Balls and Vase problem - final solution
Earie
even been granted a name by itself (the Littlewood-Ross paradox). Time
to close it, I think.
In short, this is the problem: We have a vase and an infinite number
of numbered balls. At t=1, we insert balls 1-10 and take out number 1.
At t=1.5 we insert 11-20 and take out number 2. At t=1.75 we insert
21-30 and take out 3, and so on. How many are left inside at t=2? one
argument says that infinite balls will be inside, as at each of the
(infinite) actions we inserted more that taken out. Another argument
says that none will be left, as for each ball we can show the instant
in which it was taken out. The first argument strengthens itself by
this variation: at t=1 we insert 1-10 and take out number 10, at t=1.5
we insert 11-20 and take out number 20 etc. Clearly aleph0 balls will
be left, and this scenario is equivalent to the formar one up to
permutation of the numbers on the balls, which doesn't really change
anything.
Solution: I don't know why this is presented as a supertask with
infinite number of actions. Clearly the outcome is not changed if we
do all the actions at once: at t=1 we insert aleph0 balls, and at t=2
we take out aleph0 balls. Now we have aleph0-aleph0 left, which is
undetermined, and can be anything between [0,aleph0]. For example, if
we take out only the even balls, then we have aleph0-aleph0=aleph0
left (all the odd ones). If we take out all the balls whose number is
greater than 35, we have aleph0-aleph0=35 balles left. In our case,
however, we have taken them all out, so none have left. The first
argument, that permutation of the numbers on the balls is meaningless,
is now clearly wrong. It is very meaningfull.
The problem is completely equivalent to Hilbert's hotel: one has
aleph0 occupied rooms, yet with a small manipulation of the room
numbers one can get 1, 35 or even aleph0 rooms empty, without taking
out any of the residents. So this problem doesn't warrant a name by
itself, and certainly not being called 'a paradox', as no one really
views Hilbert's hotel as a paradox.
Sheep Net
"Earie" <par...@yahoo.com> wrote in message news:2b175476-e4d8-4eaf-a831-
Jesse F. Hughes
> The balls and vase problem has been around for much to long, and has
> even been granted a name by itself (the Littlewood-Ross paradox). Time
> to close it, I think.
Yes, I don't know why you didn't just come in and fix things sooner.
--
Jesse F. Hughes
"Truth is common stuff, ready to your hand, but lies you have to make
yourself, and you can't be sure they are any good until you've
used them --- and then it's too late." John Steinbeck
Transfer Principle
> The balls and vase problem has been around for much to long, and has
> even been granted a name by itself (the Littlewood-Ross paradox). Time
> to close it, I think.
Ah yes, a problem which has drawn several heated debates and many
five-letter insults exchanged.
> Solution: I don't know why this is presented as a supertask with
> infinite number of actions. Clearly the outcome is not changed if we
> do all the actions at once: at t=1 we insert aleph0 balls, and at t=2
> we take out aleph0 balls. Now we have aleph0-aleph0 left, which is
> undetermined, and can be anything between [0,aleph0]. For example, if
> we take out only the even balls, then we have aleph0-aleph0=aleph0
> left (all the odd ones). If we take out all the balls whose number is
> greater than 35, we have aleph0-aleph0=35 balles left. In our case,
> however, we have taken them all out, so none have left. The first
> argument, that permutation of the numbers on the balls is meaningless,
> is now clearly wrong. It is very meaningfull.
The reason that many "anti-Cantorians" and "post-Cantorians" oppose
this resolution of the L-R paradox is because in the usual statement
of the problem, the number of balls increases by nine after each step
(i.e., the number of balls at time t is -9(floor(log_2(2-t))-1) and
so at time t=2 there ought to be infinitely many balls in the vase
independent of the labels on the removed balls.
If the total number of balls is finite, then the number of balls in
the vase is independent of the labels due to the Pigeonhole Principle
for finite sets. But anti-Cantorians -- actually I take that back,
since _post_-Cantorians are the ones with this belief -- prefer that
Pigeonhole Principle apply to _all_ sets, not just the finite sets.
Since, as Earie points out, all the balls with standard natural
numbered labels are removed by time t=2, post-Cantorians conclude
that the balls still in the vase have _nonstandard_ natural numbers
on their labels.
Transfer Principle
Alan Morgan
Earie <par...@yahoo.com> wrote:
>The balls and vase problem has been around for much to long, and has
>even been granted a name by itself (the Littlewood-Ross paradox). Time
>to close it, I think.
[snip]
Your solution seems to say that the number of balls left in the vase
is somewhere between 0 and aleph_0, inclusive.
Thanks for narrowing it down.
Alan
--
Defendit numerus
FredJeffries
>
> The reason that many "anti-Cantorians" and "post-Cantorians" oppose
> this resolution of the L-R paradox is because in the usual statement
> of the problem, the number of balls increases by nine after each step
> (i.e., the number of balls at time t is -9(floor(log_2(2-t))-1) and
> so at time t=2 there ought to be infinitely many balls in the vase
> independent of the labels on the removed balls.
>
> If the total number of balls is finite, then the number of balls in
> the vase is independent of the labels due to the Pigeonhole Principle
> for finite sets. But anti-Cantorians -- actually I take that back,
> since _post_-Cantorians are the ones with this belief -- prefer that
> Pigeonhole Principle apply to _all_ sets, not just the finite sets.
Once again, Leonard, you reveal your biases by only considering one
side of an argument and ignoring the other, all the while claiming an
interest in alternative systems.
Here you give aid and comfort to the hyper-Dirichletians who wish to
imprison us in a system where the "Pigeonhole Principle appl[ies] to
_all_ sets, not just the finite sets" without even indicating the
existence of the libertarian anti-Dirichletians who point out that as
it is only the small minority of finite sets to which the principle
applies, the principle should be outlawed for all sets.
Outlawing the pigeonhole principle entirely would have the additional
advantage of allowing non-standard finite ordinals, since it is only
finite sets in our current systems which require there be only one
distinct ordinal of a given cardinality.
FredJeffries
> numbered labels are removed by time t=2, post-Cantorians conclude
> that the balls still in the vase have _nonstandard_ natural numbers
> on their labels.
Do they ever indicate when or how the balls with nonstandard natural
numbers on their labels get INTO the urn?
Ariel31
> In article <2b175476-e4d8-4eaf-a831-22ea440f1...@f42g2000yqn.googlegroups.com>,
>
> Earie <part...@yahoo.com> wrote:
> >The balls and vase problem has been around for much to long, and has
> >even been granted a name by itself (the Littlewood-Ross paradox). Time
> >to close it, I think.
>
> [snip]
>
> Your solution seems to say that the number of balls left in the vase
> is somewhere between 0 and aleph_0, inclusive.
>
> Thanks for narrowing it down.
>
the answer is 0.
> Alan
> --
> Defendit numerus
David R Tribble
> The balls and vase problem [...]
>
> The problem is completely equivalent to Hilbert's hotel: [...]
It's also equivalent to Galileo's observation of infinite sets,
wherein each natural is paired with its square.
As we proceed by counting the naturals (1, 2, 3, etc.), and at
each step pairing it with its square (1, 4, 9, etc.), we skip an
ever-increasing number of non-square naturals. Yet every
natural is paired with a square, and vice versa. Galileo's
conclusion was that infinite quantities don't follow the same
(arithmetic) rules as those of finite quantities.
I'm currently reading this fine book:
http://www.amazon.com/exec/obidos/ASIN/1841196509
Transfer Principle
I believe in one of the old ball and vase threads from last year,
someone posed the following equivalent problem:
Suppose at time 2-2^-(n-1), we add balls 10n-9 to 10n-1 (and not
10n) to another vase, and then take ball n and add a zero to the
right end of its label (so that its label is now 10n).
Then at all times t<2, the balls in this vase have the same labels
as those in the original problem's vase. But at time t=2, standard
theory states that the original vase is empty, but the new vase
can't possibly be empty -- since at no time t is any ball removed
at all! Instead, we see that each ball has as its label a number
followed by infinitely many zeros. And voila -- there are your
nonstandard natural numbers!
According to standard theory, the original vase is empty while the
new vase has infinitely many balls. According to post-Cantorians
("hyper-Dirichetians"), these two cases ought to be _equivalent_,
so that one vase is empty if and only if the other is.
And so they choose to consider that both vases are full.
Transfer Principle
> On Aug 23, 3:24 pm, Transfer Principle <lwal...@lausd.net> wrote:
> > If the total number of balls is finite, then the number of balls in
> > the vase is independent of the labels due to the Pigeonhole Principle
> > for finite sets. But anti-Cantorians -- actually I take that back,
> > since _post_-Cantorians are the ones with this belief -- prefer that
> > Pigeonhole Principle apply to _all_ sets, not just the finite sets.
> imprison us in a system where the "Pigeonhole Principle appl[ies] to
> _all_ sets, not just the finite sets" without even indicating the
> existence of the libertarian anti-Dirichletians who point out that as
> it is only the small minority of finite sets to which the principle
> applies, the principle should be outlawed for all sets.
I remember someone giving this sort of argument before -- possibly,
probably Jeffries himself. Since there are only countably many
finite ordinals and countably many hereditarily finite sets, while
there are uncountably many infinite ordinals and sets (indeed, too
many to be a set), the finite sets are the exceptions, so that if
anything, one should make finite sets act more like infinite sets
and not vice versa, so the argument goes.
So we eliminate the Pigeonhole Principle for finite sets, since
infinite sets don't adhere to it. By that logic, since ZFC proves
that the sum and product of any two infinite cardinals is their
maximum, why don't we also declare the sum and product of two
_finite_ cardinals to be their max, as well?
Now, let's see if these "anti-Dirichletians" will put their money
where their mouth is.
So let's arrange for the anti-Dirichletians to have a new
compensation plan. Their annual salary will no longer consist of
the total of all of their biweekly of monthly paychecks. Instead,
we'll only have to pay them the highest paycheck once -- since
they believe that the sum equals the max -- and then we never
have to pay them again.
Of course, some anti-Dirichletians might say that they don't
mind this arrangement, as long as they only have to pay the max
of their _bills_, instead of the total. But here we say no -- it
doesn't work that way. They still have to pay the total of all
of their bills. The only exception is if their creditor also
happens to be an anti-Dirichletian -- only then does one only
have to pay the max instead of the total. But those who aren't
anti-Dirichletian will expect the total payment.
This is the point I'm trying to make here. Changing finite
arithmetic so that it works like infinite arithmetic has dire
consequences on the real world, including finances. On the
other hand, changing infinite arithmetic so that it works like
finite arithmetic has no effect on the real world, where for one
thing, people are paid finite amounts of money.
Even ultrafinitists who reject the existence of large finite
numbers should at least set their upper bound greater than the
total lifetime amount of their paychecks. AP's upper bound of
10^500 is large enough to avoid financial consequences. One can
increase the bound to include RSA and other applications.
And before people start asking for a new theory in which the
smallest natural number is one billion -- in order to become an
instant billionaire -- all I have to say is, fat chance
convincing an employer to go along with that. On the other hand,
there'll be plenty of takers for anti-Dirichletians who only
need one paycheck.
FredJeffries
Once again the anti-Cantorian shows his fixation on cardinal numbers,
ignoring ordinals and is thus unable to see the trees for the forest.
An anti-Cantorian and an anti-Dirichletian go into a bar. Each has K
dollars.
The anti-Cantorian buys the first round. It costs K dollars.
The anti-Dirichletian buys the second round. It also costs K dollars.
They say "Let's have a third round." But the anti-Cantorian has spent
his K dollars and could buy no more. The anti-Dirichletian, however,
had ordered the K dollars in her wallet so that she had only spent
every other dollar. Thus she had K dollars left with which to buy the
third round.
When later a group of people from work came into the bar, the anti-
Cantorian could only look sheepish whereas the anti-Dirichletian, by
careful budgeting, was able to buy a round for the whole group.
And leave a tip.
mjc
>
> - Show quoted text -
And I, being a math/comp-sci type person, look at from a physical
point of view.
Eventually, as the number of balls gets sufficiently large,
if the distance to move the balls into and out of the container is
bounded below,
the balls will have to move faster than the speed of light.
This will provide a upper limits of the number of balls that can be
moved
in a amount of time.
The amount of energy to move the balls might also be a consideration.
FredJeffries
> we see that each ball has as its label a number
> followed by infinitely many zeros. And voila -- there are your
> nonstandard natural numbers!
Of course. Perhaps I was momentarily fooled by your chosen sobriquet
into nor realizing that you were referring to the nonstandard
nonstandard natural numbers of the anti-cantorians wherewith they show
themselves unable/unwilling to distinguish between cardinal and
ordinal but also unable/unwilling to distinguish either of those from
nonstandard natural numbers.
Let us dip into the urn and pull out one of these balls with a number
followed by infinitely many zeros. It says 1745 with infinitely many
zeros following. Was this the ball that was originally numbered 1745?
Or is it the one that was originally numbered 17450 or the one
originally numbered 1745000 or...
Thus we have the irony that the persons who complain about not being
able to distinguish the cardinalities of the natural numbers and the
even numbers have constructed their own system with (infinitely many)
classes each having infinitely many members none of which can be
distinguished from each other.
FredJeffries
> On Aug 25, 9:18 am, FredJeffries <fredjeffr...@gmail.com> wrote:
>
> > On Aug 23, 3:24 pm, Transfer Principle <lwal...@lausd.net> wrote:
> > > Since, as Earie points out, all the balls with standard natural
> > > numbered labels are removed by time t=2, post-Cantorians conclude
> > > that the balls still in the vase have _nonstandard_ natural numbers
> > > on their labels.
> > Do they ever indicate when or how the balls with nonstandard natural
> > numbers on their labels get INTO the urn?
>
> I believe in one of the old ball and vase threads from last year,
> someone posed the following equivalent problem:
Equivalent according to whose standards?
>
> Suppose at time 2-2^-(n-1), we add balls 10n-9 to 10n-1 (and not
> 10n) to another vase, and then take ball n and add a zero to the
> right end of its label (so that its label is now 10n).
>
> Then at all times t<2, the balls in this vase have the same labels
> as those in the original problem's vase.
Here you display the same problem you have with the Yessenin-Volpin
story -- you are only interested in the final result of a process and
throwing away the actual process itself. The snapshots you take indeed
are the same but what happens between the snapshots is entirely
different. So we have another irony where the anti-cantorian claim is
that Cantor considered infinite sets as completed totalities and
ignored that infinity is a process, yet here you support the anti-
cantorian by only considering the completed totality and ignoring the
process.
FredJeffries
> And I, being a math/comp-sci type person, look at from a physical
> point of view.
>
> Eventually, as the number of balls gets sufficiently large,
> if the distance to move the balls into and out of the container is
> bounded below,
> the balls will have to move faster than the speed of light.
> This will provide a upper limits of the number of balls that can be
> moved
> in a amount of time.
>
> The amount of energy to move the balls might also be a consideration.
While in no way disputing your reasoning, perhaps you are being too
literal. Physics may indeed outlaw the supertasks. But physics has
found use for mathematical objects like self-adjoint operators on
infinite dimensional Hilbert spaces to represent observables. Some of
us may still hope that someday in some form the mathematics of the
transfinite may find some application, if not in physics then perhaps
in psychology, linguistics, ethics, aesthetics, ...
But, of course, all of that is mere wishful thinking on my part.
Tony Orlow
Hi Transfer et al -
Been taking a break from posting and just looking in occasionally, and
noticed this discussion on what I consider the most egregious example
of transfinitological "logic". I went through the thread, and I think
your understanding of the objections is fairly good, but there seems
to be an argument not yet mentioned. I guess I'll do that now.
