A stupid question about infinity

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?

MrAnchovy

No.

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

micromass

See our 3-part FAQ about infinity: http://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

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

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

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

You do know about countable and uncountable right?

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.

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...

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.

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.

http://wiki.answers.com/Q/Which_weig...nd_of_feathers

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

Norwegian

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.

MrAnchovy

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

Clearly the shelf spans an uncountable number of dimensions!