Re: An uncountable countable set

In article <453b44a4@xxxxxxxxxxxxxxxxxxx>,
Tony Orlow <tony@xxxxxxxxxxxxx> wrote:

cbrown@xxxxxxxxxxxxxxxxx wrote:
Tony Orlow wrote:
cbrown@xxxxxxxxxxxxxxxxx wrote:
Tony Orlow wrote:
cbrown@xxxxxxxxxxxxxxxxx wrote:
Tony Orlow wrote:
cbrown@xxxxxxxxxxxxxxxxx wrote:
Tony Orlow wrote:
cbrown@xxxxxxxxxxxxxxxxx wrote:
Tony Orlow wrote:

<snip>

You have agreed with everything so far. At every point before noon balls
remain.
To be precise, the assertions above all imply that at every time t =
-1/n, where n is a natural number, there are balls in the vase.

But that *alone* does not even include every time t before noon; let
alone every time t. For example, notice that nowhere above do you or I
/explicitly/ assert: "at t=-2/3, the number of balls in the vase is a
positive finite number".

We assert something specific about t = -2/4, and something specific
about t = -1/3, but nowhere do we directly state somthing about t =
-2/3.

On the other hand, given the problem statement, I think we would both
/agree/ that there "should be" an obvious (perhaps even unique)
well-defined answer to the question : "what is the number of balls in
the vase at time t = -1/pi?"

Assuming in the remaining statements that one agrees with the previous
statement, this leads us to the question: what are the unstated
assumptons that allow to agree that this must be the case?

I attempted to describe those assumptions in my previoius post. Did you
read those assumptions? If so, do you agree with those assumptions?
At this point I don't recall your previous post. I've been off a bit.

Well, allow me to repeat them here (with two minor changes):

In order to interpret the problem

"At each time t = -1/n where n is a (strictly positive) natural number,
we place the balls labelled 10*(n-1)+1 through 10*n inclusive in the
vase, and remove the ball labelled n from the vase. What is the number
of balls in the vase at time t=0?"

I make the following simple (and I would claim, fairly uncontroversial
and natural) assumptions:

--- (object permanence)

(1) When we speak of a time t, we mean some real number t.

(2) If a ball is in the vase at any time t0, there is a time t <= t0
for which we can say "that ball was placed in the vase at time t".

(3) If a ball is placed in the vase at time t1 and it is not removed
from the vase at some time t where t1 <= t <= t2, then that ball is in
the vase at time t2.

(4) If a ball is removed from the vase at time t1, and there is no time
t such that t1 < t <= t2 when that ball is placed in the vase, then
that ball is not in the vase at time t2.

---- (obedience to the problem constraints)

(5) If a ball is placed in the vase at some time t, it must be in
accordance with the description given in the problem: it must be a ball
with a natural number n on it, and the time t at which it is placed in
the vase must be -1/floor(n/10).

(6) If a ball is removed from the vase at some time t, it must be in
accordance with the description given in the problem: it must be a ball
with a natural number n on it, and the time t at which it is removed
from the vase must be -1/n.

(7) If n is a natural number with n > 0, then the ball labelled n is
placed in the vase at some time t1; and it is removed from the vase at
some time t2.

--- (very general definition of "the vase is empty at noon")

(8) the number of balls in the vase at time t=0 is 0 if, and only if,
the statement "there is a ball in the vase at time t=0" is false.

---

Perhaps you would add other assumptions (9), (10), etc.; but my
question is:

Given the problem statement, do you agree that /each/ of these
assumptions, /on its own/, is reasonable and not just some arbitrary
statement plucked out of thin air?

If not, which assumption(s) is(are) not reasonable or is(are)
unneccessarily arbitrary?

<snip>

Those all look reasonable to me as I read them. I don't see any
statement regarding the fact that ten balls are added for every one
removed, though that can be surmised from the insertion and removal
schedule. That's the salient fact here. You never remove as many as you
add, so you can't end up empty.

On the other hand, the vase can end up empty if every ball inserted into
it is also removed from it.


You claim nothing changes at noon.
Where, exactly, above do I claim that "nothing changes at noon"?
Do you disagree with the other standard-bearers, and claim that
something DOES occur at noon?

That is not a response to /my/ question "where, exactly, above do I
claim that 'nothing changes at noon'?"

/I/ don't claim that "something occurs at noon"; nor do /I/ claim that
"nothing occurs at noon".

Uh, what would be your opinion on the matter. CAN something occur at
noon in this experiment or not? Either way, you have a problem.

Since cbrown does not express any opinion, only TO has that problem.


/YOU/ are claiming that the truth of these statements follows logically
from our assumptions; but until you make some kind of mathematical
statement which corresponds to "something occurs at noon", I really
can't address it as a /mathematical/ question.