Noon does not exist in the gedanken. Nothing happens at noon because
every ball is labeled with a finite number that denotes times of
insertion and removal for that ball which occur *before* noon. At
every moment before noon there are an increasing number of balls in
the vase, and at noon nothing can happen to the balls to make them
disappear. The Zeno Paradox Engine does not accurately represent any
supposed completion of N and nothing else, since no completion exists.
If N has been completed, indeed one must have generated nonstandard
numbers, or one would still be enumerating standard natural numbers.
If noon exists, then all the standard naturals are indeed gone, and
replaced with an uncountable number of nonstandard numbers.
But, noon doesn't exist, because all we have are standard regulation
natural ping pong balls.
Peace,
Tony
David R Tribble
> This is the point I'm trying to make here. Changing finite
> arithmetic so that it works like infinite arithmetic has dire
> consequences on the real world, including finances. On the
> other hand, changing infinite arithmetic so that it works like
> finite arithmetic has no effect on the real world, where for one
> thing, people are paid finite amounts of money.
Zimbabwe could learn a thing or two about that:
http://en.wikipedia.org/wiki/Zimbabwean_dollar#Hyperinflation
http://upload.wikimedia.org/wikipedia/en/3/3e/Zimbabwe_%24100_trillion_2009_Obverse.jpg
-drt
David R Tribble
> Noon does not exist in the gedanken. Nothing happens at noon because
> every ball is labeled with a finite number that denotes times of
> insertion and removal for that ball which occur *before* noon. At
> every moment before noon there are an increasing number of balls in
> the vase, and at noon nothing can happen to the balls to make them
> disappear. The Zeno Paradox Engine does not accurately represent any
> supposed completion of N and nothing else, since no completion exists.
> If N has been completed, indeed one must have generated nonstandard
> numbers, or one would still be enumerating standard natural numbers.
> If noon exists, then all the standard naturals are indeed gone, and
> replaced with an uncountable number of nonstandard numbers.
>
> But, noon doesn't exist, because all we have are standard regulation
> natural ping pong balls.
That's certainly one objection to the problem.
But it contradicts the original conditions of the problem,
which states that there is a time t=noon, so it fails to
answer the question.
-drt
Tony Orlow
Perhaps there is a time t=noon, however, none of the finitely numbered
balls moves at noon, and at every moment before noon there are balls
in the vase. Think carefully here. If there are always balls before
noon (and after one minute before noon), and nothing occurs *at* noon,
when do all the balls that were there a moment before noon leave the
vase? You have a self contradiction in your conclusion.
1. Balls are there.
2. Nothing happens.
3. The balls are gone.
What happens *at* noon to make things different than they were before
noon? Nothing.
Peace,
Tony
FredJeffries
> arithmetic so that it works like infinite arithmetic has dire
> consequences on the real world, including finances. On the
> other hand, changing infinite arithmetic so that it works like
> finite arithmetic has no effect on the real world, where for one
> thing, people are paid finite amounts of money.
>
You want to be a revolutionary as long as it has no effect on the real
world.
FredJeffries
>
> Perhaps there is a time t=noon, however, none of the finitely numbered
> balls moves at noon, and at every moment before noon there are balls
> in the vase. Think carefully here. If there are always balls before
> noon (and after one minute before noon), and nothing occurs *at* noon,
> when do all the balls that were there a moment before noon leave the
> vase? You have a self contradiction in your conclusion.
>
> 1. Balls are there.
> 2. Nothing happens.
> 3. The balls are gone.
>
> What happens *at* noon to make things different than they were before
> noon? Nothing.
>
Another victim of the stroboscopic effect which makes the spokes of a
wagon wheel go backwards and the blades of a fan stand still, so our
hero fearlessly sticks his head into the fan...
Virgil
<afaab674-e58b-4ec4...@a36g2000yqc.googlegroups.com>,
Tony Orlow <to...@lightlink.com> wrote:
What happens "at" noon is that finally all the balls are gone.
If you claim otherwise, point out a ball that is not gone!
David R Tribble
>> Noon does not exist in the gedanken. Nothing happens at noon because
>> every ball is labeled with a finite number that denotes times of
>> insertion and removal for that ball which occur *before* noon. At
>> every moment before noon there are an increasing number of balls in
>> the vase, and at noon nothing can happen to the balls to make them
>> disappear. The Zeno Paradox Engine does not accurately represent any
>> supposed completion of N and nothing else, since no completion exists.
>> If N has been completed, indeed one must have generated nonstandard
>> numbers, or one would still be enumerating standard natural numbers.
>> If noon exists, then all the standard naturals are indeed gone, and
>> replaced with an uncountable number of nonstandard numbers.
>> But, noon doesn't exist, because all we have are standard regulation
>> natural ping pong balls.
>
David R Tribble wrote:
>> That's certainly one objection to the problem.
>> But it contradicts the original conditions of the problem,
>> which states that there is a time t=noon, so it fails to
>> answer the question.
>
Tony Orlow wrote:
> Perhaps there is a time t=noon, however, none of the finitely numbered
> balls moves at noon, and at every moment before noon there are balls
> in the vase. Think carefully here. If there are always balls before
> noon (and after one minute before noon), and nothing occurs *at* noon,
> when do all the balls that were there a moment before noon leave the
> vase? You have a self contradiction in your conclusion.
> 1. Balls are there.
> 2. Nothing happens.
> 3. The balls are gone.
> What happens *at* noon to make things different than they were before
> noon? Nothing.
Ah, Tony, Tony, Tony.
You left out one small, but critical, step there. You forgot
the part where a ball is removed from the vase at every step.
So far be it from "nothing" happening before noon, actually
"quite a lot" of movement is going on before the clock
strikes noon.
To be sure, you are correct when you say nothing happens
*at* noon. But that's because everything that happens
has happened *before* noon. Especially the part about
each ball being removed from the vase. Before noon.
And, yes, you're right when you point out that all the action
takes place before noon. Each event (inserting balls, taking
them out, etc.) occurs at some finite time before noon.
What you're missing, though, is the basic premise of the
question, that *all* events take place by the time noon
arrives. An infinite number of insertions and removals, all
occurring at some finite time, and all of them transpiring
by the time noon arrives.
Consider the timing of the events:
1/2 + 1/4 + 1/8 + 1/16 + ...
Surely you agree that these intervals sum up to 1, right?
So when you add all of the intervals, an infinite number of
ever-decreasing but always non-zero intervals, they total 1.
Well, it's exactly the same with the balls in the vase.
When you have performed an infinite number of insertion and
removal steps, each one at some ever-decreasing but always
non-zero time from the previous one, you will have reached
noon. And every ball will have been removed from the vase.
Earie
> On Aug 26, 9:58 pm, Transfer Principle <lwal...@lausd.net> wrote:
>
> > we see that each ball has as its label a number
> > followed by infinitely many zeros. And voila -- there are your
> > nonstandard natural numbers!
>
> Of course. Perhaps I was momentarily fooled by your chosen sobriquet
> into nor realizing that you were referring to the nonstandard
> nonstandard natural numbers of the anti-cantorians wherewith they show
> themselves unable/unwilling to distinguish between cardinal and
> ordinal but also unable/unwilling to distinguish either of those from
> nonstandard natural numbers.
>
> Let us dip into the urn and pull out one of these balls with a number
> followed by infinitely many zeros. It says 1745 with infinitely many
> zeros following. Was this the ball that was originally numbered 1745?
> Or is it the one that was originally numbered 17450 or the one
> originally numbered 1745000 or...
>
inserted, as no ball with a zero-ending number was ever inserted.
> Thus we have the irony that the persons who complain about not being
> able to distinguish the cardinalities of the natural numbers and the
> even numbers have constructed their own system with (infinitely many)
> classes each having infinitely many members none of which can be
> distinguished from each other.
You have infinitely many sets, with one member each.
You can change the scheme to match your analisys of infinite*infinite,
but I don't see the point.
Earie
> On Aug 25, 9:18 am, FredJeffries <fredjeffr...@gmail.com> wrote:
>
> > On Aug 23, 3:24 pm, Transfer Principle <lwal...@lausd.net> wrote:
> > > Since, as Earie points out, all the balls with standard natural
> > > numbered labels are removed by time t=2, post-Cantorians conclude
> > > that the balls still in the vase have _nonstandard_ natural numbers
> > > on their labels.
> > Do they ever indicate when or how the balls with nonstandard natural
> > numbers on their labels get INTO the urn?
>
> I believe in one of the old ball and vase threads from last year,
> someone posed the following equivalent problem:
>
> Suppose at time 2-2^-(n-1), we add balls 10n-9 to 10n-1 (and not
> 10n) to another vase, and then take ball n and add a zero to the
> right end of its label (so that its label is now 10n).
>
Here you end up with 9*aleph0=aleph0 balls, while in the standart
problem you end up with 0 balls. See below.
> Then at all times t<2, the balls in this vase have the same labels
> as those in the original problem's vase. But at time t=2, standard
> theory states that the original vase is empty, but the new vase
> can't possibly be empty -- since at no time t is any ball removed
> at all! Instead, we see that each ball has as its label a number
> followed by infinitely many zeros. And voila -- there are your
> nonstandard natural numbers!
>
In the standart problem, each ball has a natural number on it, as each
ball, when inserted, had a natural number, and the numbers were never
changed. For each number, we can show the moment in which its ball was
taken out, so at t=2 the vase in empty.
In your problem, you are left with aleph0 balls. The first's number is
'1' followed by aleph0 zeroes, the second's number is '2' followed by
aleph0 zeroes etc. So arithmetically, the first's number is equivalent
to 10^aleph0 which is more than aleph0 (it's the continuum - C). That
is, no ball is left which has a natural number (you insist saying that
they have nonstandart natural numbers, but evidently 10^aleph0 isn't a
natural number, standart or nonstandart. There're exactly aleph0
natural numbers (by definition), and since 10^aleph0>aleph0, it
follows what 10^aleph0 isn't natural).
So if we look at the natural numbers, both problems are euqivalent, as
each moment a natural number is deleted, and so no natural number is
left at t=2. In the standart vase-balls problem, natural numbers
correspond to balls, so no balls are left. However in your problem,
balls don't necessarily correspond to natural numbers, so that
(aleph0) balls are left although no natural number is left.
So, Jeffries' question remains for you to answer: in the standart
balls-vase problem, who inserted the Continuum numbered balls (or the
nonstandart naturals, if you *still* insist)? and when??? recall that
each inserted ball has a natural number, and these are not allowed to
be changed. unlike your problem.
Tony Orlow
Yes, there is a lot of movement before noon, and with each of those
steps occurring before noon, nine more balls are in the vase. There
are always balls in the vase at every moment before noon, nothing
happens at noon, and you believe that "by" noon, all balls have
disappeared? Do you not see the absurdity of such a conclusion? At
what moment do the balls disappear? Not before noon, and not at noon.
When does your magic occur? If you can answer that question, you win.
If not, concede.
Smiles,
Tony
Jesse F. Hughes
> So if we look at the natural numbers, both problems are euqivalent, as
> each moment a natural number is deleted, and so no natural number is
> left at t=2. In the standart vase-balls problem, natural numbers
> correspond to balls, so no balls are left. However in your problem,
> balls don't necessarily correspond to natural numbers, so that
> (aleph0) balls are left although no natural number is left.
No, there's still a difference.
In the latter situation, at each of our critical moments, each ball has
a natural number on it, which is replaced by another natural number.
There is no time prior to noon at which any ball has an infinite string
of digits, though at noon, they all have an infinitely long label.
--
Jesse F. Hughes
"I post for many reasons [...] and there's no reason to think that
I'll stop." -- James S. Harris
Jesse F. Hughes
> Yes, there is a lot of movement before noon, and with each of those
> steps occurring before noon, nine more balls are in the vase. There
> are always balls in the vase at every moment before noon, nothing
> happens at noon, and you believe that "by" noon, all balls have
> disappeared? Do you not see the absurdity of such a conclusion? At
> what moment do the balls disappear? Not before noon, and not at noon.
> When does your magic occur? If you can answer that question, you win.
> If not, concede.
There is no point prior to noon at which the vase is empty, but it is
empty at noon. Hence, the magic occurs at noon (the limit).
It's not a very satisfying answer, but it is the answer that the
mathematics gives. If you fool around with ridiculously
counter-intuitive situations, then sometimes you get ridiculously
counter-intuitive results.
--
Jesse F. Hughes
"Casting [Demi] Moore as a woman who has come to the New World so that
she can 'worship without fear or persecution' in _The_Scarlet_Letter_
is like casting Bruce Willis as Young Rene Descartes." -Joe Queenan
David R Tribble
>> Perhaps there is a time t=noon, however, none of the finitely numbered
>> balls moves at noon, and at every moment before noon there are balls
>> in the vase. Think carefully here. If there are always balls before
>> noon (and after one minute before noon), and nothing occurs *at* noon,
>> when do all the balls that were there a moment before noon leave the
>> vase? You have a self contradiction in your conclusion.
>> 1. Balls are there.
>> 2. Nothing happens.
>> 3. The balls are gone.
>> What happens *at* noon to make things different than they were before
>> noon? Nothing.
>
David R Tribble wrote:
>> You left out one small, but critical, step there. You forgot
>> the part where a ball is removed from the vase at every step.
>> So far be it from "nothing" happening before noon, actually
>> "quite a lot" of movement is going on before the clock
>> strikes noon.
>>
>> To be sure, you are correct when you say nothing happens
>> *at* noon. But that's because everything that happens
>> has happened *before* noon. Especially the part about
>> each ball being removed from the vase. Before noon.
>>
>> And, yes, you're right when you point out that all the action
>> takes place before noon. Each event (inserting balls, taking
>> them out, etc.) occurs at some finite time before noon.
>> What you're missing, though, is the basic premise of the
>> question, that *all* events take place by the time noon
>> arrives. An infinite number of insertions and removals, all
>> occurring at some finite time, and all of them transpiring
>> by the time noon arrives.
>
Tony Orlow wrote:
> Yes, there is a lot of movement before noon, and with each of those
> steps occurring before noon, nine more balls are in the vase. There
> are always balls in the vase at every moment before noon, nothing
> happens at noon, and you believe that "by" noon, all balls have
> disappeared? Do you not see the absurdity of such a conclusion?
Yes, it does seem counterintuitive, almost absurd. But then
that's typical of the results you get when dealing with infinities.
Recall that Galileo recognized the apparent disconnect between
finite and infinite sets, when he described the bijection between
naturals and their squares. He concluded, correctly, that the rules
governing finite numbers simply don't apply to infinite numbers
(or infinite sets).
I know you're a big proponent of trying to make the rules of finite
arithmetic work with infinities, but we've know for quite some
time now that they do not. Infinite entities really are different
from finite entities.
> At what moment do the balls disappear?
> Not before noon, and not at noon.
The description of the problem states quite clearly when each
ball is removed. But they are not removed all at once, though.
Ball 1 disappears (is removed) at step 1, time 1/2.
Ball 2 is removed at step 2, time 1/2+1/4.
Ball 3 is removed at step 3, time 1/2+1/4+1/8.
And so on.
My question to you is, why do you think that any ball is
not removed from the vase? At what moment is any ball,
any ball at all, left in the vase? Please identify just one
single ball that is not removed, or equivalently, identify one
single moment when a ball is not removed from the vase,
and then we'll start to understand your objections.
> When does your magic occur? If you can answer that question, you win.
> If not, concede.
The magic occurs at each step, like I said.
But it does not occur all at once, which you seem to be
implying. Do you think "the magic" occurs all at once, at some
special moment?
