Re: infinity

William Hughes wrote:

<snip> 


I is the union of a bunch of sets.

Define I_n to be the set of balls added at step n.

I is the set defined by:

   e is an element of I if and only if e is an element of I_n for
   at least one n.

Since we know know how to determine if any given thing is an
element of I, we have defined the set I. (This is the standard
definition for union. There is no change whether or not there
are an infinite number of sets I_n.)

Simlarly, we define set 0.  We define set I\O to be the set
of all elements of I that are not elements of O.  Since this
is the empty set, by definition the vase is empty at state E.

But this is not correct for any step n. 


As to state E, what else could the state of the vase be at noon be,
but the state after all finite steps?

                       - William Hughes


Hi again, William. A couple of things above trouble me. Maybe three. When you say "There is no change whether or not there
are an infinite number of sets I_n", I'm unconvinced. When I add one to one, I get two. When I add one to infinity, I get infinity. I get suspicious at attempts to relate the finite to the infinite, or at casual transits from one to the other, as appears to happen when time passes from that path being followed by the defined sequence (which continues indefinitely before noon) to noon, and the number of balls is said to pass from a number without limit to a perfectly finite and ordinary zero. And that is supposed to be accomplished by subtracting only one ball at a time from that limitless number, albeit a limitless number of times, and somehow bypassing the sequence 3,2,1,0 in reaching zero.
And the phrase, "after all finite steps" is murky to me. Each step is finite, but the number of finite steps has no limit, so "after" the limitless steps is... murky to me.
And, finally, the arguement seems to be that commutivity in an infinite sum makes it (order) ignorable. Fact is, I believe the commutivity used in this particular infinite sum, by which 9+9+9... is rewritten as 0+0+0... is fallacious. I haven't articulated *exactly* why yet, even to myself, but I'm slow.
Hmm, maybe I did, and forgot. The sum describing the addition and subtraction of balls is (10-1)+(10-1)+... It is proposed that the sum be reordered as (10-1-1-1-1-1-1-1-1-1-1)+... The only proper reordering would be, it seems to me, (10-1-1-1-1-1-1-1-1-1-1+10+10+10+10+10+10+10+10+10)+... I mean, the first sum reflects what happens at each step. The proposed reordering, 0+0+0... reflects something that never happens at any known or proposed step. Therefore, it could only occur, in some indefinable way, "after all finite steps". I don't believe it.

Thanks, by the way, for taking the extra time.

Kirby
	
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