An uncountable countable set
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From: Tony Orlow on 28 Oct 2006 19:47 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
From: David Marcus on 28 Oct 2006 19:57 Tony Orlow wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> I am beginning to realize just how much trouble the axiom of > >> extensionality is causing here. That is what you're using, here, no? The > >> sets are "equal" because they contain the same elements. That gives no > >> measure of how the sets compare at any given point in their production. > >> Sets as sets are considered static and complete. However, when talking > >> about processes of adding and removing elements, the sets are not > >> static, but changing with each event. When speaking about what is in the > >> set at time t, use a function for that sum on t, assume t is continuous, > >> and check the limit as t->0. Then you won't run into silly paradoxes and > >> unicorns. > > > > There is a lot of stuff in there. Let's go one step at a time. I believe > > that one thing you are saying is this: > > > > |IN\OUT| = 0, but defining IN and OUT and looking at |IN\OUT| is not the > > correct translation of the balls and vase problem into Mathematics. > > > > Do you agree with this statement? > > Yes. OK. Since you don't like the |IN\OUT| translation, let's see if we can take what you wrote, translate it into Mathematics, and get a translation that you like. You say, "When speaking about what is in the set at time t, use a function for that sum on t, assume t is continuous, and check the limit as t->0." Taking this one step at a time, first we have "use a function for that sum on t". How about we use the function V defined as follows? 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 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). Next you say, "assume t is continuous". Not sure what you mean. Maybe you mean assume the function is continuous? However, it seems that either the function we defined (e.g., V) is continuous or it isn't, i.e., it should be something we deduce, not assume. Let's skip this for now. I don't think we actually need it. Finally, you write, "check the limit as t->0". I would interpret this as saying that we should evaluate the limit of V(t) as t approaches zero from the left, i.e., lim_{t -> 0-} V(t). Do you agree that you are saying that the number of balls in the vase at noon is lim_{t -> 0-} V(t)? -- David Marcus
From: David Marcus on 28 Oct 2006 20:15 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 corr
From: Virgil on 29 Oct 2006 01:02 In article <4543ec1b(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > > 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. Which balls from the original problem does it omit? The only relevant question in the original gedankenexperiment is "According to the rules of the the problem, is each ball which is inserted into the vase before noon also removed from the vase strictly before noon?" The statement of the problem requires an affirmative answer. So TO's response is to invent mythical impossibly numbered balls that appear out of nowhere at noon to fill the supposed vacuum he alleges is caused by the regular balls being removed. While nature allegedly abhors a vacuum, this problem does not take place in that physical world.
From: Ross A. Finlayson on 29 Oct 2006 11:14
Tony Orlow wrote: > imaginatorium(a)despammed.com wrote: > > Tony Orlow wrote: > >> Virgil wrote: > >>> In article <454286e8(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>> Like, perhaps, the Finlayson Numbers? :) > >>> Any set of numbers whose properties are known. Are the properties of > >>> "Finlayson Numbers" known to anyone except Ross himself? > >> Uh, yeah, I think I understand what his numbers are. Perhaps you've seen > >> our recent exchange on the matter? They are discrete infinitesimals such > >> that the sequence of them within the unit interval maps to the naturals > >> or integers on the real line. Is that about right, Ross? > > > > Do they form a field? > > > > Brian Chandler > > http://imaginatorium.org > > > > Good question. Ross? What says you to this? > > Here's what Wolfram says applies to fields: > http://mathworld.wolfram.com/FieldAxioms.html > > My understanding, looking at each of these axioms, is that they apply to > this system, and that it's a field. I suppose you would want proof of > each such fact, but perhaps you could move the process along by > suggesting which of the ten axioms you think the Finlayson Numbers might > violate? After all, if you find only one, then you've proved your point. > Not that I am necessarily concerned with whether they form a ring or a > field or whatever, until that becomes important. Is it? Why the question? > > Tony It's a point of consideration that without some ball labelled infinity the process doesn't complete, for the completion of the "supertask." Consider Achilles and the tortoise, again, every distance between the start and finish line is covered by each, and ten times as much by Achilles. Virgil is a troll, 'tis true. The "Finlayson numbers" as coined by some other fellow, contain all the numbers in the "Finlayson numerical model", named by somebody else, the origin is (0, 0, 0, ....). That about sums them. The "Finlayson real numbers", or as I generally call them the "real numbers", have characteristics of being at once complete ordered field, and contiguous points on a line. In terms of their scalar value, those "indefinite" reals are defined as the immdiate neighbors, and "definite" reals basically as Dedekind/Cauchy, which is insufficient (using Dedekind cuts / Cauchy sequences, the "standard" method) to describe all real numbers. The real numbers are, macroscopically complete ordered field, and microscopically partially ordered ring, and I see there being something along the lines of a "rather restricted transfer principle", in the words of Schmieden and Laugwitz, in terms of transitions or transitive application of their form as predicates. With a least positive real, for example as is illustrated in a counterexample to standard real analysis, Dedekinf/Cauchy can't be sufficient to describe a real number. There is no set of numbers in ZF. There is no universe in ZF, that is sufficient reason for many to abandon regularity and promote alternative theories with alternative resolution of the Russell, Cantor, and other "paradoxes" seen to result otherwise from unrestricted comprehension. There are understandably less who would say that regularity is a "false axiom", I do where there are only primary objects in a pure object theory, saying that unrestricted comprehension is natural. The null axiom theory has as primary objects variously sets or numbers and geometric forms. There are only and everywhere real numbers between zero and one. Ross |