Consider this next question, which will help you find the
answer you seek:
At what point in
s = 1/2 + 1/4 + 1/8 + 1/6 + ...
does s = 1?
-drt
Jesse F. Hughes
>> When does your magic occur? If you can answer that question, you win.
>> If not, concede.
>
> The magic occurs at each step, like I said.
>
> But it does not occur all at once, which you seem to be
> implying. Do you think "the magic" occurs all at once, at some
> special moment?
Obviously, David and I gave contradictory answers here.
If "the magic" is "the vase is empty" or "the moment the number of balls
decrease", then my answer is correct -- it happens at noon and not
before.
If "the magic" is that balls are being removed, then this happens at
each step.
--
Scientists have calculated that the chance of anything so patently
absurd actually existing are millions to one. But magicians have
calculated that million-to-one chances crop up nine times out of ten.
-- Terry Pratchett on Intelligent Design. Or something.
Transfer Principle
After seeing this and several subsequent posts in this thread,
perhaps the problem is my use of the phrase "nonstandard
natural number."
If the phrase "nonstandard natural number" means something like
"hypernatural" (i.e., the analog of the naturals in Robinson's
hyperreals), then someone already pointed out in another of the
many ball and vase threads earlier that the nonstandard natural
numbered balls are entered at an infinitesimal time before the
critical time -- but then those balls are removed an even
smaller infinitesimal time before the critical time, so that at
the critical time itself the vase is still empty.
Once again, the answer to this question is often dependent on
the poster whose belief I'm representing. Some posters (most
notably RF, though I'm not sure whether he's participated in
ball and vase threads or not) belief in the existence of a
smallest positive infinitesimal, iota. Since it's the smallest
infinitesimal, the balls entered at time iota before critical
time can't be removed, since there is no smaller infinitesimal
time before critical time. Thus, those balls would still be in
the vase at noon.
Other posters (and unfortunately, I forgot who first brought up
this idea in the first place) rely on the equivalence of
removing balls and adding zeros to their labels, as I mentioned
in an earlier post. Then the answer would be that when they are
entered, they have standard natural numbered labels, but at the
critical time, infinitely many zeros have been added to them.
But then someone might point out that one can hardly call
labels with infinitely many zeros "nonstandard natural numbers"
since even the post-Cantorians believe that one should be able
to add one to a natural number -- but how can add one to
something with infinitely many zeros?
So perhaps it would be better if I didn't use the phrase
"nonstandard natural number" to describe the labels. To the
post-Cantorians, all that matters is that there are infinitely
many balls in the vase. It doesn't matter what the labels on
those balls are -- but if they're convinced that all the balls
with standard natural numbers have been removed, then something
other than standard natural numbers must be on the balls
(whether it's infinitely many zeros, or labels like "infinity,"
"infinity-1," "infinitely-2," etc.).
Transfer Principle
Even with hyperinflation, the amounts of money involved are still
less than AP's upper bound of 10^500, much less googolplex or
Graham's number. Perhaps I'll reconsider my position if there's
a hyperinflationary situation involving googolplex dollars.
And the amounts of money are still very much finite. One can't go
to a bar, buy drinks for all the money in one's pocket, K dollars,
and still have K dollars in one's pocket as Jeffries describes,
not even in Zimbabwe.
Transfer Principle
Because the instant that someone proposes a theory that does have
an effect on the real world, another poster will use that effect
as a reason to oppose the revolutionary theory. Indeed, this is
exactly how RSA numbers were first brought up in these threads --
because someone proposed a revolutionary theory will would affect
the RSA numbers, and so someone else opposed the theory on the
grounds that it would ruin cryptography.
But if Jeffries insists on having this "anti-Dirichletian" theory,
then one can attempt to see how to axiomatize it.
So we want finite sets to act like infinite sets. Presumably, the
infinite sets would be unaffected, but we note that an infinite
set like omega contains finite elements -- and so it would be
altered in the new theory as well. (For one thing, Jeffries
mentions nonstandard finite ordinals, and so these nonstandard
finite ordinals must be elements of omega, being the set of all
finite ordinals, whether standard or nonstandard.)
And of course, the sum or product of our anti-Dirichletian finite
cardinals must equal their max.
In that case, a good place to start is 0 and 1. We notice that in
standard theory, the sum of 0 and any nonempty cardinal is that
cardinal -- meaning that 0 is already anti-Dirichletian as far as
addition is concerned! So we only have to find a way to make 0
anti-Dirichetian wrt multiplication. Similarly, 1 is already
anti-Dirichetian wrt multiplication since the product of 1 and a
larger cardinal is the larger cardinal, so we only need to make 1
anti-Dirichetian wrt addition.
But this is difficult. We suddenly want the cardinal product of 0
and a nonempty cardinal to be nonempty -- and recall that this
product is based on the Cartesian product. How can the Cartesian
product of an empty set and a nonempty set be nonempty? Thus is
the anti-Dirichletian's challenge.
Perhaps a better tactic would be simply to write an axiom which
declares by fiat that all sets are anti-Dirichetian. We then add
this axiom to as many axioms of ZFC as possible without there
being an inconsistency, then see what turns up.
A few more thoughts about an anti-Dirichletian theory:
Jeffries states that in an anti-Dirichetian theory, there exist
nonstandard finite wellorder types. But given a finite cardinal,
how many wellorder types should there be?
Well, since finite sets are supposed to act like infinite sets,
we should look at the infinite cardinals and ordinals. We know
that the countably infinite cardinal omega has uncountably many
possible wellorder types -- indeed, there are exactly aleph_1,
the successor cardinal of aleph_0, wellorder types. Indeed, we
know that for an infinite cardinal kappa, the number of possible
wellorder types is the successor cardinal kappa+.
So we extend this to finite sets as well. Thus, the number of
order types on the empty set 0 should be 0+ or 1. This, of
course, already matches standard theory. But now the number of
wellorders on the singleton 1 should be 1+ or 2. The challenge
once again is on the anti-Dirchletians to find the second
wellorder on 1.
Transfer Principle
> Tony Orlow wrote:
> > Yes, there is a lot of movement before noon, and with each of those
> > steps occurring before noon, nine more balls are in the vase. There
> > are always balls in the vase at every moment before noon, nothing
> > happens at noon, and you believe that "by" noon, all balls have
> > disappeared? Do you not see the absurdity of such a conclusion?
> arithmetic work with infinities, but we've know for quite some
> time now that they do not.
...in ZFC, that is. But that doesn't exclude the possibility of
a theory other than ZFC in which the rules of finite arithmetic
really do work with infinities, a theory which would suit TO
better than ZFC.
In these TO threads, as I continue to think about how to come
up with a suitable axiomatization for TO, I consider some of
the differences between ZFC and TO.
In ZFC, the primitive is "e," elementhood or membership. Thus,
the question that ZFC users consider is, for any natural number
n, is ball n an element or member of the set of balls inside
the vase at noon? If not, then they conclude that the vase must
be empty.
But to TO, membership isn't primitive. The primitive concept
for TO appears to be cardinality or Bigulosity -- in other
words, set size. Thus, the question that TO asks is, for any
time t, is the size of the set of balls inside the vase
increasing at all times up to noon? If so, then he concludes
that the vase must be full at noon.
I was searching for some old ball and vase threads, and I found
one where the late Dik Winter mentions that what is needed is
some sort of "limit" of sets:
{2,3,...,9,10}
(3,4,...,19,20}
{4,5,...,29,30}
...
What is the limit here? Well, in order to define a limit, there
needs to be some sort of topology, and the relevant topology
that's given by most ZFC users is the cofinite topology.
This tells us that if {a_1, ..., a_t, ...} is a sequence of
sets, then the limit a is given as follows -- a natural number
n is an element of a iff it's an element of a_t for cofinitely
many values of t. Now for the sequence above, each natural
number is in only finitely many sets a_t. Thus, the limit of
the sets must be 0 and the vase is empty.
(Of course, not every sequence of sets converges. The usual
example of an oscillating sequence would be to put ball 1 in
the vase, then remove it, then add it again, then remove it,
etc., infinitely many times before noon. Then the cofinite
topology can't tell us where ball 1 is at noon. This is the
Thomson's Lamp Paradox.)
But according to TO, the limit of this sequence must preserve
set size/Bigulosity (i.e., the size of the limit must equal
the limit of the sizes), and so 0 is unacceptable as a limit
of this sequence of sets. Thus, TO would reject the cofinite
topology in favor of some as yet unknown topology.
But so far, I haven't come up with the correct axioms that
will describe TO's theory. We want an axiom which tells us
that limits preserve set size.
What if we were to simplify the problem? Let's say we had the
following scenario:
11:59:00 -- Put ball 1 in the vase.
11:59:30 -- Remove ball 1 and put ball 2 in the vase.
11:59:45 -- Remove ball 2 and put ball 3 in the vase.
11:59:52.5 -- Remove ball 3 and put ball 4 in the vase.
...
Now once again, ZFC will say that the vase is empty at noon,
but now TO will say that the vase has exactly 1 ball. We can
then define the label on this ball to be Big'Un, or tav, or
zillion, or whatever label TO wants to give it, and then we
proceed from there.
David R Tribble
>> When does your magic occur? If you can answer that question, you win.
>> If not, concede.
>
Jesse F. Hughes wrote:
>> There is no point prior to noon at which the vase is empty, but it is
>> empty at noon. Hence, the magic occurs at noon (the limit).
>
David R Tribble writes:
>> The magic occurs at each step, like I said.
>> But it does not occur all at once, which you seem to be implying.
>> Do you think "the magic" occurs all at once, at some special moment?
>
Jesse F. Hughes wrote:
> Obviously, David and I gave contradictory answers here.
Not surprising, since Tony is asking about when the "magic"
occurs.
There is no magic, just logic.
Like you said, we're both right. Depends on what, exactly,
Tony means by "magic".
David R Tribble
>> This is the point I'm trying to make here. Changing finite
>> arithmetic so that it works like infinite arithmetic has dire
>> consequences on the real world, including finances. On the
>> other hand, changing infinite arithmetic so that it works like
>> finite arithmetic has no effect on the real world, where for one
>> thing, people are paid finite amounts of money.
>
David R Tribble wrote:
>> Zimbabwe could learn a thing or two about that:
>> http://en.wikipedia.org/wiki/Zimbabwean_dollar#Hyperinflation
>> http://upload.wikimedia.org/wikipedia/en/3/3e/Zimbabwe_%24100_trillio...
>
Transfer Principle (L Walker) wrote:
> Even with hyperinflation, the amounts of money involved are still
> less than AP's upper bound of 10^500, [...]
Read the Wikipedia article more closely. It says:
| By December 2008, inflation was estimated at 6.5 quindecillion
| novemdecillion percent (65 followed by 107 zeros).
At an annual inflation rate of 65 x 10^105, it would take
only a few short years to exceed 10^500 Z dollars.
But I'll be the one to say it:
AP is a crank, and his definition of 10^500 as an infinite
number is a crank idea, just like all of his other ideas.
David R Tribble
>> Yes, there is a lot of movement before noon, and with each of those
>> steps occurring before noon, nine more balls are in the vase. There
>> are always balls in the vase at every moment before noon, nothing
>> happens at noon, and you believe that "by" noon, all balls have
>> disappeared? Do you not see the absurdity of such a conclusion?
>
David R Tribble wrote:
>> I know you're a big proponent of trying to make the rules of finite
>> arithmetic work with infinities, but we've know for quite some
>> time now that they do not.
>
Transfer Principle (L Walker) wrote:
> ...in ZFC, that is. But that doesn't exclude the possibility of
> a theory other than ZFC in which the rules of finite arithmetic
> really do work with infinities, a theory which would suit TO
> better than ZFC.
It would, but that's not what he's looking for.
He wants his "unit infinities" and "infinite arithmetic" to
work on the normally accepted "real number line", i.e.,
within standard arithmetic.
Go back and read his extensive posts on the matter if
this isn't clear to you.
-drt
David R Tribble
> following scenario:
> 11:59:00 -- Put ball 1 in the vase.
> 11:59:30 -- Remove ball 1 and put ball 2 in the vase.
> 11:59:45 -- Remove ball 2 and put ball 3 in the vase.
> 11:59:52.5 -- Remove ball 3 and put ball 4 in the vase.
> ...
> Now once again, ZFC will say that the vase is empty at noon,
> but now TO will say that the vase has exactly 1 ball. We can
> then define the label on this ball to be Big'Un, or tav, or
> zillion, or whatever label TO wants to give it, and then we
> proceed from there.
Why not simplify it even further? Thus:
a. Begin with a vase containing an infinite number of balls,
each one labeled with a unique natural number.
b. At each step, remove the ball with the least natural number.
c. An infinite number of steps are executed by noon.
We conclude that at each finite step before noon, an infinite
number of balls remain in the vase. But at noon, the vase
is empty (because every ball has been removed by noon).
This simplification of the problem still has the "magic"
aspect that Tony doesn't quite comprehend.
-drt
David R Tribble
> What if we were to simplify the problem? Let's say we had the
> following scenario:
> 11:59:00 -- Put ball 1 in the vase.
> 11:59:30 -- Remove ball 1 and put ball 2 in the vase.
> 11:59:45 -- Remove ball 2 and put ball 3 in the vase.
> 11:59:52.5 -- Remove ball 3 and put ball 4 in the vase.
> ...
> Now once again, ZFC will say that the vase is empty at noon,
> but now TO will say that the vase has exactly 1 ball. We can
> then define the label on this ball to be Big'Un, or tav, or
> zillion, or whatever label TO wants to give it, and then we
> proceed from there.
The next step is to then answer the question:
What is the label on the last ball that was removed?
Tim Little
> The next step is to then answer the question:
> What is the label on the last ball that was removed?
Big'Un - 1, duh.
- Tim
Earie
Please clarify, as this last paragraph of yours seems to be reverse
logic, that is, simply begging the question.
We seem to agree that no balls with standart natural numbers will be
in the vase at noon. Now, I know of no "nonstandart" numbers (or
infinity-1 or however you choose to call them), thus I conclude that
the vase is simply empty. You *assume* that the vase is nonempty, and
thus conclucde the existance of such numbers. But clearly, you should
supply a better proof of their existance than simply saying "since the
vase is full, these numbers exist".
You further (in other posts) seem to derive the "full vase" conclusion
from the *assumtion* that the size of the limit set is the limit of
the set sizes. But yet again, that is the very thing we question - I
take my logic (that shows an empty vase) to refute this assumtion. So
so far you've only begged the question.
FredJeffries
>
> Because the instant that someone proposes a theory that does have
> an effect on the real world, another poster will use that effect
> as a reason to oppose the revolutionary theory. Indeed, this is
> exactly how RSA numbers were first brought up in these threads --
> because someone proposed a revolutionary theory will would affect
> the RSA numbers, and so someone else opposed the theory on the
> grounds that it would ruin cryptography.
>
YES! It's all a conspiracy by the rich who have invested in RSA, who
realize that the merest hint of finitism would open their vaults to
plunder by the poor and needy.
FredJeffries
> then one can attempt to see how to axiomatize it.
>
> So we want finite sets to act like infinite sets.
Not necessarily. We could just do away with finite sets altogether. I
would imagine that just as you speak finitists and ultra-finitists and
finitists-who-allow-potential-but-not-actual-infinity, we could have
infinitists, ultra-infinitists and potentially finite sets.
After all, if finite sets have exactly the same properties as infinite
sets, what's the point of the adjectives? How would one even know
which adjective to apply to which set anyhow?
Notice that in the "An anti-Cantorian and an anti-Dirichletian walked
into a bar..." no mention was made as to whether the K dollars was
finite or infinite.
