Re: infinity

William Hughes wrote:

Tony Orlow (aeo6) wrote:

William Hughes said:
<snip> 
This is true, but you only add 10 balls at a finite step.  No, you
never get the empty set at any finite step.  No one disputes this.

You never get an empty set at ANY step. 


-You never get the empty set at any finite step
-we only have finite steps
-you never get an empty set at ANY step

looks like we agree.  What I also have is the concept of the
state after all the finite steps. Call this state, state E
Which balls are in the vase at state E.
We do not know, this has to be defined.

I balk at "*all* the finite steps". Easy to say, hard to define. 

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.

Actually, I - O, right? And that is not at E (which has yet to be defined, IMO) but at any given specific step.

How many balls are in the vase at state E?

Definition 2:

   Define the number of balls in the vase at state E
   to be the number of balls in the set I\0.

Now you may not like these definitions, but they seem natural to
me.  Furthermore, any definitions that are inconsistent with these
seem unnatural to me.

Using the above definitions

   1.  The vase is empty at state E

   2.  The number of balls in the vase at state E
       does not depend on the number of balls in the vase
       at any finite step  (in particular the number of
       balls in the vase  at state E is not any sort of
       infinite sum).

You may not like the conclusions, but if so you need to
disagree with the definitions.



At EACH step, when you remove 1 ball,
you have just added 10 balls. Unless, of course, you are saying that at
infinite steps you are following a different procedure, but that was never
stated. So, you can NEVER, at any finite OR infinite step, have an empty set.



Nothing was said about infinite steps because they do not exist.



here they are arguing that there is a difference between whether the
one ball removed is labeled one way or another. If we add 10x
through 10x+9 (for x=0 to N) and then remove 10x, then 10x+1 through
10x+9 are NEVER removed. However, if you say you are adding 10x
through 10x+9, and taking away x+1, then you claim that you remove
all elements. There is a serious inconsistency here between claiming
that order doesn't matter for sets in general, and then claiming
that here it does. What difference does the label on the removed
ball really make?


Without "labels" or other identifying features, you can't decide
whether a given ball, once put in, is taken out again at some time.


It doesn't matter about any particular ball. Whenever you remove a ball, you
have just added 10, so the vase can't be empty. QED


But you only remove a ball af a finite step.  So you have just
concluded that the base is not empty at any finite step.


Then you never remove balls at an infinite step? How can it ever become empty
then? Do you also stop adding balls at infinite steps? It sounds like infinite
steps don't even exist, but that seems like par for the course with the current
misunderstanding of infinity.



Check the definitions.
I have no need of the hypothesis that infinite steps exist.
(In other contexts it is convenient to call state E, step omega,
but I don't have to do that here.)


Since it is feasible to remove just a subset (we are talking about
infinite sets here), there is no way except checking the balls.  I
could equally well forget ball 0 and start the removal action with
ball 1.  In that case, at the end ball 0 will be left.  I could remove
just the odd-numbered balls, in which case the even-numbered balls
will be left.  And so on.


So, the same operation results in entirely different numbers of remaining
balls, depending on the labels on the balls? This is sheer nonsense.


The solution to the problem lies in forgetting about your
bijections, and noting that at each of an infinite number of steps
you are adding a net 9 balls to the vase,

Oh, but counting to 9 is all about bijections. 

It's about counting. What purpose is this statement supposed to serve?

and the overall number never decreases from one step to the next.
You see, it's not that I am missing anything, but simply noting that
your system produces contradictory results, none of which make very
good sense.


Once you realize that an infinite set can be mapped 1:1 to a proper
subset of itself, it becomes clear that at the "end of the process",
we can no longer rely on the pigeon hole principle: the number of
things put in and taken out alone does not suffice for determining
anything (well, it is sufficient for determining that the remaining
number of balls must be countable, but that is not really impressive).


When every removal is accompanied by a larger addition, the vase will never be
empty, but continually more full. Does the infinite series (10,-1,10,-1,10....)
converge to zero? Do the terms have a limit of zero? No, sorry.

Does the problem have anything to do with infinite series. No sorry. 

Sorry, but the problem has EXACTLY to do with a sum of an infinite series of
terms, which is what infinite series IS. Otherwise, what do you think infinite
series are all about? Do you, or do you not, add 10, then remove 1, as
described above, an infinite number of times? Are you not summing an infinite
number of terms, and if so, don't the rules regarding infinite series apply?



No check the definitions above.  I am looking at the number of elements
in a set difference.  I am not using any sort of infinite series.

             <snip>


                     -William Hughes

Kirby
.

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", ...
    (sci.math)
  • 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. ...
    (sci.math)
  • Re: infinity
    ... the set of balls in the vase at state E ... >> consists of a finite number of sets or an infinite number of sets. ... The sum of an infinite series does depend on the number ...
    (sci.math)
  • Re: infinity
    ... Since there are an infinite ... >>> in the vase and state E, I don't have any problem with the vase ... >>> of balls in the vase at state E, ... the missing something that happens is the preservation of the ratio ...
    (sci.math)
  • Re: infinity
    ... >>>You never get an empty set at ANY step. ... >> Which balls are in the vase at state E. ... >> the vase at any finite step. ... So, you can NEVER, at any finite OR infinite step, have an empty set. ...
    (sci.math)