A lot of balls

Originally posted by AThousandYoung
You may be able to boil an egg, but it is impossible to put balls in and take them out that quickly (i.e. infinitely fast). You can only see what the limit is as you approach infinity (t=0 in this case). That limit is

B = 2S - S = S

The number of balls equals the number of steps. As time approaches zero, the number of balls put in approaches ...[text shortened]... , not 1. That's because the numerator approaches infinity twice as quickly as the denominator.

Lets take EXACTLY the same problem except that the ball removed is the highest number in the bag, not the lowest.

We are still putting balls in and taking balls out at the same rate.

how many balls left at time t=0

explain.

Originally posted by wolfgang59
Lets take EXACTLY the same problem except that the ball removed is the highest number in the bag, not the lowest.

We are still putting balls in and taking balls out at the same rate.

how many balls left at time t=0

explain.

Originally posted by Palynka
Again you completely ignore my posts and decide to rephrase the same arguments.

In that case:
Set of balls inserted : {1,2,3,4,...}
Set of balls removed: {2,4,6,...}
Set of balls remaining {1,3,5,...}

The rate is irrelevant, it's about the operations creating a one-to-one correspondence or not. In your new case, although one can create a one-to-on ...[text shortened]... t in this case inserting 2 or 10 balls at a time will lead to the same balls remaining at t=0?

There is no such thing as t = 0 in your problem because you've tied the existence of t = 0 to the existence of infinities. You cannot do math with infinities. You can only do math with limits as variables approach infinity.

The technically correct way to approach this is to say "what is the limit of the number of balls in the bag as time approaches 0".

I believe your argument, Palynka, boils down to "infinity minus infinity = 0" which is fallacious.

There is no one to one correspondence between the two sets until t=0 and t cannot = 0, t can only approach 0 - because of how you've set up the problem.

Originally posted by AThousandYoung
There is no such thing as t = 0 in your problem because you've tied the existence of t = 0 to the existence of infinities. You cannot do math with infinities. You can only do math with limits as variables approach infinity.

The technically correct way to approach this is to say "what is the limit of the number of balls in the bag as time approach ...[text shortened]... il t=0 and t cannot = 0, t can only approach 0 - because of how you've set up the problem.

Originally posted by AThousandYoung
There is no such thing as t = 0 in your problem because you've tied the existence of t = 0 to the existence of infinities. You cannot do math with infinities. You can only do math with limits as variables approach infinity.

The technically correct way to approach this is to say "what is the limit of the number of balls in the bag as time approach il t=0 and t cannot = 0, t can only approach 0 - because of how you've set up the problem.

Originally posted by Palynka
I also don't see how I do infinity - infinity at any point. I just say that if I removed all elements of set {1,2,3,4,...} from {1,2,3,4,...} (the same set!) then I get {}. Not Inf-Inf=0. These are not the same.

Originally posted by AThousandYoung
There are infinite elements in the first set. There are infinite elements in the second set. If I remove (subtract) all infinity elements of set 2 from set 1's infinite supply of elements I will get a set with 0 elements.

Infinity minus infinity equals zero.

Originally posted by Palynka
I don't do any subtraction.

Originally posted by Palynka
That is the wrong operation because we're talking about elements in a set (balls in the bag). With a set with countably infinite cardinality, if you insert a new element for each element that exists the number of elements remains the same because you can still do a one-to-one correspondence (bijection) between them.
http://en.wikipedia.org/wiki/Galileo%27s_ ...[text shortened]... here is a one-to-one correspondence between the two sets then all balls are eventually removed.

Originally posted by Palynka
Set of ALL balls inserted : {1,2,3,...}
Set of ALL balls removed: {1,2,3,...}

Same set. You're also welcome to *sigh* disprove the argument that it's impossible that there are infinite balls in the bag and there isn't a lowest numbered ball.

Again *sigh* your mistake is extrapolating cardinality follows from infinite sequences, but anyone would agree ...[text shortened]... n show that the sets
{1,2,3,...} ; {2,4,6,...}
have exactly the same cardinality.

*sigh*

Originally posted by iamatiger
Ye gods.

Once again you are saying that the series (num of balls in bag):

(1,2,3,4,5,6,7,8,9,10...}

converges to 0!!

Originally posted by PBE6

"If you've done six impossible things this morning, why not round if off with breakfast at Millyways?" - Douglas Adams

Suppose you and your friend come into possession of a bag of infinite capacity, along with an infinite supply of numbered balls and a stopwatch with an infinite number of gradations. After watching the universe explode ...[text shortened]...
[b]Question: How many balls will be left in the bag when the stopwatch reads 0 seconds?[/b]

Originally posted by Palynka
Of course there is t=0, Achilles reaches the tortoise doesn't he? But you are right in the sense that the limit cannot help you there because cardinality is not continuous with respect to set theoretic limits.

And in t=0, any ball that you can think of has been removed. If there were infinite balls, I would take one out (any one) and realized that it shou ,...} from {1,2,3,4,...} (the same set!) then I get {}. Not Inf-Inf=0. These are not the same.