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In article <454771cc@xxxxxxxxxxxxxxxxxxx>,
Tony Orlow <tony@xxxxxxxxxxxxx> wrote:
> David Marcus wrote:
> > Tony Orlow wrote:
> >> David Marcus wrote:
> >>> Tony Orlow wrote:
> >>>> David Marcus wrote:
> >>>>> Tony Orlow wrote:
> >>>>>> David Marcus wrote:
> >>>>>>> Tony Orlow wrote:
> >>>>>>>> David Marcus wrote:
> >>>>>>>>> Tony Orlow wrote:
> >>>>>>>>>> David Marcus wrote:
> >>>>>>>>>>> Tony Orlow wrote:
> >>>>>>>>>>>> David Marcus wrote:
> >>>>>>>>>>>>> Tony Orlow wrote:
> >>>>>>>>>>>>>> David Marcus wrote:
> >>>
> >>>>>>>>>>>>>>> You are mentioning balls and time and a vase. But, what
> >>>>>>>>>>>>>>> I'm asking is completely separate from that. I'm just
> >>>>>>>>>>>>>>> asking about a math problem. Please just consider the
> >>>>>>>>>>>>>>> following mathematical definitions and completely ignore
> >>>>>>>>>>>>>>> that they may or may not be relevant/related/similar to
> >>>>>>>>>>>>>>> the vase and balls problem:
> >>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>> --------------------------
> >>>>>>>>>>>>>>> For n = 1,2,..., let
> >>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>> A_n = -1/floor((n+9)/10),
> >>>>>>>>>>>>>>> R_n = -1/n.
> >>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>> For n = 1,2,..., define a function B_n: R -> R by
> >>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>> B_n(t) = 1 if A_n <= t < R_n,
> >>>>>>>>>>>>>>> 0 if t < A_n or t >= R_n.
> >>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>> Let V(t) = sum_n B_n(t).
> >>>>>>>>>>>>>>> --------------------------
> >>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>> Just looking at these definitions of sequences and
> >>>>>>>>>>>>>>> functions from R (the real numbers) to R, and assuming
> >>>>>>>>>>>>>>> that the sum is defined as it would be in a Freshman
> >>>>>>>>>>>>>>> Calculus class, are you saying that V(0) is not equal to
> >>>>>>>>>>>>>>> 0?
> >>>>>>>>>>>>>> On the surface, you math appears correct, but that doesn't
> >>>>>>>>>>>>>> mend the obvious contradiction in having an event occur in
> >>>>>>>>>>>>>> a time continuum without occupying at least one moment. It
> >>>>>>>>>>>>>> doesn't explain how a divergent sum converges to 0.
> >>>>>>>>>>>>>> Basically, what you prove, if V(0)=0, is that all finite
> >>>>>>>>>>>>>> naturals are removed by noon. I never disagreed with that.
> >>>>>>>>>>>>>> However, to actually reach noon requires infinite
> >>>>>>>>>>>>>> naturals. Sure, if V is defined as the sum of all finite
> >>>>>>>>>>>>>> balls, V(0)=0. But, I've already said that, several times,
> >>>>>>>>>>>>>> haven't I? Isn't that an answer to your question?
> >>>>>>>>>>>>> I think it is an answer. Just to be sure, please confirm
> >>>>>>>>>>>>> that you agree that, with the definitions above, V(0) = 0.
> >>>>>>>>>>>>> Is that correct?
> >>>>>>>>>>>> Sure, all finite balls are gone at noon.
> >>>>>>>>>>> Please note that there are no balls or time in the above
> >>>>>>>>>>> mathematics problem. However, I'll take your "Sure" as
> >>>>>>>>>>> agreement that V(0) = 0.
> >>>>>>>>>> Okay.
> >>>>>>>>>>> Let me ask you a question about this mathematics problem.
