Re: infinity

William Hughes said:
> Tony Orlow (aeo6) wrote:
> > William Hughes said:
> > >
> > > Tony Orlow (aeo6) wrote:
> > > > David Kastrup said:
> > > > > Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> writes:
> > > > >
> > > > > > If you want to think of it that way, I suppose you can, even though
> > > > > > there is no point at which you are taking away more balls than you
> > > > > > put in. While set theorists claim that order is irrelevant within a
> > > > > > set,
> > > > >
> > > > > For determining the cardinality of a set the order is irrelevant. It
> > > > > turns out that the cardinality of the set of balls put into the vase
> > > > > and the cardinality of the set of balls taken out again are the same.
> > > > And yet, as Cantorians have admitted in the past, cardinality of infinite sets
> > > > is not the same as set size.
> > > >
> > > > > What does this mean? That there is _one_ way of mapping the balls
> > > > > taken in and taken out 1:1. But since this cardinality happens to be
> > > > > infinite, and infinite sets can be mapped to a proper subset of
> > > > > themselves, this does not mean that there might not also be a mapping
> > > > > 1:10. So the question "do balls remain, and how many?" can't be
> > > > > answered by looking just at the set sizes, because the sets are
> > > > > infinite. Instead one has to take a look at every ball individually
> > > > > and has to check whether it will be put in, but not removed. Because
> > > > > we are talking about infinite sets, and the pigeon hole principle is
> > > > > no longer valid.
> > > > So, you look to see if you can ever remove the last ball. When you have just
> > > > added 10 balls, removing 1 ball will never result in an empty set. Isn't that
> > > > true?
> > >
> > > 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.
>
> 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.
>
> How many balls are in the vase at state E?
An infinite number. A larger infinity than the infinite number that have been
removed.
>
> Definition 2:
>
> Define the number of balls in the vase at state E
> to be the number of balls in the set I\0.
That follows from the above definition.
>
> 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.
They don't seem wrong to me.
>
> Using the above definitions
>
> 1. The vase is empty at state E
This does not follow from the definitions above. How do you derive this,
exactly? Certainly not mathematically.
>
> 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).
How can the number of balls in the vase at state E NOT depend on how the balls
are added and removed at each finite step? Are you saying that if I add 10
balls and then take half away at each point, that I will get the same answer,
because it doesn't matter what happens at each step. Indeed not. At state E
given that procedure, I will end up with 10 balls (provided my set can include
fractions of balls). Of course the final outcome depends on what happens at
each finite step. And of course it is EXACTLY what an infinite series is, the
sum of an infinite number of terms. So, I don't know WHY you think this is any
different, or requires labelling of balls and mind contortions. You add a net
of 9 balls per iteration, for an infinite number of iterations. The vase has 9
times some infinity of balls in it.
>
> You may not like the conclusions, but if so you need to
> disagree with the definitions.
Those conclusions do not follow from the above definitions. If you insist they
do, please provide some derivation of those conclusions from the definitions,
so I can see what other axioms/rues/assumptions you are applying in your logic,
because the idea that the vase is empty and that this has nothing to do with
any finite step does not follow logically from statements that generally define
the relationship between balls in vs. out of the vase at state E.

Now, I know how Cantorians tend to equate things. They ignore the mapping
functions that they use to establish bijections. If you can establish a
bijection, voila! the sets have the same "cardinality". This is a prime example
of why this doesn't work. At every point, you have added ten times more balls
than you have removed. I still haven't seen any Cantorians comment on the
scenario where all the balls removed are placed in a second vase. Maybe I just
haven't seen it yet. I won't hold my breath.
>
>
> >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.
(sigh) The don't talk about state E, if it doesn't exist, because state E is
what happens at infinity. If you can't count your way there, don't pretend to
know anything about what you will find there.
>
>
> > >
> > > > >
> > > > > > 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.)
So, what the heck is it you ARE doing?? If you could derive your conclusions
step-by-step, that might be interesting.
>
> > > >
> > > > > 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.
So, your conclusion is derived from some unspoken axiom that aleph_0-aleph_0=0?
That's what it sounds like, and yet, I don't think Cantorians agree on any such
thing. Are you working from YOUR own personal set of axioms?
>
> <snip>
>
>
> -William Hughes
>
>

--
Smiles,

Tony
.

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