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David Marcus wrote:
Tony Orlow wrote:
Randy Poe wrote:
Tony Orlow wrote:
stephen@xxxxxxxxxx wrote:
Tony Orlow <tony@xxxxxxxxxxxxx> wrote:
stephen@xxxxxxxxxx wrote:
Tony Orlow <tony@xxxxxxxxxxxxx> wrote:
stephen@xxxxxxxxxx wrote:
<snip>
What does that have to do with the sets IN and OUT? IN and OUT are
the same set. You claimed I was losing the "formulaic relationship"
between the sets. So I still do not know what you meant by that
statement. Once again
IN = { n | -1/(2^(floor(n/10))) < 0 }
OUT = { n | -1/(2^n) < 0 }
I mean the formula relating the number In to the number OUT for any n.
That is given by out(in) = in/10.
What number IN? There is one set named IN, and one set named OUT.
There is no number IN. I have no idea what you think out(in) is
supposed to be. OUT and IN are sets, not functions.
OH. So, sets don't have sizes which are numbers, at least at particular
moments. I see....
If that is what you meant, then you should have said that.
And technically speaking, sets do not have sizes which are numbers,
unless by "size" you mean cardinality, and by "number" you include
transfinite cardinals.
So, cardinality is the only definition of set size which you will
consider.....your loss.
In any case, it still does not make any sense. I am not sure
what |IN| is for any n. IN is a single set. There is only
one set, and it does not depend on n. In fact, there isn't
an n specified in the problem. Yes I used the letter n in
the set description, but that does not define an entity named 'n'.
There most certainly is an 'n'. The problem describes a repeating
process, each repetition of which is indexed with a successive n in n,
and during each repetition of which ball n is removed. What do you mean
there's no n???
The ORIGINAL problem. This is a new one, inspired by
the original, but it is one with no balls, no vases, no
time steps, no iterations. Just a definition of two subsets
of the natural numbers, one called IN and one called
OUT.
The definition of the set IN does not include a definition
of something called IN(n).
You are being asked to characterize these two subsets.
- Randy
Set-theoretically, they are the same set, by the axiom of
extensionality. That doesn't mean the axioms of ZFC account for all the
information in the problem. There is the combining of +10 and -1 in each
iteration n at time t=-1/n, a coupling that is not being addressed by
your method. You are considering the two sets statically, outside of
time, as completed, but for any n, the max of in is 10 times the max of
out. Just because these both have some limit at oo, even though they
don't reach it, doesn't mean they reach it at the same time. They DON'T
reach it, and if they did, if noon occurred, in would reach it in 1/10
the time as out. But,there is no such ending to the finites, and so the
set-theoretic approach using the set N is bogus at its core.
It is remarkable that you seem to be unable to answer any post without
mentioning the ball and vase problem. Why is this? Are you afraid that
if you do, you will be trapped into an inconsistentcy?
Is the following an accurate description of what you are saying?
1. You agree that (given the definitions above) IN = OUT and that |IN
\OUT| = 0.
2. You don't agree that, in the ball and vase problem, the number of
balls in the vase at noon is |IN\OUT|.
2
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