> >>>>>>>>>>> Please answer without using the words "balls", "vase",
> >>>>>>>>>>> "time", or "noon" (since these words do not occur in the
> >>>>>>>>>>> problem).
> >>>>>>>>>> I'll try.
> >>>>>>>>>>> First some discussion: For each n, B_n(0) = 0 and B_n is
> >>>>>>>>>>> continuous at zero.
> >>>>>>>>>> What??? How do you conclude that anything besides time is
> >>>>>>>>>> continuous at 0, where yo have an ordinal discontinuity????
> >>>>>>>>>> Please explain.
> >>>>>>>>> I thought we agreed above to not use the word "time" in
> >>>>>>>>> discussing this mathematics problem?
> >>>>>>>> If that's what you want, then why don't you remove 't' from all
> >>>>>>>> of your equations?
> >>>>>>> It is just a letter. It stands for a real number. Would you
> >>>>>>> prefer "x"? I'll switch to "x".
> >>>>>> It is still related to n in such a way that x<0.
> >>>>>>>>> As for your question, let's look at B_2 (the argument is
> >>>>>>>>> similar for the other B_n).
> >>>>>>>>>
> >>>>>>>>> B_2(t) = 1 if A_2 <= t < R_2,
> >>>>>>>>> 0 if t < A_2 or t >= R_2.
> >>>>>>>>>
> >>>>>>>>> Now, A_2 = -1 and R_2 = -1/2. So,
> >>>>>>>>>
> >>>>>>>>> B_2(t) = 1 if -1 <= t < -1/2,
> >>>>>>>>> 0 if t < -1 or t >= -1/2.
> >>>>>>>>>
> >>>>>>>>> In particular, B_2(t) = 0 for t >= -1/2. So, the value of B_2
> >>>>>>>>> at zero is zero and the limit as we approach zero is zero. So,
> >>>>>>>>> B_2 is continuous at zero.
> >>>>>>>> Oh. For each ball, nothing is happening at 0 and B_n(0)=0.
> >>>>>>>> That's for each finite ball that one can specify.
> >>>>>>> I thought we agreed to not use the word "ball" in discussing this
> >>>>>>> mathematics problem? Do you want me to change the letter "B" to a
> >>>>>>> different letter, too?
> >>>>>> Call it an element or a ball. I don't care. It doesn't matter.
> >>>>>>>> However, lim(t->0: sum(B_n| B_n(t)=1))=oo. Why do you
> >>>>>>>> conveniently forget that fact?
> >>>>>>> Your notation is nonstandard, so I'm not sure what you mean. Do
> >>>>>>> you mean to write
> >>>>>>>
> >>>>>>> lim_{x -> 0-} sum_n B_n(x) = oo ?
> >>>>>>>
> >>>>>>> If so, I don't understand why you think I've forgotten this fact.
> >>>>>>> If you look in my previous post (or below), you will see that I
> >>>>>>> wrote, "Now, V is the sum of the B_n. As t approaches zero from
> >>>>>>> the left, V(t) grows without bound. In fact, given any large
> >>>>>>> number M, there is an e < 0 such that for e < t < 0, V(t) > M."
> >>>>>> Then don't you see a contradiction in the limit at that point
> >>>>>> being oo, the value being 0, and there being no event to cause
> >>>>>> that change? I do.
> >>>>>>>>>>> In fact, for a given n, there is an e < 0 such that B_n(t) =
> >>>>>>>>>>> 0 for e < t <= 0.
> >>>>>>>>>> There is no e<0 such that e<t and B_n(t)=0. That's simply false.
> >>>>>>>>> Let's look at B_2 again. We can take e = -1/2. Then B_2(t) = 0
> >>>>>>>>> for e < t <= 0. Similarly, for any other given B_n, we can find
> >>>>>>>>> an e that does what I wrote.
> >>>>>>>> Yes, okay, I misread that. Sorry. For each ball B_n that's true.
