An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: Lester Zick on 30 Oct 2006 17:19 On 30 Oct 2006 10:44:29 -0800, "Randy Poe" <poespam-trap(a)yahoo.com> wrote: > >Lester Zick wrote: >> On 30 Oct 2006 08:34:21 -0800, "Randy Poe" <poespam-trap(a)yahoo.com> >> wrote: >> >> > >> >Lester Zick wrote: >> >> On 28 Oct 2006 12:54:51 -0700, "Randy Poe" <poespam-trap(a)yahoo.com> >> >> wrote: >> >> >> >> > >> >> >Lester Zick wrote: >> >> >> On Fri, 27 Oct 2006 14:23:58 -0400, David Marcus >> >> >> <DavidMarcus(a)alumdotmit.edu> wrote: >> >> >> >> >> >> >Lester Zick wrote: >> >> >> >> On Fri, 27 Oct 2006 16:30:04 +0000 (UTC), stephen(a)nomail.com wrote: >> >> >> >> >A very simple example is that there exists a smallest positive >> >> >> >> >non-zero integer, but there does not exist a smallest positive >> >> >> >> >non-zero real. >> >> >> >> >> >> >> >> So non zero integers are not real? >> >> >> > >> >> >> >That's a pretty impressive leap of illogic. >> >> >> >> >> >> "Smallest integer" versus "no smallest real"? Seems pretty clear cut. >> >> > >> >> >You must be joking. I can't believe even you can be this dense. >> >> >> >> Oh I dunno. I can be pretty dense. Just not as dense as you, Randy, >> >> but that's nothing new. >> >> >> >> >Is 1 the smallest positive non-zero integer? Yes. >> >> > >> >> >Is it the smallest positive non-zero real? No. 1/10 is smaller. >> >> >Ah well, then is 1/10 the smallest positive non-zero real? No, >> >> >1/100 is smaller. Is that the smallest? No, 1/1000 is smaller. >> >> > >> >> >Does that second sequence have an end? Can I eventually >> >> >find a smallest positive non-zero real? >> >> > >> >> >How about the first? Is there something smaller than 1 which >> >> >is a positive non-zero integer? >> >> >> >> See the problem here, Randy, is that you're explaining an issue I >> >> didn't raise then pretending you're addressing the issue I raised. >> > >> >I'm providing explicit descriptions of why, as the original >> >quote said, there is a least positive non-zero integer, >> >but not a least positive non-zero real. >> >> Who cares? That has nothing to do with the question I asked. > >The question you asked was whether the fact that there is >a smallest non-positive integer (i.e., 1) but no smallest >non-positive real implies that integers are not real. > >The answer is no. Thanks, slick. Then maybe you can show us all exactly how it is that the propositions "least integer" "no least real" imply that? >> >> I >> >> don't doubt there is no smallest real except in the case of integers. >> >> But that is not what was said originally. What was said is that there >> >> is a least integer but no least real. >> > >> >Since the original text is above, we can actually see whether >> >that is what was said. >> >"A very simple example is that there exists a smallest positive >> >non-zero integer, but there does not exist a smallest positive >> >non-zero real." >> > >> >Are you equating "least integer" with "smallest positive >> >non-zero integer" and "least real" with "smallest >> >positive non-zero real"? >> >> It certainly seemed to me that I was just asking a question regarding >> implications of the point made. And I reinterpreted that question just >> above for clarification. > >Well, it is not so implied. OK? Well then maybe you could do us all the courtesy of explaining why it is not so implied? > There is indeed a smallest >non-positive integer. It is 1. There is no smallest non-positive real. >That does not imply the integers are non-real. It doesn't? My mistake. So there's "no least real" but a "least integer real"? Hmmm. Curiouser and curiouser. ~v~~
From: Lester Zick on 30 Oct 2006 17:22 On 30 Oct 2006 10:46:23 -0800, "Randy Poe" <poespam-trap(a)yahoo.com> wrote: > >Lester Zick wrote: >> On 30 Oct 2006 08:34:21 -0800, "Randy Poe" <poespam-trap(a)yahoo.com> >> wrote: >> >> > >> >Lester Zick wrote: >> >> On 28 Oct 2006 12:54:51 -0700, "Randy Poe" <poespam-trap(a)yahoo.com> >> >> wrote: >> >"There is a least integer" and "there is a least real" >> >are both false. >> >> They are? > >Yes. If you disagree, perhaps you can name me the minimum >element of the sets Z and R. Or perhaps you can show us how it is "there is a least integer" and "there is a least real" are both false since it's your contention not mine. >> Perhaps you should take that up with theologians then. > >We are discussing mathematics. In the mathematical objects >called "the set of integers" and "the set of reals" there is no >least member. Well perhaps you can just prove that since it's your contention not mine? ~v~~