> Presumably, the
> infinite sets would be unaffected, but we note that an infinite
> set like omega contains finite elements
Only if you take too seriously the myth that zero IS the empty set, 1
IS the set consisting of the empty set, omega IS the set of finite
ordinals...
If one takes, say, a category theory approach, we need only consider
the necessary properties of omega and its relationships.
> -- and so it would be
> altered in the new theory as well. (For one thing, Jeffries
> mentions nonstandard finite ordinals, and so these nonstandard
> finite ordinals must be elements of omega, being the set of all
> finite ordinals, whether standard or nonstandard.)
>
> And of course, the sum or product of our anti-Dirichletian finite
> cardinals must equal their max.
>
> In that case, a good place to start is 0 and 1.
You haven't yet proved that 0 and 1 exist, but for the sake of
argument let us continue.
> We notice that in
> standard theory, the sum of 0 and any nonempty cardinal is that
> cardinal -- meaning that 0 is already anti-Dirichletian as far as
> addition is concerned! So we only have to find a way to make 0
> anti-Dirichetian wrt multiplication. Similarly, 1 is already
> anti-Dirichetian wrt multiplication since the product of 1 and a
> larger cardinal is the larger cardinal, so we only need to make 1
> anti-Dirichetian wrt addition.
>
> But this is difficult. We suddenly want the cardinal product of 0
> and a nonempty cardinal to be nonempty -- and recall that this
> product is based on the Cartesian product. How can the Cartesian
> product of an empty set and a nonempty set be nonempty? Thus is
> the anti-Dirichletian's challenge.
But you're forgetting about the non-standard infinitesimal finite
ordinals -- those between zero and one.
>
> Perhaps a better tactic would be simply to write an axiom which
> declares by fiat that all sets are anti-Dirichetian. We then add
> this axiom to as many axioms of ZFC as possible without there
> being an inconsistency, then see what turns up.
To ZFC!? What is this fixation with ZFC?
The axiom of extensionality was proven wrong in the 1960's by the
Phylis Diller theorem: I've got everything Raquel Welch has, just
arranged differently.
>
> A few more thoughts about an anti-Dirichletian theory:
>
> Jeffries states that in an anti-Dirichetian theory, there exist
> nonstandard finite wellorder types. But given a finite cardinal,
> how many wellorder types should there be?
>
> Well, since finite sets are supposed to act like infinite sets,
> we should look at the infinite cardinals and ordinals. We know
> that the countably infinite cardinal omega has uncountably many
> possible wellorder types -- indeed, there are exactly aleph_1,
> the successor cardinal of aleph_0, wellorder types. Indeed, we
> know that for an infinite cardinal kappa, the number of possible
> wellorder types is the successor cardinal kappa+.
>
> So we extend this to finite sets as well. Thus, the number of
> order types on the empty set 0 should be 0+ or 1. This, of
> course, already matches standard theory. But now the number of
> wellorders on the singleton 1 should be 1+ or 2. The challenge
> once again is on the anti-Dirchletians to find the second
> wellorder on 1.
Your use of the phrase "THE second" (my emphasis) already begs the
question. Even if there objects like 0, 1 and 2 they would ex
hypothesi not have the properties you are so attached to.
FredJeffries
> On 28 אוגוסט, 09:50, FredJeffries <fredjeffr...@gmail.com> wrote:
>
> > On Aug 26, 9:58 pm, Transfer Principle <lwal...@lausd.net> wrote:
>
> > > we see that each ball has as its label a number
> > > followed by infinitely many zeros. And voila -- there are your
> > > nonstandard natural numbers!
>
> > followed by infinitely many zeros. It says 1745 with infinitely many
> > zeros following. Was this the ball that was originally numbered 1745?
> > Or is it the one that was originally numbered 17450 or the one
> > originally numbered 1745000 or...
>
> Not really. In the scheme Transfer described, 17450 was never
> inserted, as no ball with a zero-ending number was ever inserted.
>
You are correct, I am wrong here.
Hmm. Infinitely long strings with no multiples of 10. That ought to be
fuel for the (.999...!=1)'ists
Earie
>
> Hmm. Infinitely long strings with no multiples of 10. That ought to be
> fuel for the (.999...!=1)'ists
>
doesn't beg the question. Limit simply *isn't* sum, and if you can
prove otherwise without begging the question, please let me know.
For example, one rgument goes this way: a=0.9999 10a=9.9999 9a=10a-
a=9 => a=1
This is wrong.
Let's just note that we can't write sum{1,inf) or so - we can only
write things like lim(T->inf)sum{1,T}. Now let's repeat the argument:
a=0.99999=lim{T->inf}sum{1,T}{9*10^-i}
10a=9*lim{T->inf}sum{0,T-1}{10^-i}
9a=10a-a=9*lim{T->inf}{1-10^-T}
Now to argue that this really equals 9 is to argue that this
infinitesimal really equals 0, which is begging the question. You
prove that the limit equals the sum (0.9999=1) by assuming that the
limit equals the sum (lim{T->inf}{10^-T}=0). Likewise all such proofs
beg the question, I think.
Virgil
<d74e3b1c-3d64-4cff...@s15g2000yqm.googlegroups.com>,
Earie <par...@yahoo.com> wrote:
> On 31 אוגוסט, 17:57, FredJeffries <fredjeffr...@gmail.com> wrote:
> >
> > Hmm. Infinitely long strings with no multiples of 10. That ought to be
> > fuel for the (.999...!=1)'ists
> >
> Of course 0.999...!=1. I never saw an argument of their equality that
> doesn't beg the question. Limit simply *isn't* sum, and if you can
> prove otherwise without begging the question, please let me know.
How about an argument for their inequality?
If, as you claim, 0.999...!=1 then, presumably, 0.999...< 1.
Let A = 0.999.... then one has A < (A+1)/2 < 1
And letting B = (A+1)/2, one has A < (A+B)/2 < B < (B+1)/2 < 1
And so on, ad infinitum.
So that one finds that there are infinitely many numbers between
0.999... and 1, all of which must have some sort of decimal expansions,
or at least decimal approximations, but for which no decimal expansions
or even reasonable decimal approximations can possibly exist.
Or if assuming 0.999... to be less than 1 produces nonsense, and if it
is still not equal to 1, it must be greater than one.
Then 0.999... - 1 must have a positive decimal representation, or at
least positive decimal approximations.
Which I challenge Earie to find.
Tim Little
> You prove that the limit equals the sum (0.9999=1) by assuming that
> the limit equals the sum (lim{T->inf}{10^-T}=0).
It doesn't need to be assumed, it is easily proved. Look up the
mathematical definition of limit. Here, I'll help you with a handy
link:
http://en.wikipedia.org/wiki/Limit_(mathematics)#Limit_of_a_sequence
Next time you use a mathematical term, find out what it means first.
- Tim
FredJeffries
You just don't understand, Tim. Using a mathematical definition is
begging the question.
Earie
> In article
> <d74e3b1c-3d64-4cff-97c5-35059732f...@s15g2000yqm.googlegroups.com>,
>
> Earie <part...@yahoo.com> wrote:
> > On 31 אוגוסט, 17:57, FredJeffries <fredjeffr...@gmail.com> wrote:
>
> > > Hmm. Infinitely long strings with no multiples of 10. That ought to be
> > > fuel for the (.999...!=1)'ists
>
> > Of course 0.999...!=1. I never saw an argument of their equality that
> > doesn't beg the question. Limit simply *isn't* sum, and if you can
Okay, I grant that you gave a 'proof' that doesn't beg the question.
Yet, you'd better try a true proof....
> > prove otherwise without begging the question, please let me know.
>
> How about an argument for their inequality?
>
Well, that's obvious. I said (and you seem to agree, as you didn't
object) that 0.999...=1-e, e being an infinitesimal. Declaring it
equal to one simply means to declare that e=0, thus nullifying the
whole Calculus. You know, that was the problem in the first versions
of Calculus (mainly Newton's), which led Cauchy to define the 'limit'
concept, to avoid exactly this kind of things. A limit is *not* a sum
(in case of sequences) or a value (in case of functions). I may
elaborate on this in my reply to Tim, of I have the time.
> If, as you claim, 0.999...!=1 then, presumably, 0.999...< 1.
> Let A = 0.999.... then one has A < (A+1)/2 < 1
> And letting B = (A+1)/2, one has A < (A+B)/2 < B < (B+1)/2 < 1
> And so on, ad infinitum.
>
0.999... is not a number - it's a sum of an infinite series. It may be
better written as lim{T->inf}sum{1,T}{10^-i}. So, let's define it as
A, and the (A,1) average as B, and look what happens when the T gets
larger and larger:
A B
0.9 0.95
0.99 0.995
0.999 0.9995
0.9999 0.99995
0.999.... 0.999....95
So yes, B exists and different from both A and 1. But to write it
expelicitely I have to write an infinite number of 9's. Sorry I don't
have the time for that now.
(that's one answer to your 'proof').
> So that one finds that there are infinitely many numbers between
> 0.999... and 1, all of which must have some sort of decimal expansions,
Now that's very wrong, and that's a second answer to your 'proof'.
*Not* every real number has a decimal expansion. At least not a usable
one. See, when I write 0.999...., you really understand what I mean,
and can tell me immediately what the n'th digit is, and everything
else on this number (even if it is a number - actually it's only an
infinite sum). But that's only because it's a very simple number to
consider. Consider PI instead. The expansion 3.14... doesn't really
tell everything. You can't tell me what the 10^12'th digit is, no
matter how many digits I supply (technically, I can't get anywhere
near supplying 10^12 digits). However PI is a known number, and
formulas exist for calculating its n'th digit. But most real numbers
aren't. Just throw a zero-head-sized arrow on the [0,1] line, and
chances are you'd hit a number for which no '...' representation can
be written, that is, a number whose digits don't have any apparent
pattern.
So your expectation that every real number will have a '...'
representation is simply wrong. Indeed there're infinite number of
numbers between 0.999.... and 1 (or between any two different real
numbers), not for all of whom a decimal expansion exists.
Earie
> On 2010-09-01, Earie <part...@yahoo.com> wrote:
>
> > You prove that the limit equals the sum (0.9999=1) by assuming that
> > the limit equals the sum (lim{T->inf}{10^-T}=0).
>
> It doesn't need to be assumed, it is easily proved. Look up the
> mathematical definition of limit. Here, I'll help you with a handy
> link:
>
> http://en.wikipedia.org/wiki/Limit_(mathematics)#Limit_of_a_sequence
>
it would have been wrong. Limit simply *isn't* sum. Read that
definition of wikipedia again, carefully, and this time pay attention.
Actually, the whole notion of a a 'limit' was invented exactly for
that - to allow managing infinite series (and infinitesimal objects)
without pretending any knowledge on their sum, or value, as that leads
to plain paradoxes.
Here, I'll give you some history (and basic math knowledge). The first
Calculus, Newton's, looked something like that: he defined the
derivative as the slope of a function, or its rate of change in a
specific point, or any such intuitive definition, and calculated it as
follows. Suppose we want to calculate the derivative of the function
f(x)=x^2. We define an infinitesimal dx, and
f(x+dx) = (x+dx)^2
{f(x+dx)-f(x)}/dx = {2xdx+dx^2}/dx = 2x+dx = 2x
So the derivative is 2x. This analysis was citicized by many (e.g.
Berkeley) to be self-contradictory, as in {2xdx+dx^2}/dx we
explicitely state that dx!=0 (otherwise the derivative is
meaningless), but in 2x+dx=2x we explicitely declare that dx=0. This
points out a very basic difficulty in the notion of 'derivative' - the
'slope of a function', or 'rate of change', has meaning *only* over
intervals, and is meaningless at a specific point. Yet the
'derivative' notion wants to apply this idea to a point, not an
interval. And that's an inherent contradiction.
This contradiction bothered Caushy, among others, and so he has
developed his own version of Calculus, which is the standart today. He
introduced the notion of a limit (and that's a perfect time to re-read
its definition. You'll note that it speaks *only* on intervals, and
never on a point), and so the derivative is defined as a limit: lim(dx-
>0){{f(x+dx)-f(x)}/dx}. Note now that the derivative *isn't* defined
on a point - we say that the derivative approaches a certain value as
we approach a certain point, but it may never really get to this
value, as we never really get to this point.
Generally, the limit of a function isn't connected to its value on
this point. Consider this funciton:
f(x)= x^2 if x!=4, and 123132 if x=4
The limit at x=4 will be 4^2=16, which isn't even close to the value.
That's what limit is all about. Now of course you will remind me that
the limit equals the value for continuous functions, and I will remind
you that it's the very *definition* of continuous functions - a
continuous function is a function for which the limit equals the value
at each point. And our discussion (concerning the 0.999...) has
nothing to do with it, unless you prove it all acts continuously.
Which is the very thing I question.
> Next time you use a mathematical term, find out what it means first.
>
2u2. Hope I've helped you here.
> - Tim
Virgil
<a3ecd25b-6fad-4742...@q2g2000yqq.googlegroups.com>,
Earie <par...@yahoo.com> wrote:
> Well, that's obvious. I said (and you seem to agree, as you didn't
> object) that 0.999...=1-e, e being an infinitesimal.
It has been proved that the set of infinite decimals, without any
infinite sequences of nines, is isomorphic as a complete ordered field,
to the Dedekind cut complete ordered field and the equivalence classes
of Cauchy sequences complete ordered field, neither of which requires or
even allows any infinitesimals.
Jesse F. Hughes
> Well, that's obvious. I said (and you seem to agree, as you didn't
> object) that 0.999...=1-e, e being an infinitesimal. Declaring it
> equal to one simply means to declare that e=0, thus nullifying the
> whole Calculus.
Don't you find it remarkable that every working mathematician fails to
notice this amazing consequence?
How is it that you understand that 0.999... < 1, while all them high 'n
mighty "professors" seem to miss it?
Note: this is not a counterargument to your claim. I'm just curious how
you reconcile the fact that you see the truth and no trained
mathematician sees it.
--
"There are some dark forces among you though[...] They are, quite
simply, anti-mathematicians. They pretend to be mathematicians, but
show their true colors when discovery is about, as their role is to
block human progress!!!" -- James Harris on the beast of the number.
Transfer Principle
> Transfer Principle (L Walker) wrote:
> > Even with hyperinflation, the amounts of money involved are still
> > less than AP's upper bound of 10^500, [...]
> only a few short years to exceed 10^500 Z dollars.
Thus, the hyperinflation as described by Tribble does involve
the number googol, but not googol_plex_. He'd be hard pressed
to find any economic situation involving googolplex.
If one wants to stay away from AP's upper bound of 10^500,
then there are people other than AP with other bounds. For
example, the ultrafinitist Doron Zeilberger proves that some
large prime number p be chosen (prime, presumably so that we
replace the field R with another field, Z/pZ). We might as
well choose the largest known prime, 2^43112609-1, for now
(but we can increase this as new primes are found). Then this
bound is well larger than any number that appears in RSA
cryptography, economics, or physics.
> But I'll be the one to say it:
> AP is a crank, and his definition of 10^500 as an infinite
> number is a crank idea, just like all of his other ideas.
And it's possible that one might call Zeilberger a so-called
"crank" as well.
But to me, there exists a finite number that's larger than any
finite number that appears in cryptography, ecomonics, and
physics, and that one's not necessarily a "crank" just for
believing that such a number exists.
So if we avoid 10^500 just because it's associated with a
"crank," then let's choose an upper bound that has nothing to
do with "cranks" at all. Actually, what's wrong with just
plain ol' googolplex? I doubt that anyone, even Tribble, could
find a use for numbers exceeding googolplex in cryptography,
economics, or physics. And if he does, then we'll just jump to
googolduplex (i.e., 10^googolplex) -- or even Graham's Number.