> >>>>>>>> For the sum of balls n such that B_n(t)=1, it diverges as t->0.
> >>>>>>>>>>> In other words, B_n is not changing near zero.
> >>>>>>>>>> Infinitely more quickly but not. That's logical. And wrong.
> >>>>>>>>> Not sure what you mean.
> >>>>>>>> The sum increases without bound.
> >>>>>>>>>>> Now, V is the sum of the B_n. As t approaches zero from the
> >>>>>>>>>>> left, V(t) grows without bound. In fact, given any large
> >>>>>>>>>>> number M, there is an e < 0 such that for e < t < 0, V(t) >
> >>>>>>>>>>> M. We also have that V(0) = 0 (as you agreed).
> >>>>>>>>>>>
> >>>>>>>>>>> Now the question: How do you explain the fact that V(t) goes
> >>>>>>>>>>> from being very large for t a little less than zero to being
> >>>>>>>>>>> zero when t equals zero even though none of the B_n are
> >>>>>>>>>>> changing near zero?
> >>>>>>>>>> I'll consider answering that when you correct the errors
> >>>>>>>>>> above. Sorry.
> >>>>>>> I believe we now agree that what I wrote is correct. So, let me
> >>>>>>> repeat my question:
> >>>>>>>
> >>>>>>> How do you explain the fact that V(x) goes from being very large
> >>>>>>> for x a little less than zero to being zero when x equals zero
> >>>>>>> even though none of the functions B_n are changing near zero?
> >>>>>> How do I account for it? I don't, because I don't believe it's
> >>>>>> true. I "account" for it by saying there is a logical flaw in the
> >>>>>> argument that says so. That's what I've been saying all along.
> >>>>>> Cardinality is not a fine enough measure even to detect the
> >>>>>> different of a finite fraction of an infinite set, much less the
> >>>>>> finite addition or subtraction applied to an infinite set.
> >>>>> When I posed this mathematics (Calculus) problem (see the text
> >>>>> quoted above), the first thing I asked you was what the value of
> >>>>> V(0) was. You replied that it was zero. I asked you to confirm that
> >>>>> you meant that V(0) = 0. You replied "Sure".
> >>>>> Now, you say that you "don't believe it's true". Does this mean you
> >>>>> now say V(0) <> 0? Have you changed your mind?
> >>>> Sorry, I wasn't clear. I don't believe that V(0) represents all the
> >>>> balls in the vase at t=0 in the original problem.
> >>> Tony,
> >>>
> >>> If you reread what is written above, you will see that the very first
> >>> thing I wrote is "You are mentioning balls and time and a vase. But,
> >>> what I'm asking is completely separate from that. I'm just asking
> >>> about a math problem. Please just consider the following mathematical
> >>> definitions and completely ignore that they may or may not be
> >>> relevant/related/similar to the vase and balls problem."
> >>>
> >>> So, please do as I asked and consider this as a completely separate
> >>> math problem. There are no balls, vases, time, or noon. Here is the
> >>> problem again (with slightly changed notation):
> >>>
> >>> For j = 1,2,..., let
> >>>
> >>> a_j = -1/floor((j+9)/10),
> >>> b_j = -1/j.
> >>>
> >>> For j = 1,2,..., define a function f_j: R -> R by
> >>>
> >>> f_j(x) = 1 if a_j <= x < b_j,
> >>> 0 if x < a_j or x >= b_j.
> >>>
> >>> Let g(x) = sum_j f_j(x). What is g(0)?
> >>>
> >>> Do you say that g(0) = 0?
> >> No, I say that lim(x->0: g(x))=oo.
> >
> > I assume you mean lim_{x->0-} g(x) = oo. However, that isn't the
> > question I asked. The question I asked is "What is g(0)?". Please answer
> > the question that I asked.
> >
> oo
So now TO is claiming that the sum of infinitely many zeros adds up to
more than any finite quantity?
All his Reimann integrals must diverge.
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