us put forward an argument because we think we are right. Regardless of the reason (which may include completely selfless and non-selfless reasons with everything inbetween), it doesn't make sense on a psychological level to provide a very convincing argument only to point out all of its flaws.

I can understand why it isn't done, but then again I can also say that doing this would have very profound consequences on how we see the world and especially how we act in the world especially towards each other.

 

[pwsnafu]

this underscores, in my opinion, that sets are not, perhaps, the "ideal" concept for expressing things we wish to be undeniably true. they are just the best we've come up with...we may hit upon a better idea at some point (or: what i feel is more likely, come up with a different primitive concept that has sets as one possible variant).

Sounds like you would be interested in topos theory.

personally, i believe that ideation is subtler than the symbolism we come up with to express it. that "what is out there" and "what we expereience in here" does not admit of a complete description.

This is more the difference between first order and second order logic rather than set theory versus alternatives.

 

[Deveno]

Sounds like you would be interested in topos theory.



This is more the difference between first order and second order logic rather than set theory versus alternatives.

i believe i would be interested in topoi. i have a book somewhere, but it's slow going with my deteriorated brain cells. also, it's self-study, so if i feel like stopping, i do.

i "almost" understand your second remark. as i understand it, to even create a foundational theory of math, we need to decide first, "what rules are legal". this starts to get complicated, although "classical logical" (or any of its various incarnations using logic gates, boolean structures, or what-not) seems to be preferred for it's natural affinity with our "common language syntax". we have so far been unsuccessful in endowing a logical system with the ability to think, although there's been a lot of progress (the best computer chess players can beat the best human chess players, but robo-cop is still a few years away).

from what i gather, second-order logics are more expressive than first-order logics, but i do not believe that "more expressive power" is what we need. my belief is:

all formal systems of sufficient expressive power are incomprehensive. that is, we cannot model EVERYTHING with ANYTHING. that doesn't mean, to me, we should stop trying. a formal system that allows "a lot" of stuff to be modeled, is of greater utility than a more limited one. and i believe we may be able to eventually have "local formal systems" that "link together" that may be able to do things a single formal system cannot do (the analogy here is like the way a riemann surface allows a one-to-many map to be regarded as a function).