Sooner or later, we'll reach a finite number that even Tribble
accepts as exceeding any finite number that appears in the
sciences, and that will be our upper bound.
Transfer Principle
> On 30 אוגוסט, 21:50, Transfer Principle <lwal...@lausd.net> wrote:
> > So perhaps it would be better if I didn't use the phrase
> > "nonstandard natural number" to describe the labels. To the
> > post-Cantorians, all that matters is that there are infinitely
> > many balls in the vase. It doesn't matter what the labels on
> > those balls are -- but if they're convinced that all the balls
> > with standard natural numbers have been removed, then something
> > other than standard natural numbers must be on the balls
> > (whether it's infinitely many zeros, or labels like "infinity,"
> > "infinity-1," "infinitely-2," etc.
> logic, that is, simply begging the question.
> We seem to agree that no balls with standart natural numbers will be
> in the vase at noon. Now, I know of no "nonstandart" numbers (or
> infinity-1 or however you choose to call them), thus I conclude that
> the vase is simply empty. You *assume* that the vase is nonempty, and
> thus conclucde the existance of such numbers.
Yes, I agree that this is only an _assumption_. What I want to do
is to rewrite this _assumption_ as a rigorous _axiom_.
In other words, I agree that _ZFC_ proves that no "infinity-1" or
whatever is a natural number, and that the vase is empty. But TO
rejects this conclusion and hence he rejects ZFC. So we want to
find a theory other than ZFC in which one concludes both that the
vase is full and that "infinity-1" (or whatever one wishes to call
it) is one of the labels still in the vase.
But as we've seen before, trying to write TO's assumption as a
rigorous axiom is not easy. Perhaps this is because TO's ideas
differ greatly from ZFC -- in particular, he's more concerned with
how many balls are in the vase, than with which balls are in the
vase (which is more in accord with ZFC's primitive "e").
I was thinking, since TO seems to treat his "Bigulosity" as a
primitive notion, perhaps we should do the same. So instead of
having "e" as a primitive, we use "Bigulosity" instead -- or
maybe we have both as primitives.
And then we write axioms relating Bigulosity to equivalence and
order relations in the expected manner, including most importantly
that no set has the same Bigulosity as its proper subset.
Tony Orlow
Hi Jesse -
I have to say, your comments on this matter show a lot of maturation
since last we discussed any such matters. Nice to see :)
The "magic" occurs when all the balls that have accumulated before
noon disappear from the vase and it becomes empty. It does not occur
before noon, nor does it occur at noon.
"happens by noon" = t<=noon = (t<noon v t=noon)
t<noon? no, there are balls there still.
t=noon? no, nothing occurs anymore.
Peace,
Tony
Tony Orlow
ZFC does not prove any such thing. This "proof" relies on the
countably infinite model of the von Neumann ordinals and the concept
of omega as some kind of distinctly completed quantity. N is completed
sometime between morning and noon. You know, one of those extra
moments in there. Pretty distinct....
:)
TOny
Transfer Principle
> On Aug 30, 12:55 pm, Transfer Principle <lwal...@lausd.net> wrote:
> > So we want finite sets to act like infinite sets.
> Not necessarily. We could just do away with finite sets altogether.
OK. So we would need an axiom telling us that every set is an
infinite set. We write thusly:
Axiom of Anti-Dirichetlian Infinity
Ax (x infinite)
where "infinite" is some standard definition of infinite
(Tarski, Dedekind, etc.) reduced to primitives.
> > We then add
> > this axiom to as many axioms of ZFC as possible without there
> > being an inconsistency, then see what turns up.
> To ZFC!? What is this fixation with ZFC?
Experience has shown that the closer a theory is to the standard
theory ZFC, the more difficult it will be for an opponent to
criticize the theory. I'm thinking in terms of those posters who
regularly use five-letter insults. Since such posters regularly
defend ZFC, they're less likely to attack a theory that is very
similar to ZFC.
> The axiom of extensionality was proven wrong in the 1960's by the
> Phylis Diller theorem: I've got everything Raquel Welch has, just
> arranged differently.
I'm not sure whether we really need to drop Extensionality for
our anti-Dirichletian theory to work. I must warn Jeffries that
according to some posters (most notably MoeBlee and Virgil),
objects that don't adhere to Extensionality ought not to be
called "sets" at all. Thus, if Jeffries wants Extensionality to
be dropped, he should be prepared to come up with a name other
than sets ("zets," perhaps) for his objects.
And so let's look at some of the axioms of ZFC and see which
ones to keep and which to drop or modify.
Infinity:
To be replaced by Anti-Dirichletian Infinity Axiom.
Extensionality:
No need to drop despite Jeffries's comments -- though we may
need to drop it further down the road.
Empty Set and Pairing:
The most obvious axioms to drop. These axioms assert the
existence of finite sets with at most two elements, while we
want every set to be infinite.
Powerset and Union:
Keep them. The powerset of an infinite set is itself infinite
(indeed uncountable), so we may keep it. As it turns out, the
union of an infinite set is also infinite, but this isn't
quite as obvious. Proof: Let x be a set, and y = U(x). By the
definition of union, y is a set (indeed the smallest set) such
that every element of x is a subset of y. That is, every
element of x is in the _powerset_ of y. That is, x is a subset
of P(y). Now the powerset of a finite set is again finite, and
a subset of a finite set is again finite. So if y is finite,
then x is finite too. QED
Foundation/Regularity:
A tricky one. At first, we note that in ZFC, every descending
e-chain is finite and ends in the empty set. But in the
anti-Dirichletian theory there is no empty set. So it would
appear that every e-chain is infinite, and so Regularity would
need to be dropped.
But we point out that Regularity merely states that every
(nonempty) set have an element disjoint from it. (It's only
when we add other ZFC axioms to Regularity can we conclude the
nonexistence of an infinitely descending chain -- see some old
David Libert threads for more info.) We see no reason why sets,
even infinite sets, can't be disjoint from their elements.
(And besides, who's to say that we don't have urelements, so
that our descending chains can end in them? But I prefer to
stick as close to ZFC as possible, and so we avoid urelements
unless they become necessary down the road.) Thus, we should
wait until later to decide on Regularity.
Replacement and Separation Schemata:
The schemata are the biggies. The problem is that we can apply
Separation to an infinite set and obtain a finite set -- indeed,
it's been pointed out many times that from the existence of any
set, we can derive the empty set via Separation! So it appears
that we must reject Separation.
But what about Replacement? At first, Replacement appears to be
harmless, since it basically states that if the domain of a
function is a setm then so is its image. But no one said
anything about this function being injective, so what if we
let this function be a _constant_ function? Then its image,
even with infinite domain, would be a finite singleton! And so
we must reject Replacement as well.
At this point, with both schemata gone, this theory doesn't
sound very powerful at all. What we need is some sort of schema
that produces only infinite sets. So why don't we have the
following schema, which sounds very zuhair-like:
Infinite Comprehension Schema:
{x|phi} exists exactly when infinitely many sets satisfy phi.
The only problem here is that I'm not sure whether this can be
written as a first-order schema containing "e".
At this point we wonder, what about Russell's Paradox? Since
by Russell, the set {x|~xex} can't exist, we conclude that only
finitely many sets satisfy ~xex. Thus, the collection of all
sets that don't contain themselves would be too _small_ to be a
set, so it doesn't exist. (In an anti-Dirichletian theory, we
have collections that are too _small_ to be sets whereas some
collections are too _large_ to be sets in ZFC.) So we conclude
that in fact, most sets _do_ contain themselves.
We do notice that if any set exists, then the set {x|x=x} must
necessarily exist. Why? If any set S exists, then by the
anti-Dirichletian Infinity Axiom, S is infinite. So all the
elements of S satisfy x=x. So there are infinitely many sets
satisfying x=x. So Infinite Comprehension holds, and thus the
set {x|x=x} must exist. QED
By Extensionality, this set must be unique. And so we might as
well call this set V, since it is indeed the universal set.
We notice that if any object exists, then V exists. Notice how
in ZFC, if any object exists, then the empty set exists. The
two conclusions are exactly parallel. Many ZFC users avoid the
free logic where it's possible that no object exists, and
instead use classical logic where at least one object must
necessarily exist. Thus, they assert that the Empty Set Axiom
is superfluous -- 0 exists by Separation alone. Similarly, we
assert that unless we use the free logic, V exists by Infinite
Comprehension and anti-Dirichletian Infinity alone, and adding
a Universal Set Axiom is redundant.
We also note that the set V can't possibly be disjoint with
any set other than the empty set (or a urelement), which we've
already disallowed. So now we know to reject Regularity.
All this discussion about the empty set and V should certainly
sound familiar. In another thread, I started writing axioms
for a theory described by RF, in which the universal set V
exists, but the empty set 0 doesn't. Indeed, every set in RF's
theory was said to have infinitely many elements ("parts").
So we conclude that all along, RF was actually discussing an
anti-Dirichletian theory! But according to RF, he was trying
to discuss the infinitely divisible theory of TO -- who's as
far from anti-Dirichletian as we can get!
Let me think about this for a while.
Jesse F. Hughes
> On Aug 30, 11:50 am, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
>> David R Tribble <da...@tribble.com> writes:
>>
>> >> When does your magic occur? If you can answer that question, you win.
>> >> If not, concede.
>>
>> > The magic occurs at each step, like I said.
>>
>> > But it does not occur all at once, which you seem to be
>> > implying. Do you think "the magic" occurs all at once, at some
>> > special moment?
>>
>> Obviously, David and I gave contradictory answers here.
>>
>> If "the magic" is "the vase is empty" or "the moment the number of balls
>> decrease", then my answer is correct -- it happens at noon and not
>> before.
>>
>> If "the magic" is that balls are being removed, then this happens at
>> each step.
>>
>> --
>> Scientists have calculated that the chance of anything so patently
>> absurd actually existing are millions to one. But magicians have
>> calculated that million-to-one chances crop up nine times out of ten.
>> -- Terry Pratchett on Intelligent Design. Or something.
>
> Hi Jesse -
>
> I have to say, your comments on this matter show a lot of maturation
> since last we discussed any such matters. Nice to see :)
No idea what you mean.
I still have no respect for your "theory". Sorry if I made it seem
otherwise.
> The "magic" occurs when all the balls that have accumulated before
> noon disappear from the vase and it becomes empty. It does not occur
> before noon, nor does it occur at noon.
>
> "happens by noon" = t<=noon = (t<noon v t=noon)
> t<noon? no, there are balls there still.
> t=noon? no, nothing occurs anymore.
There is no moment prior to noon at which the vase is empty.
Nonetheless, it is empty at noon. If you have any problems with this
simple, clear consequence, it is likely because the situation itself is
a bit silly (infinitely many tasks in a finite time).
Regardless, if we accept the hypothetical situation, it is clear that
there is no ball in the vase, because every ball is labeled with a
natural number and every ball so labeled has been removed prior to noon.
I really don't see anything more to discuss.
--
Jesse F. Hughes
"Mathematicians don't fit in with a consistent view, unless you accept
that to a strangely large extent they are acting under the influence
of something very powerful, dark, and negative." -- James S. Harris
David R Tribble
> Now that's very wrong, and that's a second answer to your 'proof'.
> *Not* every real number has a decimal expansion. At least not a usable
> one. See, when I write 0.999...., you really understand what I mean,
> and can tell me immediately what the n'th digit is, and everything
> else on this number (even if it is a number - actually it's only an
> infinite sum). But that's only because it's a very simple number to
> consider. Consider PI instead. The expansion 3.14... doesn't really
> tell everything. You can't tell me what the 10^12'th digit is, no
> matter how many digits I supply (technically, I can't get anywhere
> near supplying 10^12 digits). However PI is a known number, and
> formulas exist for calculating its n'th digit. But most real numbers
> aren't. Just throw a zero-head-sized arrow on the [0,1] line, and
> chances are you'd hit a number for which no '...' representation can
> be written, that is, a number whose digits don't have any apparent
> pattern.
>
> So your expectation that every real number will have a '...'
> representation is simply wrong. Indeed there're infinite number of
> numbers between 0.999.... and 1 (or between any two different real
> numbers), not for all of whom a decimal expansion exists.
Can you provide an example of a real number that doesn't
have a decimal expansion?
David R Tribble
>> At an annual inflation rate of 65 x 10^105, it would take
>> only a few short years to exceed 10^500 Z dollars.
>
Transfer Principle (L Walker) wrote:
> Thus, the hyperinflation as described by Tribble does involve
> the number googol, but not googol_plex_. He'd be hard pressed
> to find any economic situation involving googolplex.
> [...]
> So if we avoid 10^500 just because it's associated with a
> "crank," then let's choose an upper bound that has nothing to
> do with "cranks" at all. Actually, what's wrong with just
> plain ol' googolplex? I doubt that anyone, even Tribble, could
> find a use for numbers exceeding googolplex in cryptography,
> economics, or physics. And if he does, then we'll just jump to
> googolduplex (i.e., 10^googolplex) -- or even Graham's Number.
>
> Sooner or later, we'll reach a finite number that even Tribble
> accepts as exceeding any finite number that appears in the
> sciences, and that will be our upper bound.
And so then anything larger is deemed "infinite"?
Why do you think that is useful, or even meaningful?
David R Tribble
>> order relations in the expected manner, including most importantly
>> that no set has the same Bigulosity as its proper subset.
>
Tony Orlow wrote:
> ZFC does not prove any such thing. This "proof" relies on the
> countably infinite model of the von Neumann ordinals and the concept
> of omega as some kind of distinctly completed quantity.
So then what is a "distinctly incompleted" quantity?
> N is completed sometime between morning and noon.
> You know, one of those extra moments in there.
Please show us the definition of N that depends on time.
Because I've never seen one.
David R Tribble
>> The "magic" occurs when all the balls that have accumulated before
>> noon disappear from the vase and it becomes empty. It does not occur
>> before noon, nor does it occur at noon.
>>
>> "happens by noon" = t<=noon = (t<noon v t=noon)
>> t<noon? no, there are balls there still.
>> t=noon? no, nothing occurs anymore.
>
Jesse F. Hughes wrote:
> There is no moment prior to noon at which the vase is empty.
> Nonetheless, it is empty at noon. If you have any problems with this
> simple, clear consequence, it is likely because the situation itself is
> a bit silly (infinitely many tasks in a finite time).
>
> Regardless, if we accept the hypothetical situation, it is clear that
> there is no ball in the vase, because every ball is labeled with a
> natural number and every ball so labeled has been removed prior to noon.
I suggested a simplified version of the problem wherein you
begin with a vase filled with an infinite number of balls, each
one labeled with a unique natural; at each step you remove
the lowest numbered ball, and there are an infinite number
of steps [performed by some given time, e.g., noon].
It's exactly the same situation as you describe: at every
point prior to noon, the vase is not empty. Yet by noon,
the vase is completely empty. (In fact, it's a little more
extreme than the original problem, since at any time
before noon the vase contains an infinite number of balls.
But the outcome is the same.)
And like you say, if you accept the premise of the problem,
the result is obvious.
Tony Orlow
> Transfer Principle (L Walker) wrote:
>
> >> And then we write axioms relating Bigulosity to equivalence and
> >> order relations in the expected manner, including most importantly
> >> that no set has the same Bigulosity as its proper subset.
>
> Tony Orlow wrote:
> > ZFC does not prove any such thing. This "proof" relies on the
> > countably infinite model of the von Neumann ordinals and the concept
> > of omega as some kind of distinctly completed quantity.
