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A stupid question about infinity 
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#1
Feb2112, 06:12 AM

P: 1

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? 


#2
Feb2112, 06:21 AM

P: 545

No.
For a further explanation of the concepts involved, google 'Hilbert hotel infinity'. 


#3
Feb2112, 06:38 AM

Mentor
P: 18,346

See our 3part 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 2n1 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. 


#4
Feb2212, 12:09 PM

P: 1

A stupid question about infinity
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
Feb2212, 12:14 PM

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P: 18,346

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
Feb2212, 01:15 PM

PF Gold
P: 1,957

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
Feb2212, 05:42 PM

P: 800

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


#8
Feb2212, 06:32 PM

P: 225

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
Feb2212, 06:35 PM

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P: 18,346

You do know about countable and uncountable right? 


#10
Feb2212, 08:15 PM

P: 4,579

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


#11
Feb2412, 06:33 PM

P: 225

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


#12
Feb2412, 06:39 PM

Mentor
P: 18,346

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


#13
Feb2412, 06:50 PM

P: 800

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! 


#14
Feb2412, 07:00 PM

P: 144




#15
Feb2412, 07:16 PM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,683

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
Feb2412, 07:55 PM

P: 545

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
Feb2412, 08:25 PM

Sci Advisor
P: 839




#18
Feb2412, 08:57 PM

P: 225

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. 


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