From: David Marcus on 30 Oct 2006 17:27 Tony Orlow wrote: > David Marcus wrote: > > 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)? > > > > Find limits of formulas on numbers, not limits of sets. I have no clue what you mean. There are no "limits of sets" in what I wrote. > Here's what I said to Stephen: > > out(n) is the number of balls removed upon completion of iteration n, > and is equal to n. > > in(n) is the number of balls inserted upon completion of iteration n, > and is equal to 10n. > > contains(n) is the number of balls in the vase upon completion of > iteration n, and is equal to in(n)-out(n)=9n. > > n(t) is the number of iterations completed at time t, equal to floor(-1/t). > > contains(t) is the number of balls in the vase at time t, and is equal > to contains(n(t))=contains(floor(-1/t))=9*floor(-1/t). > > Lim(t->-0: 9*floor(-1/t)))=oo. The sum diverges in the limit. You seem to be agreeing with what I wrote, i.e., that you say that the number of balls in the vase at noon is lim_{t -> 0-} V(t). Care to confirm this? -- David Marcus
From: David Marcus on 30 Oct 2006 17:29 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
From: David Marcus on 30 Oct 2006 17:32
Lester Zick wrote: > On Sun, 29 Oct 2006 15:10:18 -0500, David Marcus > <DavidMarcus(a)alumdotmit.edu> wrote: > > >Lester Zick wrote: > >> On 28 Oct 2006 12:54:51 -0700, "Randy Poe" <poespam-trap(a)yahoo.com> > >> wrote: > >> > >> > > >> >Lester Zick wrote: > >> >> On Fri, 27 Oct 2006 14:23:58 -0400, David Marcus > >> >> <DavidMarcus(a)alumdotmit.edu> wrote: > >> >> > >> >> >Lester Zick wrote: > >> >> >> On Fri, 27 Oct 2006 16:30:04 +0000 (UTC), stephen(a)nomail.com wrote: > >> >> >> >A very simple example is that there exists a smallest positive > >> >> >> >non-zero integer, but there does not exist a smallest positive > >> >> >> >non-zero real. > >> >> >> > >> >> >> So non zero integers are not real? > >> >> > > >> >> >That's a pretty impressive leap of illogic. > >> >> > >> >> "Smallest integer" versus "no smallest real"? Seems pretty clear cut. > >> > > >> >You must be joking. I can't believe even you can be this dense. > >> > >> Oh I dunno. I can be pretty dense. Just not as dense as you, Randy, > >> but that's nothing new. > >> > >> >Is 1 the smallest positive non-zero integer? Yes. > >> > > >> >Is it the smallest positive non-zero real? No. 1/10 is smaller. > >> >Ah well, then is 1/10 the smallest positive non-zero real? No, > >> >1/100 is smaller. Is that the smallest? No, 1/1000 is smaller. > >> > > >> >Does that second sequence have an end? Can I eventually > >> >find a smallest positive non-zero real? > >> > > >> >How about the first? Is there something smaller than 1 which > >> >is a positive non-zero integer? > >> > >> See the problem here, Randy, is that you're explaining an issue I > >> didn't raise then pretending you're addressing the issue I raised. I > >> don't doubt there is no smallest real except in the case of integers. > >> But that is not what was said originally. What was said is that there > >> is a least integer but no least real. Now these strike me as mutually > >> exclusive predicates. But then who am I to analyze mathematical > >> predicates in logical terms especially when there are self righteous > >> neomathematikers around who prefer to specialize in name calling > >> rather than keep their arguments straight in reply to simple queries. > > > >I'll probably regret asking, but what the heck. Are you saying that the > >following two statements are contradictory? > > > >1. There is a smallest positive integer. > >2. There is no smallest positive real. > > No. In response to these two propositions I'm simply asking whether > you consider integers real? Yes, integers are real. > Or you might try reading what I originally > asked which you pronounced illogical without apparently bothering to > read what I wrote. You wrote, "So non zero integers are not real?". I've no idea why you would think that. It doesn't seem to follow from anything that was said. -- David Marcus |