Re: infinity

David R Tribble said:
> 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.
Right, you still get (9+9+9+9...)
>
> 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.
I assume no such thing. One can add 1 and get W+1. One can have W-1 and W/2.
So, you lost me there. You claim all (countable) infinities are the same. I
reject that notion.
>
> 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?
Yes. You declared all infinities equal, assumed you had some smallest
infinity,w hich is impossible, and then claimed W-W=0, which is hardly agreed
upon, and generally considered to be undefined, because infinities actually DO
come in different forms.
>
> -drt
>
>

--
Smiles,

Tony
.

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