1/n=0. Happy? Que pasa aqui? Is n in N?

How does that correspond to "something occurs at noon"?


<snip>

Do you accept the above statements, or do you still claim that there
is
/no/ valid proof that ball 15 is not in the vase at t=0?
I said that any specific ball was obviously out of the vase at noon.


That's good: we at least agree that it logically follows from (1) - (8)
that there are no labelled balls in the vase at t=0.

No finite balls.

And in the total absence of any justification for their existence, where
would any others come from?


What I honestly find baffling is your repeated claim that it doesn't
then logically follow from assumptions (2) and (5), that if a ball is
in the vase at /any/ time, it is a ball which is labelled with a
natural number; and so therefore the above statement is logically
equivalent to "there are no balls in the vase at t=0".

At every time before noon, there are balls in the vase. If nothing
occurs at noon, there are still balls in the vase.

Non sequitur. You are assuming a finiteness property must apply to
non-finite process.

If balls are removed
at noon, then balls are inserted such that 1/n=0, infinite balls.

If balls are removed at noon, it is a different GE.

What really baffling is that a divergent sum could be considered to
evaporate, and by so many apparently bright people. Are you all
Christians or something?

We are only accepting the conditions of the GE as originally stated.
TO keeps trying to ring changes on it to allow his assumptions to be
valid.


<snip>

Therefore ball 15 is not in the vase at noon; and nothing you
said above challenges the logic of this conclusion.

Of course not. Ball 15 is gone.

You say here, "of course not", but you previously stated that it is
"not in my purview" to claim /anything at all/ about the state of
affairs at noon; because "noon does not occur". Therefore, we can
conclude that you now retract these statements, is that correct?

The state of affairs with regard to the VASE. Ball 15 is obviously out
of the vase. All finitely numbered balls are out. But not before noon.

Assuming "But not before noon" = "But not at any time t < 0"; these
statements follow from (1)-(8).

Ja. Corrrect.


And if the experiment continues UNTIL noon, infinitely-numbered balls
are added. There's no way around that. There's nothing in between.


I don't see how you justify your assertion "if the experiment continues
UNTIL noon, infinitely-numbered balls are added", if we only assume (1)
- (8) above (in particular, it directly contradicts (5)).

0=1/n. n=?

Relevance?


So my guess is that you are appealing to some /other/ assumption in
your argument. What is that assumption?

t=1/n ^ t=0 -> 1/n=0 -> n=oo

if t = 1/n and t = 0 then one can also prove 2 = 1 and all sorts of
other interesting things.


<snip>

All of this is logical. The question is which ideas are logically
consistent with each other. I don't think I'm being illogical, or that
many here are. It's just that we're working with incompatible assumptions.


Well, to assist in distingushing between "(some statement) doesn't
follow logically from our assumptions" and "(some statement) shows that
we're working with incompatible assumptions", it is neccessary to know
/what/ assumptions we are using.

I gave a list of 8 that seem very uncontroversial and intuitively
natural; it would be helpful if you would do something similar where
you disagree with my assumptions.

For example, how does your response above /in any way/ address my
statement "because t=0 is /not/ a time such that t=-1/n for some
natural n"?
All events happen before noon. At all times before noon, there are balls
in the vase. At noon they are gone.

Okay; so you choose to ignore my argument. So let's consider yours, in
light of the only explicit assumptions we have - which are the ones I
give above.

What am I ignoring? I am answering your question. You ask what my
comment has to do with t=-1/n=0 not having a solution for n in the
naturals, and I respond that the implication is that nothing happens at
t=0. Is that not an answer to the question?


Assuming that by "all events happen before noon", you mean "balls are
only placed in or removed from the vase at times before noon"; then
these are logical conclusions from assumptions (1) - (8).

All finite balls are inserted and removed at times before 0, at which
times there are balls in the vase.

Assuming that there are balls in the vase at noon is not justified by
any requirement in the original statement of the problem.


Either something happened to the
balls at noon, or it happened sometime between then and every moment
before noon, which doesn't even make sense.

I agree; to the extent that I don't know /what/, exactly, you mean to
imply by your statements. They don't make sense to me either.

Either something happens an noon, or it doesn't. Where do you stand on
the matter?


To start with, since by (1), by "a time t", we mean "a real number t",
there is no time t which is both after /every time/ before noon (i.e.,
for all t'< 0, t' < t), and yet still is before noon itself (i.e., t <
0); by virtue of the fact that t would then have to come before itself:
it would require that t < t; and that is not true of any real number t,
and so it is not true for any time t.