>
> So then what is a "distinctly incompleted" quantity?
If there's no completion, it can't be very distinct, can it? The cute
little Zeno Paradox Engine that makes some infinite sequences of
events occur "by" some moment is a bit of an illusion, since the
sequence is never completed before, nor at the moment of, completion.
That's what's indistinct about this completion of omega. Normally, it
just never ends......
>
> > N is completed sometime between morning and noon.
> > You know, one of those extra moments in there.
>
> Please show us the definition of N that depends on time.
> Because I've never seen one.
Ask Zeno.
Tony Orlow
Sure. If it's not empty before noon, but then it is empty at noon,
then it became empty at noon, right?
>
> Regardless, if we accept the hypothetical situation, it is clear that
> there is no ball in the vase, because every ball is labeled with a
> natural number and every ball so labeled has been removed prior to noon.
Well, sure. It's kind of ridiculous to consider it a proof of
anything.
>
> I really don't see anything more to discuss.
Fine. Be that way, I'm going out for some smokes.
;) Tony
Jesse Hughes
> Jesse F. Hughes wrote:
>> There is no moment prior to noon at which the vase is empty.
>> Nonetheless, it is empty at noon. If you have any problems with this
>> simple, clear consequence, it is likely because the situation itself is
>> a bit silly (infinitely many tasks in a finite time).
>>
>> Regardless, if we accept the hypothetical situation, it is clear that
>> there is no ball in the vase, because every ball is labeled with a
>> natural number and every ball so labeled has been removed prior to noon.
>
> I suggested a simplified version of the problem wherein you
> begin with a vase filled with an infinite number of balls, each
> one labeled with a unique natural; at each step you remove
> the lowest numbered ball, and there are an infinite number
> of steps [performed by some given time, e.g., noon].
>
> It's exactly the same situation as you describe: at every
> point prior to noon, the vase is not empty. Yet by noon,
> the vase is completely empty. (In fact, it's a little more
> extreme than the original problem, since at any time
> before noon the vase contains an infinite number of balls.
> But the outcome is the same.)
>
> And like you say, if you accept the premise of the problem,
> the result is obvious.
Yes, I like that simplification.
--
Jesse Hughes
Adjunct professor,Philosophy
Bentley College, Waltham MA
Transfer Principle
> On 1 ספטמבר, 10:06, Virgil <Vir...@home.esc> wrote:
> > So that one finds that there are infinitely many numbers between
> > 0.999... and 1, all of which must have some sort of decimal expansions,
> representation is simply wrong. Indeed there're infinite number of
> numbers between 0.999.... and 1 (or between any two different real
> numbers), not for all of whom a decimal expansion exists.
Somehow, this suddenly turned into yet another 0.999... thread. Of
course, 0.999... and the ball-and-vase problem go hand in hand,
since some people believe that the balls without standard natural
numbered labels are put into the vase at 11:59:59.999...
Once again, let me give the link to the Metamath proof of the
classical result 0.999..., proved in the theory ZF-Regularity:
http://us.metamath.org/mpegif/0.999....html
But that being said, just because ZF-Reg and classical analysis
prove that 0.999...=1, it doesn't mean that there can't be a
nonstandard theory or analysis in which 0.999...<1.
Both Earie and TO believe in nonzero infinitesimals. Indeed,
both Earie and TO, unlike RF, believe that there is no smallest
positive infinitesimal. This is interesting since most advocates
of 0.999...<1 believe that 1-0.999... is the smallest positive
infinitesimal (most notably MR). Thus, Earie allows some reals
not to have decimal expansions.
Because of this, I see no problem with setting Earie's 0.999...
equal to TO's 1-1/zillion. (We choose zillion since according
to TO there are a zillion reals in the unit interval.) We note
that unlike Earie, TO isn't concerned with assigning decimal
representations to any of his reals.
Earie might be glad to know that he has the support of 40% of
the participants in a poll given at the Metamath link. Of course,
as we've seen when I first mentioned that poll, most classical
analysts will reject any poll which gives anything less than
near-unanimous support for 0.999...=1.
Transfer Principle
Not necessarily. Most ultrafinitists simply assume that no
number larger than the largest finite number exists at all,
whether finite or infinite. I must admit that AP defies
classification as a(n ultra)finitist due to his beliefs
that sufficiently large finite numbers according to the
standard theory are infinite in his proposed theory.
So if I were coming up with an ultrafinitist theory, I'd
just choose the upper bound M and then declare M to be the
largest number. The numbers larger than M wouldn't be
considered "infinite," as they wouldn't exist at all.
Transfer Principle
I don't believe that TO is saying that N depends on time.
To me, TO is saying that every ball with a label in N has
completed (its entry into the vase) sometime between
morning and noon. Similarly, we can say that the set
{1,2,3,4,5,6,7,8,9,10} is completed at 11:59, since every
ball with a label in that set has completed its entry into
the vase by 11:59. This doesn't mean that (construction of)
the set {1,2,3,4,5,6,7,8,9,10} itself depends on time --
it just relates it to that time in this _particular_ problem.
So TO is telling us that every ball with a label in N has
completed its entry into the vase by noon. He states that
it is "one of those extra moments in there" -- and most
likely it is some positive infinitesimal time before noon.
Virgil
<c0c1f880-d5d6-49b0...@f42g2000yqn.googlegroups.com>,
Tony Orlow <to...@lightlink.com> wrote:
> The "magic" occurs when all the balls that have accumulated before
> noon disappear from the vase and it becomes empty. It does not occur
> before noon, nor does it occur at noon.
>
> "happens by noon" = t<=noon = (t<noon v t=noon)
> t<noon? no, there are balls there still.
> t=noon? no, nothing occurs anymore.
That just requires that each ball "disappears" before noon but that
there is no last ball. No problem.
Transfer Principle
> So we conclude that all along, RF was actually discussing an
> anti-Dirichletian theory! But according to RF, he was trying
> to discuss the infinitely divisible theory of TO -- who's as
> far from anti-Dirichletian as we can get!
> Let me think about this for a while.
OK, I've thought it some more. At first, when Jeffries first
described his anti-Dirichletian theory, I wasn't sure whether
to consider it, since I don't always like considering a theory
with only a single advocate. But now we see that RF had
mentioned this theory much earlier. With two advocates, I now
want to proceed full steam to work on this theory,
What about TO? Frankly, I'm not sure why RF tried to tie this
anti-Dirichletian theory to TO's infinitesimals anyway. Indeed,
we see that if we look at that theory as an anti-Dirichetian
theory, more of it suddenly clicks into place -- and some of
the problems I had in trying to axiomatize it disappear.
Now we have our set V, we can form {x|~x=V}. How do we know
that this set exists? Well, we know that V contains infinitely
many elements since every set is infinite. In particular, it
must contain infinitely many non-V elements, since we can't
remove a single element from an infinite set and expect to
obtain a finite set. So by Infinite Comprehension the set
{x|~x=V} must exist.
So what exactly is {x|~x=V} anyway? It must be a set which
contains every other set except V. In the other thread, I
called this set V_1. Earlier, I was mystified as to whether
this set even exists. But now we know that it does. Also, we
know that V_1 must contain itself -- which makes perfect sense
in this theory, since we've proved that nearly every set must
contain itself!
Similarly, we can form {x|~x=V & ~x=V_1} -- but we can also
form the set {x|~x=V_1} as well. This latter set contains V
as an element, yet is not itself equal to V. In the other
thread, we stated that no set other than V can contain V, but
in this thread, we now allow such sets to exist.
But there is a problem with this universal set V. In the other
post, I decided to keep the Powerset Axiom -- so naturally, we
ask, what exactly is P(V)? Since there are no urelements, the
only plausible answer is that P(V)=V. But then we wonder, what
about Cantor's Theorem? We already know that if we try to
diagonalize in P(V), we obtain the collection of sets that
don't contain themselves, which is too small to be a set. So
we conclude that P(V) really is V. Thus, this theory proves
the existence of non-Cantorian sets.
Now this point might unsettle us a bit. When we came up with
this theory, Jeffries was trying to make it so that every set
is infinite and has the properties shared by all infinite sets
in standard theory. It's doubtful that he intended anything
like V to appear, with its non-Cantorian properties. And
recall that V instantly appeared only when we wanted to add a
Separation/Comprehension Schema that works for infinite sets.
What to do? I'm not sure.
But whether Jeffries wants the set V to appear or not, he
certainly stated that he wanted the set omega to appear -- or
at least a set with the essential properties of omega (most
notably, its order type).
But do order types exist in this theory? The problem is that
the notion of order type, in standard theory, is intertwined
with the notions of (order) relation, Cartesian product, and
ultimately, ordered pair. And if "ordered pair" means the
Kuratowski pair, then we have a problem, since the Kuratowski
pair is evidently a finite set.
Hmmm. It's possible that ordered pairs might exist in this
theory, if we can find a way to connect sets a and b to some
set (a,b) so that it satisfies the key property of ordered
pairs, namely that (a,b)=(c,d) iff (a=c & b=d). Once we have
that, only then can we figure out order relations, order
types, and omega.
So far, there's one more axiom of ZFC that we have yet to
consider, and that's the Axiom of Choice. At this point, we
see no reason to drop Choice, so we keep it. It's possible
that AC, just as in standard theory, can assist us in coming
up with order types for our sets.
Virgil
<c6ec42dd-64bf-4741...@x42g2000yqx.googlegroups.com>,
Tony Orlow <to...@lightlink.com> wrote:
> On Sep 2, 10:43 pm, David R Tribble <da...@tribble.com> wrote:
> > Transfer Principle (L Walker) wrote:
> >
> > >> And then we write axioms relating Bigulosity to equivalence and
> > >> order relations in the expected manner, including most importantly
> > >> that no set has the same Bigulosity as its proper subset.
> >
> > Tony Orlow wrote:
> > > ZFC does not prove any such thing. This "proof" relies on the
> > > countably infinite model of the von Neumann ordinals and the concept
> > > of omega as some kind of distinctly completed quantity.
> >
> > So then what is a "distinctly incompleted" quantity?
>
> If there's no completion, it can't be very distinct, can it? The cute
> little Zeno Paradox Engine that makes some infinite sequences of
> events occur "by" some moment is a bit of an illusion, since the
> sequence is never completed before, nor at the moment of, completion.
> That's what's indistinct about this completion of omega. Normally, it
> just never ends......
Then a Zeno-like analysis would prove that all motion is impossible,
which experience disproves.
>
> >
> > > N is completed sometime between morning and noon.
> > > You know, one of those extra moments in there.
> >
> > Please show us the definition of N that depends on time.
> > Because I've never seen one.
>
> Ask Zeno.
That will have to wait at least until we are as dead as he is, and even
then, only if we have souls surviving our bodies, which, while widely
believed, has not been established beyond doubt.
Virgil
<76cf9dd6-6a61-4a68...@i13g2000yqd.googlegroups.com>,
Tony Orlow <to...@lightlink.com> wrote:
> Sure. If it's not empty before noon, but then it is empty at noon,
> then it became empty at noon, right?
If not having happened before noon and having happened by noon is your
definition of something happening "at" noon, then yes!
Tim Little
> No, you look at it. It says nothing about a sum.
Of course it doesn't. You're the only one who said the limit
expression "lim{T->inf}{10^-T} = 0" was a sum.
- Tim
K_h
"Transfer Principle" <lwa...@lausd.net> wrote in message
news:21dea821-1d7e-4908...@x25g2000yqj.googlegroups.com...
> On Aug 31, 5:57 am, FredJeffries <fredjeffr...@gmail.com> wrote:
>> On Aug 30, 12:55 pm, Transfer Principle <lwal...@lausd.net> wrote:
>> > So we want finite sets to act like infinite sets.
>> Not necessarily. We could just do away with finite sets altogether.
>
>
> necessarily exist. Why? If any set S exists, then by the
> anti-Dirichletian Infinity Axiom, S is infinite. So all the
> elements of S satisfy x=x. So there are infinitely many sets
> satisfying x=x. So Infinite Comprehension holds, and thus the
> set {x|x=x} must exist. QED
{x|x=x} looks like a proper class to me. If x is any set then x=x is trivially
true for all sets and so {x such that x=x} will be the class of all sets.
> Let me think about this for a while.
+
K_h
"Tony Orlow" <to...@lightlink.com> wrote in message
news:76cf9dd6-6a61-4a68...@i13g2000yqd.googlegroups.com...
On Sep 2, 9:57 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
> Tony Orlow <t...@lightlink.com> writes:
> > On Aug 30, 11:50 am, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
> >> David R Tribble <da...@tribble.com> writes:
>
> Sure. If it's not empty before noon, but then it is empty at noon,
> then it became empty at noon, right?
I would like to ask you a question on an easier problem. Suppose we have a vase
with one ball in it. At noon the ball is removed. Now, AT noon is the ball in
the vase or not?
+
Jesse F. Hughes
> So if I were coming up with an ultrafinitist theory, I'd
> just choose the upper bound M and then declare M to be the
> largest number. The numbers larger than M wouldn't be
> considered "infinite," as they wouldn't exist at all.
What advantage would you find in such a theory?
--
Jesse F. Hughes
"If you hadn't noticed, basically every result I have destroys some
precious belief of mathematicians and they have from what I've gathered
basically gone collectively bonkers." -- James S. Harris
Jesse F. Hughes
> Earie might be glad to know that he has the support of 40% of
> the participants in a poll given at the Metamath link. Of course,
> as we've seen when I first mentioned that poll, most classical
> analysts will reject any poll which gives anything less than
> near-unanimous support for 0.999...=1.
No, most sensible people will reject any forum-based poll requiring
readers to choose whether to respond or not.
I have no idea how many people believe (wrongly) that 0.999... < 1
(understood as an inequality in the theory of real numbers). Perhaps
the poll came to the right estimate, though, if so, it is merely
coincidental.
You pretend that the complaint is that the results are offensive. The
complaint is instead that the problems with the poll are obvious.
I should note that your conclusion is carefully worded: 40% of the
participants in that poll said such-and-such[1]. That's true, but that
tells us nothing about the general public.
By the way, Earie didn't say he was working in a non-standard theory.
Did you notice that? He said that proofs that 0.999... = 1 were
*wrong*. He didn't say that they were invalid in some alternative
theory, but that they were invalid -- specifically, circular.
Footnotes:
[1] Actually, you said they support Earie, which is false. We don't
know whether their misconceptions are similar to Earie's or not.
--
Jesse F. Hughes
"Students said they wanted to make people feel more comfortable by not
having to choose a gender at the bathroom door."
-- Boston Globe article on gender-neutral bathrooms at universities.
Tony Orlow
You cannot fully complete N without going beyond it. It has no end.
When I say it occurs between morning and noon, I am implying it
doesn't occur, since noon is the end of morning. It's like saying
there exists a point between the endpoint of a line segment and the
rest of the segment.
Now, if one wants to precisely number the iterations with some
specific infinity, like zillion (not indistinct omega or tav), then
one can express the number of balls and the numbering of the balls in
the vase in terms of that count in zillions. Specifically, a zillion
iterations leaves 9 zillion balls in the vase, numbered from zillion+1
to 10 zillion.
Have fun!
TOny
Tony Orlow
> In article
> <c0c1f880-d5d6-49b0-96dc-f4d75bd1d...@f42g2000yqn.googlegroups.com>,
> Tony Orlow <t...@lightlink.com> wrote:
>
> > The "magic" occurs when all the balls that have accumulated before
> > noon disappear from the vase and it becomes empty. It does not occur
> > before noon, nor does it occur at noon.
>
> > "happens by noon" = t<=noon = (t<noon v t=noon)
> > t<noon? no, there are balls there still.
