Re: infinity

Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> writes:

> Jesse F. Hughes said:
>> Let's change the problem slightly. Again, we have an infinite set of
>> ping pong balls, each ball labeled with a natural number. But instead
>> of the old procedure, let's put *all* of the balls into the vase at
>> 11:59 and remove the first one. At 11:59:45, we remove the second,
>> and so on.

> This is the same question Virgil posted a half dozen times in a row.

Yes it is, but that is coincidental.

>>
>> Tony: Is the vase empty or not at noon?
> Yes.
>>
>> If empty, then when was the last ball removed?
> Noon.
>>
>> If not empty, then which balls did we fail to remove?
> None.
>>
>> Can we put infinitely many balls into a vase by doing it one at a time
>> (with increasing speed)? If so, are we able to also empty a vase with
>> infinitely many balls by the same method? (Countably infinite in each
>> case, of course.)

> Yes, but not all countable infinities are the same. This is a basic problem
> with "cardinality".

But in both examples (the ten-at-a-time example and the all-at-once
example), we put the same number of balls in the vase: one ball for
each natural number. And in both examples, we removed the balls in
exactly the same way. How can the outcome be any different?

Or do you think that putting in balls ten at a time until we've
exhausted the set N of natural numbers produces a *bigger* set than
putting all of the balls in the vase at once? Where did those extra
natural numbers come from?

--
"It's an exercise in game theory[...] I've been brutally logical in
my analysis on this point.[...] My analysis indicates that the
optimal strategy for mathematicians is to acknowledge the result
today." --JSH gives practical reasons to accept his FLT proof
.

Relevant Pages

  • Re: infinity
    ... >> I is the union of a bunch of sets. ... >> Define I_n to be the set of balls added at step n. ... by definition the vase is empty at state E. ... > are an infinite number of sets I_n", ...
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  • Re: infinity
    ... >>> Which axioms allow completion of an infinite ... That's what a sequence is, by the way: ... > If you do not interrupt the process, the vase never "reaches" noon. ... > where xis the number of balls labeled i. ...
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  • Re: An uncountable countable set
    ... The number of balls approaches infinity as time ... So, David, you think the fact that balls leave the vase only by being ... from infinite series, ... Very basic logic would hold that, if the vase is not empty at any time t ...
    (sci.math)
  • Re: infinity
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  • Re: Logarithm of transfinite numbers
    ... infinite number of finite numbers. ... balls in the vase, whether n is finite or infinite. ... Yes, and at any given finite n, you have 9n balls in your vase. ... At the end of time T, Tony, is the bin empty ...
    (sci.math)