Exactly. If the vase is empty at noon, it either became empty at noon,
or at some time before noon. So, which is it?

False dichotomy. It presumes. falsely, that what happens in the limit
must happen before the limit.


Secondly, when you say "the number of balls at noon is different from
the number of balls at every time before noon", I understand you to
mean "for each t < 0, the number of balls at time t is different from
the number of balls at time 0".

Yes, and? There are balls before noon, but not at noon. That implies
something happened to them at noon, but nothing occurs in the vase at noon.

Define f(t) so that
for t >= 0, f(t) = 0
for t < 0, f(t) = Ceil(-1/t)
Then lim_{t -> 0-} f(t) = oo but f(0) = 0.

So that functions which do what TO claims is impossible are possible.


Furthermore, you seem to be relying on the assumption:

(*) If the number of balls at time t1 is different from the number of
balls at time t2, and t1 <= t2, then there is some time t such that t1
<= t <= t2, such that balls are added or removed at time t'.

Uh, yeah. What's wrong with that? Are you saying that balls leave by
some other means than removal?


But "the number of balls in the vase at time 0 is different than the
number of balls in the vase at any time -1 < t < 0" is /not/
inconsistent with (1) through (8) + (*), despite the fact that no balls
are added or removed at time 0.

Yes, it clearly is woefully inconsistent with any time continuity that
an infinite number of balls should disappear, not at a given moment, and
not before, and yet be gone from that moment forward, while having been
there up until that moment. What the heck is that. Math? No, it's hocus
pocus.

It cannot happen before 0, it cannot happen at 0, but at 0 it has happened.

Define f(t) so that
for t >= 0, f(t) = 0
for t < 0, f(t) = Ceil(-1/t)
Then lim_{t -> 0-} f(t) = oo but f(0) = 0.

That's how it does not happen for any t < 0 but "has happened" at 0.



Therefore, it's perfectly reasonable, by which I mean "logically
consistent with assumptions (1) through (8) + (*)", that the number of
balls at each time t before noon be different from the number of balls
at time 0 - it's just a possible consequence of the fact that at each
time t < 0, balls are added and removed at a time /strictly between/ t
and 0; so each t < 0 the vase can contain a different number of balls
that it does at time 0.

Proof of claim: The existence of such a time t' follows from (1) which
states that by a time t, we mean a real number t.

Since t is a real number, then by the Archimedean principle, if t < 0,
then there is a natural number n such that n*(-t) > 1.

Therefore, for each time t with -1 < t < 0, there is a natural number n
such that t < -1/n < 0.

Since n is a natural number, the values 10*(n-1) + 1 through 10*n
inclusive are natural numbers.

Therefore by (7), balls with labels n and labels 10*(n-1) through 10*n
inclusive are all placed in the vase at some times and removed from the
vase at some times.

Then by (5), balls 10*(n-1)+1 through 10*n inclusive are added to vase
at time t' = -1/n; and by (6), ball n is removed from the vase at time
t' = -1/n.

Therefore, for all t < 0, there is a time t' such that t < t' < 0 when
balls are added and removed.

Uh huh, and how many remain as each is removed?


As a sidebar, the above is an example of a mathematical argument. To
dispute it, one typically says something like "it is not logical to
conclude from (7) that the ball labelled n is placed in the vase at
some time t". It is not mathematically of interest that you simply
claim "but how could that happen?" or "that's just Cantorian nonsense".


It is Cantorian nonsense. You need to decide whether something happens
at noon, or not.

TO's objections are anti-Cantorian, anti-logical nonsense.

Define f(t) so that for t >= 0, f(t) = 0, for t < 0, f(t) = Ceil(-1/t).
Then lim_{t -> 0-} f(t) = oo, but f(0) = 0.
That's how it does not happen for any t < 0 but "has happened" at 0.


And if you claim some unrelated thing, such as "when t=0, n=oo; so noon
never occurs", that is not a refutation of the /argument/ I gave above.

Your proof above is for each particular n in N. How does that prove
anything about N itself? That's an objection I've faced countless times.
This is yet another "largest finite" argument".

Define f(t) so that for t >= 0, f(t) = 0, for t < 0, f(t) = Ceil(-1/t).
Then lim_{t -> 0-} f(t) = oo, but f(0) = 0.
That's how it does not happen for any t < 0 but "has happened" at 0.

It simply asserts that although I can prove A, you claim to be able to
prove not A. If your claim is true, it means that our system may be
inconsistent; but it is not, in the context of the argument given
above, an invalidation of the given proof of "A" - that proof still
holds until you demonstrate exactly which step in the argument is not
logically valid.

I have explained what's invalid.

You can claimed things are invalid, but those claims explain nothing.


