Re: infinity
- From: Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx>
- Date: Mon, 8 Aug 2005 10:22:45 -0400
Hi -
I've been on vacation. Before looking at any other responses, let me give my
take on this. Of course, the Cantorians will say I am wrong, but as you have
found, Cantorian theory of transfinite sets makes absolutely no sense anyway,
so don't worry about that.
Theo Jacobs said:
> Hello everyone,
>
> I'm having an argument with a friend about the following problem:
>
> Suppose you have a giant vase and a bunch of ping pong balls with an
> integer written on each one, e.g. just like the lottery, so the balls
> are numbered 1, 2, 3, ... and so on. At one minute to noon you put
> balls 1 to 10 in the vase and take out number 1. At half a minute to
> noon you put balls 11 - 20 in the vase and take out number 2. At one
> quarter minute to noon you put balls 21 - 30 in the vase and take out
> number 3. Continue in this fashion. Obviously this is physically
> impossible, but you get the idea. Now the question is this: At noon,
> how many ping pong balls are in the vase?
At each point, after adding 10 balls and removing one, you have added 9 balls,
so altogether, you have added 9 times some infinite number. The question is,
what is this infinite number? Cantorians would have us believe that, if you can
form any kind of "bijection", or symmetrical 1-1 correspondence between members
of two sets, then they have equally infinite sizes. However, a more exacting
approach involves using the function which produces the 1-1 correspondence
between members of two sets.
In this case, each successive addition of 9 balls occurs after 1/2^n minutes,
so how many such numbers are there between 0 and 1? We note that we can produce
a direct relation between {2,4,8,16,...} and {1/2,1/4,1/8,1/16,...} and can
consider those infinite sets to be the same size, much like {1,2,3,...} and {-
1,-2,-3,...} would be considered the same size; making the exponents negative
doesn't change the size of the set. So, we would compare {2,4,8,16...} to the
set of "natural" numbers {1,2,3,4...}, the size of which we use as a unit
infinity, to which we compare other infinities. Let's call that number N.
Certainly, the set of naturals numbers includes all the integral powers of 2,
but not vice versa, so we would intuitively consider our set to be smaller than
the naturals. In fact, we skip more and more natural n's for every successive 2
^n we generate, and essentially have floor(log2(n)) powers of 2, in a set of
the first n naturals. So, in my mind (and Bigulosity Theory), if N is the
number of natural counting numbers, then the size of your set of negative
powers of 2 is log2(N). Of course this is an infinite number, but it is
infinitely smaller than N. Oh, and let's not forget that at each of these log2
(N) points, we are adding 9 balls, so my final answer is 9*log2(N). I hope this
makes more sense to you than all the other nonsense you are likely to have been
told and confused by. Cantorian thought is an exercise in verbal diarrhea. I'll
try to catch up on this thread throughout the day.
>
> An 2nd experiment goes as follows:
> put no. 1 - 10 in, take no. 10 out, put no. 11-20 in, take no. 20 out, put
> no. 21-30 in, take no. 30 out, etc.
> (of course at the same moments as above)
Exactly the same. It doesn;t matter which nine balls you leave.
>
> My friend claims both experiments end up with an empty vase.
> I think however the 2nd experiment ends up with a vase with an infinite
> number of balls:
> 1-9, 11-19, 21-29 etc. are definitely in the vase.
If your friend thinks he ends up with an empty vase, then he is even confused
about Cantor, and his head might just be projecting its own emptiness on the
vase. Hopefully, your friend is a nice person with a good sense of humor,
anyway. :D
>
> He says it all has to do with Cantor's set theory, cardinality etc..., but
> browsing the internet didn't really help me much.
> Any information or relevant links are very welcome,
>
> Thanks, Theo
>
>
>
--
Smiles,
Tony
.
- References:
- infinity
- From: Theo Jacobs
- infinity
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