> > t=noon? no, nothing occurs anymore.
>
> That just requires that each ball "disappears" before noon but that
> there is no last ball. No problem.
You are missing extremely simple logic. To say event x occurs *by* a
specific moment means it either occurs *before* that moment, or *at*
that moment. If it can occur neither before nor at that moment, it
cannot happen *by* that moment.
Smiles,
Tony
Tony Orlow
> In article
> Tony Orlow <t...@lightlink.com> wrote:
>
> > Sure. If it's not empty before noon, but then it is empty at noon,
> > then it became empty at noon, right?
>
> If not having happened before noon and having happened by noon is your
> definition of something happening "at" noon, then yes!
However, no balls move anywhere at noon, so nothing happens at noon.
See the contradiction?
:)
Tony
Tony Orlow
> "Tony Orlow" <t...@lightlink.com> wrote in message
Assuming the removal of the ball begins at noon, it is in the vase at
noon, but not thereafter. This is a different problem, however. In the
original, no ball moves at noon.
:)
Tony
Tony Orlow
I wonder what the 40% of the pollees would say if they were asked
which was bigger, 0.999... decimal, or 0,111... binary?
Smiles,
Tony
David R Tribble
> Now, if one wants to precisely number the iterations with some
> specific infinity, like zillion (not indistinct omega or tav), then
> one can express the number of balls and the numbering of the balls in
> the vase in terms of that count in zillions. Specifically, a zillion
> iterations leaves 9 zillion balls in the vase, numbered from zillion+1
> to 10 zillion.
You should explain what "zillion" is supposed to mean with
a bit more precision.
Do a zillion iterations occur by noon, according to how the
problem was originally stated? Or are a zillion iterations more
(or perhaps less) than the number of iterations as originally
stated?
In other words, does the sum
s = 1/2 + 1/4 + 1/8 + 1/16 + ...
contain a zillion terms (or more, or less)?
At what point does the ball labeled with "zillion" (whatever
that looks like) get put into the vase? What does the label
of a zillion, in fact, look like?
And what happened to the balls left in the vase; are there
no times prior to noon when they were removed?
David R Tribble
> I wonder what the 40% of the pollees would say if they were asked
> which was bigger, 0.999... decimal, or 0,111... binary?
I take the poll as evidence that 40% of the web respondents
are mathematically illiterate. Which pretty much meets
normal expectations.
It's kind of like one of Jay Leno's "Jaywalks", where he asks
"ordinary" pedestrians simple questions and shows the
stunningly ignorant answers they give.
Transfer Principle
> Transfer Principle <lwal...@lausd.net> writes:
> > So if I were coming up with an ultrafinitist theory, I'd
> > just choose the upper bound M and then declare M to be the
> > largest number. The numbers larger than M wouldn't be
> > considered "infinite," as they wouldn't exist at all.
> What advantage would you find in such a theory?
For some reason, there exist mathematicians calling themselves
finitists and ultrafinitists, who simply don't accept the
existence of infinite sets. The advantage of the theory that I
propose here would be that it allows (ultra)finitists to do
all of the math required for crytography, economics, physics,
and the other sciences.
Why are some mathematicians finitists? Any answer that I can
give is likely to be challenged, so instead let's go ask the
finitists themselves.
Earlier in this thread, I mentioned Doron Zeilberger as an
example of a finitist. So I decided to Google him and see what
he has to say about finitism. And so I found this link:
http://www.math.rutgers.edu/~zeilberg/Opinion108.html
Hmmm. I noticed some interesting stuff here:
"Let me first pause and enlist an unlikely sympathizer with
_finitistm_ [sic, emphasis his], the greatest set-theorist of
our time, Paul Cohen. In the second-to-last paragraph of "The
Discovery of Forcing" (Rocky Mountain Journal of Mathematics,
v.32 (2002), 1071-1100) he said (p. 1099)
'The only reality we truly comprehend is that of our own
experience ... The laws of the infinite are extrapolations of
our experiences with the finite'
So even the great 'infinitarian' Paul Cohen was a devout finitist."
What the?!? So according to Zeilberger, Paul Cohen -- the same
Cohen who opined that the cardinality of the continuum was
greater than aleph_1, aleph_omega, and even aleph_aleph_omega --
was really a _finitist_!
Of course, we know that ZFC users are upset when the name
Abraham Robinson is used to defend finitism. Then they'll have
a field day when they see the name Paul Cohen is mentioned as
an example of a finitist!
Further down the page, we read:
"We have to kick the misleading word 'undecidable' from the
mathematical lingo, since it _tacitly_ assumes that infinity
is real. We should rather replace it by the phrase 'not even
wrong' (in other words utter nonsense), that cannot _even_ be
resurrected by talking about _symbolic_ variables."
[emphasis his]
LOL! "Not even wrong" is one of the ZFC users' favorite phrases
used to describe finitists and challengers of Cantor. So it's
poetic justice indeed to have a finitist describe _ZFC_ as
being "not even wrong." Hilarious!
"Likewise, Cohen's celebrated meta-theorem that the continuum
hypothesis is 'independent' of ZFC is a great _proof_ that none
of Cantor's alephs make any (ontological) sense."
[emphasis his]
And this should answer Hughes's question. A theory with a
largest number M allows finitists like Zeilberger (and he would
choose a large prime for his M) to do useful mathematics.
Zeilberger concludes:
"Of course, beauty is in the eyes of the beholder, and some parts
of infinite mathematics are indeed a '9' (e.g. Greg Chaitin's
utterly-fictional-yet-lovely Omega), but _finite_ mathematics is
both _real_ (in the _real_ sense of the word, not in the sense of
so-called 'real' numbers) and _beautiful_, while 'infinite'
mathematics is utterly fictional, and not-quite-as-pretty. I feel
SO SORRY, and have infinite (pardon my French) pity and compassion
for people who believe otherwise."
Transfer Principle
> Transfer Principle <lwal...@lausd.net> writes:
> > Earie might be glad to know that he has the support of 40% of
> > the participants in a poll given at the Metamath link. Of course,
> > as we've seen when I first mentioned that poll, most classical
> > analysts will reject any poll which gives anything less than
> > near-unanimous support for 0.999...=1.
> Did you notice that?
BTW, Earie didn't say he was working in a standard theory either. Did
you notice that?
> He said that proofs that 0.999... = 1 were
> *wrong*. He didn't say that they were invalid in some alternative
> theory, but that they were invalid -- specifically, circular.
But what does it mean for a proof of 0.999...=1 to be "wrong" in the
first place?
So what is a valid proof? A proof is a sequence of formulae, written
in the language of a _theory_, such that each formula is either an
axiom of a _theory_ or can be deduced from the those above it (and
whose last formula is the one that is to be proved).
In other words, unless we know what the _theory_ is, including its
language and axioms, there's no way to tell whether _any_ proof of
anything is "right" or "wrong" in the first place!
Now Earie doesn't state in which theory he's working. So Hughes
criticizes me for making the unwaranted assumption that Earie is
working in a theory other than classical analysis -- but then Hughes
is guilty of making the equally unwarranted assumption that Earie is
working _in_ classical analysis.
This is typical of classical analysts who criticize me for using
nonclassical analysis (which isn't mentioned in a post) to prove a
poster "right," then proceed to use classical analysis (which isn't
mentioned in the post either) to prove the poster "wrong."
Transfer Principle
> "Transfer Principle" <lwal...@lausd.net> wrote in message
> news:21dea821-1d7e-4908...@x25g2000yqj.googlegroups.com...
> > We do notice that if any set exists, then the set {x|x=x} must
> > necessarily exist. Why? If any set S exists, then by the
> > anti-Dirichletian Infinity Axiom, S is infinite. So all the
> > elements of S satisfy x=x. So there are infinitely many sets
> > satisfying x=x. So Infinite Comprehension holds, and thus the
> > set {x|x=x} must exist. QED
> {x|x=x} looks like a proper class to me. If x is any set then x=x is trivially
> true for all sets and so {x such that x=x} will be the class of all sets.
Yes, I am indeed describing V, the collection of all sets. And
yes, V is not a set in ZFC, and in NBG, where we have proper
classes, V is a proper class.
But I was hoping to describe a theory other than ZFC and NBG
where V can be a set after all. The existence of the set V
hinges on the following axiom schema:
{x|phi} exists iff there are infinitely many sets satisfying phi.
But the question I ask (and maybe K_h knows the answer) is, is
the above a valid schema? By "valid" I don't mean "consistent
with ZFC" (since I know that it's not). Instead, I mean is
there a way to write this as a sequence of formulae in the
language of set theory (however long the formula might be)? I
don't ask for the exact formula itself, but only whether such a
formula can exist. For if not, then I haven't written a valid
theory and it's back to the drawing board.
Come to think of it, with this schema, the anti-Dirichetian
axiom that every set is infinite is redundant. For assume that
there exists a finite set, say:
y = {y_1, y_2, y_3, ..., y_n}
Then we consider the set {x|phi}, where phi is the formula:
(x=y_1 v x=y_2 v x=y_3 v ... v x=y_n)
Then y={x|phi}. But by Infinite Comprehension, the set {x|phi}
can't exist unless infinitely many sets satisfy phi. But
evidently, only finitely many sets satisfy phi -- indeed,
exactly n such sets do. This is a contradiction. Therefore,
every set is infinite. QED
So we can drop anti-Dirichletian Infinity, and just stick to
Infinite Comprehension Schema. So here is our theory so far:
Choice
Extensionality
Infinite Comprehension Schema
Powerset
Union
We are not using the free logic, so we know that at least one
object exists, and we immediately apply Infinite Comprehension
to prove that V exists. Jeffries, whose theory I'm describing
here, doesn't mention the existence of V, but nothing that he
has written so far precludes the existence of V either.
But let's work on omega -- the set that Jeffries does mention
in his posts. We need to get back to the ordered pair.
In the other thread where I first mentioned this theory, we
mention that there is a sort of duality between this theory
and standard ZFC. In particular, the dual of the empty set in
ZFC is V to the anti-Dirichetian. We have a few other dualities:
ZFC: {0}
Dual: V_1 = {x|~x=V}
ZFC: {0,{0}}
Dual: V_2 = {x|~x=V v ~x=V_1}
ZFC: {{0}}
Dual: {x|~x=V_1}
For lack of a better name, we can call this set V_{{0}} as it
is the dual to the set {{0}} of ZFC.
So what we want is to define (a,b) to be the _dual_ of the
Kuratowski ordered pair {{a},{a,b}}. This would be something
like {x|phi}, where phi is the formula:
aex v Ay (yex v y=b)
This exists by Infinite Comprehension since it's obvious that
at most two sets can fail to satisfy this formula, both of
which have ~aex (one with bex, the other with ~bex). Of course
if a=b then there's only one possible set. In either case, all
but finitely many (either one or two) elements of the infinite
V satisfy phi. Therefore infinitely many sets satisfy phi, and
so the set exists.
We can also prove that (a,b)=(c,d) iff (a=c & b=d), with the
proof similar to that for the Kuratowski pair.
Now that we have ordered pairs, we should be able to define
Cartesian product. Now we don't need duals to define the
Cartesian product, since the Cartesian product of two infinite
sets is infinite, so it exists by Infinite Comprehension. We
have that AxB is the set {x|phi} where phi is:
Eab (aeA & beB & x=(a,b))
We can then define order relation as expected. Finally, we
can define an ordinal to be an equivalence class of order
types, as is often done in theories with a universal set (like
NFU, for instance).
Consider the ordered set:
{V, V_1, V_2, V_3, ...}
So we identify "e" with ">" (rather than "<" as usual). The
equivalence class with this order can be labeled "omega." To
prove it exists, we must show that it's infinite. But it
clearly is, since we have the isomorphic orders:
{V_1, V, V_2, V_3, ...}
{V_2, V, V_1, V_3, ...}
{V_3, V, V_1, V_2, V_4, ...}
{V_n, V, V_1, ..., V_n-1, V_n+1, ...}
Jesse F. Hughes
> For some reason, there exist mathematicians calling themselves
> finitists and ultrafinitists, who simply don't accept the
> existence of infinite sets. The advantage of the theory that I
> propose here would be that it allows (ultra)finitists to do
> all of the math required for crytography, economics, physics,
> and the other sciences.
Such mathematicians can do what most mathematicians do anyway: just do
the math and don't fuss about much with the formal theory behind the
math. I don't personally see much of a difference.
But we'll see. Perhaps you'll come up with such a theory and there will
be a renaissance in cryptography when all them formerly unemployable
ultrafinitists join the research programs. It's a noble undertaking!
In all honesty, do you *really* think that you can come up with an
ultrafinite theory of numbers that will include all of the advantages of
real analysis as it is used in, say, physics? I suppose it's good to
aim high, but still...
> Why are some mathematicians finitists? Any answer that I can
> give is likely to be challenged, so instead let's go ask the
> finitists themselves.
>
> Earlier in this thread, I mentioned Doron Zeilberger as an
> example of a finitist. So I decided to Google him and see what
> he has to say about finitism. And so I found this link:
[snip interesting but irrelevant bits from Zeilberger]
For what it's worth, I did not find Zeilberger's comments (and
especially his argument that Cohen was a finitist) to be persuasive, but
I appreciate the quotes.
> And this should answer Hughes's question. A theory with a
> largest number M allows finitists like Zeilberger (and he would
> choose a large prime for his M) to do useful mathematics.
Right. You're searching for a ultrafinitist theory that Zeilberger will
be able to contribute to, say, model fluid dynamics. Cleared it right
up, that did.
--
Jesse F. Hughes
"Well, you know as soon as you have a new number I will be happy to
add it to the list. Don't try those childish tit-for-tat games with
me." -- Ross Finlayson on Cantor's theorem.
Jesse F. Hughes
> Now Earie doesn't state in which theory he's working. So Hughes
> criticizes me for making the unwaranted assumption that Earie is
> working in a theory other than classical analysis -- but then Hughes
> is guilty of making the equally unwarranted assumption that Earie is
> working _in_ classical analysis.
Frankly, I think Earie has no idea what a mathematical theory is, and so
couldn't say what theory he means. But why not ask him?
> This is typical of classical analysts who criticize me for using
> nonclassical analysis (which isn't mentioned in a post) to prove a
> poster "right," then proceed to use classical analysis (which isn't
> mentioned in the post either) to prove the poster "wrong."
The situation is not symmetric.
The proofs that 0.999... = 1 are proofs in (classical) real analysis --
which I'll denote T_R. Unless I say otherwise, when I give such a
proof, the reader ought to assume I mean it as a proof in T_R or some
related theory (perhaps ZFC). This is clearly the default understanding
of such a proof.
Thus, if a correspondent says my proof is "wrong", it is natural to
assume that he means it is an invalid proof in the theory in which it
was presented. If he meant instead that it is invalid in some different
theory then (a) this is a pretty irrelevant observation and (b) he
should at the least be explicit on which theory he means.
Surely you can see the asymmetry in the situation. You assume that
Earie means some other theory, but he did not say so. If he meant some
other theory and he did not explicitly mention the theory, then he is
frankly incompetent at basic communication. It is clear to all parties
that the standard theorem that 0.999... = 1 is (allegedly) a theorem of
classical analysis.
--
Jesse F. Hughes
"Marriage.. ..is the union of two persons of different sex for
life-long reciprocal possession of their sexual faculties."
-- Immanuel Kant, who died an unmarried virgin
Richard Tobin
Transfer Principle <lwa...@lausd.net> wrote:
>> By the way, Earie didn't say he was working in a non-standard theory.
>> Did you notice that?
>BTW, Earie didn't say he was working in a standard theory either.
