Re: infinity

David R Tribble said:
>> But you were talking about a binary number with an infinite number
>> of digits. Which is an ill-defined concept, since N contains only
>> finite natural numbers; there is no infinite natural "number".

Tony Orlow (aeo6) wrote:
> That is your opinion, based on a flawed proof.

Okay...

Problem:
At step i, add 10 balls (which are numbered 10i+1 to 10i+10) to
the vase, and then remove the earliest ball previously added
(the one numbered i). The time from step i+1 to i+2 is half
the time from step i to i+1; by noon, all steps have been
performed.

1. You state that this is equivalent to the series:
s = (10 - 1) + (10 -1) + (10 - 1) + ...
where a (10 - 1) term is added to the series at each step i.

You further claim that the vase contains an infinite number of balls
at noon, which is equivalent to saying that sum s is infinite.

Since you allow that it is possible to rearrange the terms of the
series, provided that there are no extra terms piling up somewhere,
we'll do just that:

s = (10 + 10 + 10 + 10 + ...) - (1 + 1 + 1 + 1 + ...)

No hanky panky here, because all of the 10s and all of the 1s are
accounted for.

2. You further assert that there exists an infinite natural number,
which we'll call W. We assume that W = W, by the simple reflexive
property of arithmetic. We further assume that since W is the
largest number, W+1 does not exist; in other words, there is only
one largest number, and it is W.

3. Since W is infinite in value, any infinite series with no
negative terms that diverges must sum to W.

So both of the series:
s1 = 10 + 10 + 10 + 10 + ...
s2 = 1 + 1 + 1 + 1 + ...
must sum to W:
s1 = 10 + 10 + 10 + 10 + ... = W
s2 = 1 + 1 + 1 + 1 + ... = W

The sums s1 and s2 are both equal to W, because they are both sums
of infinite divergent series with non-negative terms.

So, substituting back into the series, we get:
s = (10 + 10 + 10 + 10 + ...) - (1 + 1 + 1 + 1 + ...)
s = s1 - s2
s = W - W
s = 0

So s = 0, which means that there are zero balls left in the vase
at noon.

Did I break any of your rules?

-drt

.

Relevant Pages

  • Re: infinity
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  • Re: infinity
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  • Re: infinity
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  • Re: infinity
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  • Re: infinity
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