You're creating a condensation point in
time to artificially give N an LUB and a completion.

Unless TO wishes to claim that time comes only in quanta of which there
is some minimal allowable size, every point in time is a condensation
point of other times, and time is continuous.




At that point in
time, if it exists, iterations are occurring infinitely rapidly, and you
blur the link between insertions and removals by pretending you have
finished an infinite process.

Define f(t) so that for t >= 0, f(t) = 0, for t < 0, f(t) = Ceil(-1/t).
Then lim_{t -> 0-} f(t) = oo, but f(0) = 0.
That's how it does not happen for any t < 0 but "has happened" at 0.


Note that the above argument does /not/ assert that the number of balls
in the vase at time is 0 or any other number; nor does it assert that
the number of balls at time 0 is /not/ the same as the number of balls
at /some/ time t < 0.

It only asserts a fact for each individual ball.


It simply demonstrates that, despite the fact that no balls are removed
or added /at/ time 0, the number of balls /at/ time 0 is not
/neccessarily/ the same as the number of balls /at/ any particular time
t < 0. This is because we can prove that /at/ any particular time t <
0, there will be a strictly /later/ time t' when balls are added and
removed, and where t' is /before/ time 0. Therefore to claim that it is
inconsistent with (*) is false.

<snip>

The fact remains that at all times -1<t<0 balls are in the vase. You
claim at t=0 no balls are in the vase, and yet nothing happened at t=0.
That's entirely nonsensical.

Define f(t) so that for t >= 0, f(t) = 0, for t < 0, f(t) = Ceil(-1/t).
Then lim_{t -> 0-} f(t) = oo, but f(0) = 0.
That's how it does not happen for any t < 0 but "has happened" at 0.



If one wants to being time into this
gedanken, then by all means, please obey the constraints of time.

We have! It is TO who wants time to be something it is not.


No, that does /not/ follow. Here's what follows (I invite you to either
confirm or debunk the claim that these statements all logically follow
from (1) through (8)):

First, if a ball is in the vase at time t = 0, we can conclude from (2)
it was placed in the vase at some time t1 <= 0.

That ball.


Since it is a ball that is placed in the vase at some time t1, then by
(5), it is a ball labelled with a natural number n, placed in the vase
at time t1 = -1/floor(n/10).

Yes, that ball.


Since n is a natural number, by (7) it is a ball which is removed at
some time t2.

That one, uh huh.


Therefore, by (6) it is a ball that is removed from the vase at time t2
= -1/n, with t1 < t2 < 0.

By (5), it is a ball which is remoevd from the vase at time t2, and not
placed in the vase at any time t with t2 < t <= 0.

Therefore by (4), it follows that it is a ball which is not in the vase
at t=0.

So if there /is/ a ball in the vase at noon, this implies it is a ball
which is /not/ in the vase at noon.

Uh, it implies it's not that naturally numbered ball that you specified.
Can you specify anything that happens arbitrarily close to noon?

Define f(t) so that for t >= 0, f(t) = 0, for t < 0, f(t) = Ceil(-1/t).
Then lim_{t -> 0-} f(t) = oo, but f(0) = 0.
That's how it does not happen for any t < 0 but "has happened" at 0.


Therefore we can conclude that "there is a ball in the vase at noon" is
false. This is by reductio absurdum: "If (A implies not A), then not
A".

No, you generalized from some specific ball to "all balls". Foul.

One can generalize from "for each x, f(x)" to "for all x, f(x)".
If in no other way, then by induction on x where x is a natural:
Let S be the set of naturals on balls removed before noon.
The first natural is in S and for each natural in S so it its successor,
therefore S = N.
.

Relevant Pages

  • Re: Balls and Vase problem - final solution
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  • Re: Balls and Vase problem - final solution
    ... Do they ever indicate when or how the balls with nonstandard natural ... I believe in one of the old ball and vase threads from last year, ... nonstandart naturals, if you *still* insist)? ...
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  • Re: An uncountable countable set
    ... makes the vase empty? ... So if you just keep on adding balls one at a time, ... You drew that from my suggestion of the number circle, ... and that's why any infinite set of naturals ...
    (sci.math)
  • Re: infinity
    ... >>> the vase keeps growing as you approach noon. ... the algorithm which describes the filling of the vase with balls ... Start with an empty vase. ... we try to connect some logical reasoning (about putting ...
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  • Re: An uncountable countable set
    ... above) and given that the "moment the vase becomes empty" means the first time t>= -1 that Vis zero, then it follows that the "vase becomes empty" at t = 0. ... No balls are added or removed at noon, but the vase becomes empty at noon. ... The fact that you have no upper bound to the naturals. ...
    (sci.math)