That's why it's called "standard". It's the one you're assumed to be
working in if you don't say otherwise. You haven't specified that
you're writing in English, but it's the standard language here, so I
won't consider the possibility that by "theory" you mean "fish".
-- Richard
David R Tribble
>> By the way, Earie didn't say he was working in a non-standard theory.
>> Did you notice that?
>
Transfer Principle (L Walker) wrote:
> BTW, Earie didn't say he was working in a standard theory either. Did
> you notice that?
Obviously if someone doesn't say which theory he is working
in, the only logical assumption is to assume he means the
standard theory. Why would anyone think otherwise?
Unless you expect everyone to state every single assumption
up front in their discussion, you must assume the default
position when you begin reading what they write. It's their
responsibility to say otherwise, not ours to assume something
other than the default (standard) position.
> Now Earie doesn't state in which theory he's working. So Hughes
> criticizes me for making the unwaranted assumption that Earie is
> working in a theory other than classical analysis
Which is a proper criticism.
> -- but then Hughes
> is guilty of making the equally unwarranted assumption that Earie is
> working _in_ classical analysis.
There is nothing unwarranted about assuming the obvious
default (standard) positions.
> This is typical of classical analysts who criticize me for using
> nonclassical analysis (which isn't mentioned in a post) to prove a
> poster "right," then proceed to use classical analysis (which isn't
> mentioned in the post either) to prove the poster "wrong."
You are only justified is assuming a non-classical position if
they actually say that's what they are using. Otherwise you are
guilty of making unwarranted assumptions, and can rightly be
accused of trying to read their minds.
It just so happens that Earie is making a statement of his
beliefs about real (standard) numbers, which is obvious from
his arguments and the tone he uses. How else do explain
a comment like "there are real numbers that do not have
decimal expansions"?
K_h
On Sep 2, 11:26 pm, "K_h" <KHol...@SX729.com> wrote:
> "Transfer Principle" <lwal...@lausd.net> wrote in message
>
>
> where V can be a set after all. The existence of the set V
> hinges on the following axiom schema:
>
> {x|phi} exists iff there are infinitely many sets satisfying phi.
As I see it, there are at least three major problems here. First, unrestricted
comprehension can lead to contradictions -- e.g. Russell's paradox. One main
reason for axiomatic set theory is to avoid such contradictions and that is why
unrestricted comprehension is not allowed. If I recall correctly, this was the
flaw that Russell pointed out in a huge work just before it went to press. The
author of that huge work was quite despondent. Second, the problem with allowing
V to be a set is that one can then have other sets not contained in V -- e.g. {V,
7} -- meaning that V is not really the set of all sets after all and that is a
contradiction. You might try to skirt around this problem by claiming that V is
a member of itself but then you are back to Russell's paradox. Third, there will
be finite sets that satisfy {x | phi} and so your schema is self-contradictory in
that area.
> So we can drop anti-Dirichletian Infinity, and just stick to
> Infinite Comprehension Schema. So here is our theory so far:
>
> Choice
> Extensionality
> Infinite Comprehension Schema
> Powerset
> Union
The "Infinite Comprehension Schema" suffers from the above flaws and so it should
be dropped or modified. Note, this is true even outside any formal theory like
ZF. If you are trying to formulate another axiomatic approach to sets then I
suggest you do an overview of the theories that are out there and see what their
general approach is. In other words, first start off with the basic ideas
totally outside any formal system and then formulate them as a formal system.
That is, define what the key ideas and motivations are.
+
K_h
On Sep 3, 2:40 am, "K_h" <KHol...@SX729.com> wrote:
> "Tony Orlow" <t...@lightlink.com> wrote in message
>
> > vase
> > with one ball in it. At noon the ball is removed. Now, AT noon is the ball in
> > the vase or not?
> >
>
> noon, but not thereafter. This is a different problem, however. In the
> original, no ball moves at noon.
If it is in the vase at noon then it has to be removed at some time after noon.
When is it removed?
+
Transfer Principle
> Transfer Principle <lwal...@lausd.net> writes:
> > For some reason, there exist mathematicians calling themselves
> > finitists and ultrafinitists, who simply don't accept the
> > existence of infinite sets. The advantage of the theory that I
> > propose here would be that it allows (ultra)finitists to do
> > all of the math required for crytography, economics, physics,
> > and the other sciences.
> Such mathematicians can do what most mathematicians do anyway: just do
> the math and don't fuss about much with the formal theory behind the
> math.
But in practice, they don't. AP doesn't just do the math -- he
tries to come up with an alternate theory. And Srinivasan
doesn't just do the math -- he tries to come up with an
alternate theory. And Zeilberger doesn't just do the math -- he
tries to come up with an alternate theory. See the pattern?
Yes, most mathematicians don't fuss about the theory -- they
have nothing to fuss about. Those who do have something to fuss
about can and do exactly that.
> In all honesty, do you *really* think that you can come up with an
> ultrafinite theory of numbers that will include all of the advantages of
> real analysis as it is used in, say, physics?
Combining this with a comment from Jeffries earlier in the thread:
"But physics has found use for
mathematical objects like self-adjoint operators on infinite
dimensional Hilbert
spaces to represent observables."
Both Hughes and Jeffries here claim that physics requires the
existence of infinite sets, including the uncountable complete
ordered field R and infinite-dimensional Hilbert spaces. This
is a commonly made argument.
But if their claim were correct, then why do see so many posters
using _physics_ to argue that infinite sets don't exist? Why
does AP use _physics_ to argue against infinite sets? Why does
Herc use _physics_ to argue against infinite sets? Why does WM
uses _physics_ to argue against infinite sets?
Were that claim isn't as obvious as Hughes and Jeffries make it
sound, then AP wouldn't be posting things like "10^500 is the
largest number used in physics." Instead, he'd be posting that
physics needs all the natural numbers and infinite sets, so we
can have those infinite-dimensional spaces. Earie wouldn't be
posting that "0.999...<1". Instead, he'd be posting that physics
requires a complete ordered field, and we can't have a complete
ordered field unless 0.999...=1. And so on.
So instead, I argue against their claim. I claim that physics
doesn't necessarily require the existence of an infinite set,
and possibly not arbitrarily large finite sets (googolplex,
Graham's number) either. But I am willing to concede the
existence of arbitrarily large finite sets if we can still have
physics without infinite sets, so that at least some of the
sci.math finitists will be satisfied.
Maybe physicists who are finitists should just do the math
needed for physics without complaining about the underlying
mathematical theories? But in reality, as evidenced right here
at sci.math, they don't.
> [snip interesting but irrelevant bits from Zeilberger]
> For what it's worth, I did not find Zeilberger's comments (and
> especially his argument that Cohen was a finitist) to be persuasive, but
> I appreciate the quotes.
Typical. Hughes snips the quotes which imply that Cohen might
be finitist, just as most ZFC users do when a finitist mentions
Abraham Robinson. Then they'll turn around and trot out Cohen
and Robinson when arguing in favor of ZFC or to prove that a
poster is "wrong" using ZFC. (Cohen's name often appears when a
poster tries to define aleph_1 as 2^aleph_0 or assumes CH
without stating that assumption.)
But I will concede to Hughes that the claim that Cohen was a
finitist is less believeable than the claim that Robinson was,
since the former referred to the continuum as being very, very
large, while Robinson merely referred to very small
infinitesimals, rather than the size of the continuum.
Jesse F. Hughes
> On Sep 4, 5:59 am, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
>> Transfer Principle <lwal...@lausd.net> writes:
>> > For some reason, there exist mathematicians calling themselves
>> > finitists and ultrafinitists, who simply don't accept the
>> > existence of infinite sets. The advantage of the theory that I
>> > propose here would be that it allows (ultra)finitists to do
>> > all of the math required for crytography, economics, physics,
>> > and the other sciences.
>> Such mathematicians can do what most mathematicians do anyway: just do
>> the math and don't fuss about much with the formal theory behind the
>> math.
>
> But in practice, they don't. AP doesn't just do the math -- he
> tries to come up with an alternate theory. And Srinivasan
> doesn't just do the math -- he tries to come up with an
> alternate theory. And Zeilberger doesn't just do the math -- he
> tries to come up with an alternate theory. See the pattern?
Oh. So then there is no point in coming up with a theory that allows
such folk to do the math for, say, physics, because these folks don't do
math. They only come up with theories[1].
> Yes, most mathematicians don't fuss about the theory -- they
> have nothing to fuss about. Those who do have something to fuss
> about can and do exactly that.
>
>> In all honesty, do you *really* think that you can come up with an
>> ultrafinite theory of numbers that will include all of the advantages of
>> real analysis as it is used in, say, physics?
>
> Combining this with a comment from Jeffries earlier in the thread:
>
> "But physics has found use for
> mathematical objects like self-adjoint operators on infinite
> dimensional Hilbert
> spaces to represent observables."
>
> Both Hughes and Jeffries here claim that physics requires the
> existence of infinite sets, including the uncountable complete
> ordered field R and infinite-dimensional Hilbert spaces. This
> is a commonly made argument.
I made no such claim.
I asked a question. I suspect that it would be *extremely* difficult to
come up with an ultrafinitist theory in which we could do large parts of
real analysis, but that's only a guess. I could certainly be wrong.
> But if their claim were correct, then why do see so many posters
> using _physics_ to argue that infinite sets don't exist? Why
> does AP use _physics_ to argue against infinite sets? Why does
> Herc use _physics_ to argue against infinite sets? Why does WM
> uses _physics_ to argue against infinite sets?
Huh? The two claims are distinct.
Claim 1: Classical real analysis involves infinite sets and also
provides a very good model for physics.
Claim 2: As it happens, there are only finitely many things in the
universe, time and space are finitely divisible and so on.
If claim 2 is correct, then claim 1 may be true but inconsequential:
perhaps a (ultra?)-finitist would do just as well, but we don't have
such a theory that does just as well at present.
Of course, claim 2 may be false. Or it may be true. I don't make a
guess either way.
> Were that claim isn't as obvious as Hughes and Jeffries make it
> sound, then AP wouldn't be posting things like "10^500 is the
> largest number used in physics." Instead, he'd be posting that
> physics needs all the natural numbers and infinite sets, so we
> can have those infinite-dimensional spaces. Earie wouldn't be
> posting that "0.999...<1". Instead, he'd be posting that physics
> requires a complete ordered field, and we can't have a complete
> ordered field unless 0.999...=1. And so on.
I didn't make any claim, much less say that it's obvious. You're
confused.
But *even if* it were obvious (as obvious as the fact, say, that the
universe is not a plutonium atom), it does not follow that Archimedes
Plutonium would see the truth of it.
> So instead, I argue against their claim. I claim that physics
> doesn't necessarily require the existence of an infinite set,
> and possibly not arbitrarily large finite sets (googolplex,
> Graham's number) either. But I am willing to concede the
> existence of arbitrarily large finite sets if we can still have
> physics without infinite sets, so that at least some of the
> sci.math finitists will be satisfied.
>
> Maybe physicists who are finitists should just do the math
> needed for physics without complaining about the underlying
> mathematical theories? But in reality, as evidenced right here
> at sci.math, they don't.
>
>> [snip interesting but irrelevant bits from Zeilberger]
>> For what it's worth, I did not find Zeilberger's comments (and
>> especially his argument that Cohen was a finitist) to be persuasive, but
>> I appreciate the quotes.
>
> Typical. Hughes snips the quotes which imply that Cohen might
> be finitist, just as most ZFC users do when a finitist mentions
> Abraham Robinson. Then they'll turn around and trot out Cohen
> and Robinson when arguing in favor of ZFC or to prove that a
> poster is "wrong" using ZFC. (Cohen's name often appears when a
> poster tries to define aleph_1 as 2^aleph_0 or assumes CH
> without stating that assumption.)
I snipped the quote for brevity. No subterfuge involved.
And I've never "trotted out" either Cohen or Robinson.
If you think I've been dishonest, then have the balls to say so. Most
readers of my post, however, can easily find your post and read the
quotes in their entirety. I didn't care to respond to Zeilberger point
by point.
> But I will concede to Hughes that the claim that Cohen was a
> finitist is less believeable than the claim that Robinson was,
> since the former referred to the continuum as being very, very
> large, while Robinson merely referred to very small
> infinitesimals, rather than the size of the continuum.
Footnotes:
[1] AP of course does *not* come up with theories at all. He just
gives an ever-changing list of vague ideas. I'm not sure whether
Srinivasan has a theory or not and I'm not familiar with Zeilberger,
though he is, as far as I know, a respected mathematician.
--
Jesse F. Hughes
"This post marks the end of an era in the world of mathematics."
-- James S. Harris and the demise of Galois theory
David R Tribble
> existence of infinite sets, including the uncountable complete
> ordered field R and infinite-dimensional Hilbert spaces. This
> is a commonly made argument.
>
> But if their claim were correct, then why do see so many posters
> using _physics_ to argue that infinite sets don't exist? Why
> does AP use _physics_ to argue against infinite sets? Why does
> Herc use _physics_ to argue against infinite sets? Why does WM
> uses _physics_ to argue against infinite sets?
AP and WM believe that math is a subset of physics, i.e.,
that all meaningful math must have some physical basis.
For them, If something doesn't have some physical manifestation,
it's not bona fide math. (I don't know about the others.)
Of course they've got it completely inside-out. Math is a
far larger domain of discourse than the relatively small
disciplines of math used to model physics.
> [...] So instead, I argue against their claim. I claim that physics
> doesn't necessarily require the existence of an infinite set,
> and possibly not arbitrarily large finite sets (googolplex,
> Graham's number) either. But I am willing to concede the
> existence of arbitrarily large finite sets if we can still have
> physics without infinite sets, so that at least some of the
> sci.math finitists will be satisfied.
>
> Maybe physicists who are finitists should just do the math
> needed for physics without complaining about the underlying
> mathematical theories? But in reality, as evidenced right here
> at sci.math, they don't.
The problem goes much deeper than that. They are not
physicists, they are cranks who firmly believe that
standard math is wrong, and that their concoctions
will fix the problems with existing math and eventually
replace it. That much is obvious from reading only a handful
of their posts.
It's not the fact that they are finitists, it's the fact that they
believe that standard math is wrong ("bogus", "fake", "lies",
etc.) that's the problem.
FredJeffries
>
> > In all honesty, do you *really* think that you can come up with an
> > ultrafinite theory of numbers that will include all of the advantages of
> > real analysis as it is used in, say, physics?
>
> Combining this with a comment from Jeffries earlier in the thread:
>
> "But physics has found use for
> mathematical objects like self-adjoint operators on infinite
> dimensional Hilbert
> spaces to represent observables."
>
> Both Hughes and Jeffries here claim that physics requires the
> existence of infinite sets, including the uncountable complete
> ordered field R and infinite-dimensional Hilbert spaces.
OK children, your assignment for today is to find how many logical
fallacies Leonard committed in the preceding sentence.
FredJeffries
> On Sep 2, 9:57 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
> > There is no moment prior to noon at which the vase is empty.
> > Nonetheless, it is empty at noon. If you have any problems with this
> > simple, clear consequence, it is likely because the situation itself is
> > a bit silly (infinitely many tasks in a finite time).
>
> Sure. If it's not empty before noon, but then it is empty at noon,
> then it became empty at noon, right?
>
No. It is a BECOMING, a PROCESS. The vase is in the process of being
emptied at every point of time before noon (even though there are
balls being added to it -- just as a tank with a flow of two liters
per minute in and a flow of 5 liters per minute out is in the process
of emptying).
You do not get a description of a process by looking at snapshots any
more than an official can referee a sporting event by looking at
snapshots of the game action.