Re: infinity

William Hughes wrote:

Kirby Cook wrote:

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. 


No, but the set of balls in the vase is not I\0 for any step
n.  By definition, the set of balls in the vase at state E
is I\O.  The vase is never empty at any step n.  The vase is
empty at state E.  Counterintuitive, Yes.  Follows from
the definitions, Yes.

	Whoa, I have to back up for a minute.  You wrote,
"Definition 1:

    Let I be the union of the sets of balls added to
    the vase at any finite step.

    Let O be the union of the sets of balls removed from
    the vase at any finite step.

    Then 0 is a subset of I and we define the set of balls
    that is in the vase at state E is the set difference I\O."

So, in any one finite case, I/O is defined by the "Let I be" and "Let O be" statements above. And, according to the definition, I/O is not empty there. And, in fact, at each and every nth step, there will be 9n more balls in I than in O, different balls. Now, the two sets maintain that relationship as they increase, in lockstep, without limit.
I've been imputed, elsewhere, with claiming a largest natural number. I don't think I've done that, but you appear to be doing so. At least, you seem to be proposing a point, some state, in which the built-in non-zero difference between I and O disappears, at which point I/O becomes empty, and you call that state E.




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.



The definition for the union of a bunch of sets:

     e is a member of the union of a bunch of sets if and only
     if e is a member of at least one of the sets.

Note this definition does not change whether the bunch of sets
consists of a finite number of sets or an infinite number of sets.
This is all that was meant by "There is no change whether or not there
are an infinite number of sets I_n".

The advantage to using the concept of an infinite union is that
it sticks very close to the concept of a finite union and
our intuition is valid.  On the other hand if we try to use
counting arguments we get things like


When I add one to
one, I get two.  When I add one to infinity, I get infinity.



which are counterintuitive.  (Indeed they are only a short form
for more exact statements about cardinality, which are themselves
counterintuitive).


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.



Most of this make no sense. One have to be very careful with
definitions when talking about infinite sets.  In particular, most
discussions of counting make no sense.


	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.



Indeed, but what else can we call noon, but "after all finite steps"?
If a ball is added to the vase before noon and not removed, don't we
have to say the ball is in the vase at noon?  If a ball is never added
to the vase don't we have to say the ball is not in the vase at noon?
If a ball is removed from the vase before noon, don't we have to say
the
ball is not in the vase at noon?  Put this together and we get the
set of balls in the vase is I\O.


	And, finally, the arguement seems to be that commutivity in an infinite
sum makes it (order) ignorable.




Note:

    1.  The number of balls in the vase at state E does not
        depend on the number of balls in the vase at finite steps

Would you mind exploring, just once more, why you say this, and how that can be? That may be all I need. All I see (dimly) right now is that it has something to do with reconciling the two observations that the number of balls in the vase increases at each step, without limit, and that each and every ball added to the vase will be, according to the established procedure, removed from the vase, but later, after the number of balls in the vase has grown.

    2.  The sum of an infinite series does depend on the number
        of balls in the vase at finite steps.

    So  The number of balls in the vase at state E is not
        given by the sum of an infinite series.

Yes the problem is set up so intuitively the answer is the sum
of an infinite series.  But the answer to the problem is
counterintuitive.



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.



The infinite series

    10 -1 + 10 - 1 + 10....

can be used to model the problem, but only sort of.  Yes, you can
rearrange the  series to get a sum of zero (sort of) and yes this
infinite rearrangement is analogous to what is happening in the problem
(sort of), but it doesn't really help.

I finally pinned my objection down. Saying the sum can be rearranged as only 0+0+0... =0 is in error, as this neglects mentioning that for each and every 0 that appears in the reordered sum there is a corresponding 90 that appears somewhere else in the same sum.


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



You're welcome

                        -William Hughes

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