An uncountable countable set
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From: Virgil on 28 Oct 2006 15:48 In article <45435898(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> Tony Orlow wrote: > >>>> David Marcus wrote: > >>>>> Tony Orlow wrote: > >>>>>> stephen(a)nomail.com wrote: > >>>>>>> What are you talking about? I defined two sets. There are no > >>>>>>> balls or vases. There are simply the two sets > >>>>>>> > >>>>>>> IN = { n | -1/(2^floor(n/10)) < 0 } > >>>>>>> OUT = { n | -1/(2^n) < 0 } > >>>>>> For each n e N, IN(n)=10*OUT(n). > >>>>> Stephen defined sets IN and OUT. He didn't define sets "IN(n)" and "OUT > >>>>> (n)". So, you seem to be answering a question he didn't ask. Given > >>>>> Stephen's definitions of IN and OUT, is IN = OUT? > >>>> Yes, all elements are the same n, which are finite n. There is a simple > >>>> bijection. But, as in all infinite bijections, the formulaic > >>>> relationship between the sets is lost. > >>> Just to be clear, you are saying that |IN - OUT| = 0. Is that correct? > >>> (The vertical lines denote "cardinality".) > >> Um, before I answer that question, I think you need to define what you > >> mean by "|IN - OUT|" =0. How are you measuring IN and OUT, and how do > >> you define '-' on these "numbers"? > > > > IN and OUT are sets, not "numbers". For any two sets A and B, the > > difference, denoted by A - B, is defined to be the set of elements in A > > that are not in B. Formally, > > > > A - B := {x| x in A and x not in B} > > > > Note that the difference of two sets is again a set. For any set, the > > notation |A| means the cardinality of A. So, saying that |A| = 0 is > > equivalent to saying that A is the empty set. In particular, for any set > > A, we have |A - A| = 0. > > > > Sure, in the sense of containing the same n's, they are the same set. Quite right. And their equality as sets is all that is relevant here.
From: stephen on 28 Oct 2006 15:47 Tony Orlow <tony(a)lightlink.com> wrote: > stephen(a)nomail.com wrote: >> Tony Orlow <tony(a)lightlink.com> wrote: >>> stephen(a)nomail.com wrote: >>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>> stephen(a)nomail.com wrote: >>>>>> David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >>>>>>> Tony Orlow wrote: >>>>>>>> David Marcus wrote: >>>>>>>>> Your question "Is there a smallest infinite number?" lacks context. You >>>>>>>>> need to state what "numbers" you are considering. Lots of things can be >>>>>>>>> constructed/defined that people refer to as "numbers". However, these >>>>>>>>> "numbers" differ in many details. If you assume that all subjects that >>>>>>>>> use the word "number" are talking about the same thing, then it is >>>>>>>>> hardly surprising that you would become confused. >>>>>>>> I don't consider transfinite "numbers" to be real numbers at all. I'm >>>>>>>> not interested in that nonsense, to be honest. I see it as a dead end. >>>>>>>> >>>>>>>> If there is a definition for "number" in general, and for "infinite", >>>>>>>> then there cannot both be a smallest infinite number and not be. >>>>>>> A moot point, since there is no definition for "'number' in general", as >>>>>>> I just said. >>>>>>> -- >>>>>>> David Marcus >>>>>> 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. If someone were to ask "does there exist a smallest >>>>>> positive non-zero number?", the answer depends on what sort >>>>>> of "numbers" you are talking about. >>>>>> >>>>>> Stephen >>>>> Like, perhaps, the Finlayson Numbers? :) >>>> If they were sensibly defined then sure you could talk about them. >>>> Nothing Ross has ever said has made any sense to me, and >>>> I severely doubt there is any sense to it, but I could be wrong. >>>> The point is, there are different types of numbers, and statements >>>> that are true of one type of number need not be true of other >>>> types of numbers. >>>> >>>> Stephen >> >>> Well, then, you must be of the opinion that set theory is NOT the >>> foundation for all mathematics, but only some particular system of >>> numbers and ideas: a theory. That's good. >> >> You do not understand what people mean when they say set theory >> is the foundation for all mathematics, do you? Set theory >> provides a set of primitives which can be used to describe >> mathematics. Integers, real numbers, hyperreal numbers, imaginary >> numbers, polynomicals, limits, functions, can all be described in >> terms of set theory. Set theory is like assembly language. You can >> use it to build up higher level concepts. Is it the only possible >> foundation for mathematics? Of course not, but it currently appears to >> be the best. >> >> Stephen >> > "Best" in which respect? Most ridiculous when it comes to oo? Yes, it's > a riot. Lame response noted. But Robinson's infinite numbers that you so love are described in terms of set theory. So set theory is perfectly capable of describing "Tony approved" infinite numbers. It can also describe other types of infinite numbers. But it is rather egomaniacal to insist that a proper foundation for mathematics be capable of only describing things you approve. Stephen
From: David Marcus on 28 Oct 2006 15:52 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 th
From: Virgil on 28 Oct 2006 15:53 In article <45435925(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > stephen(a)nomail.com wrote: > > Tony Orlow <tony(a)lightlink.com> wrote: > >> stephen(a)nomail.com wrote: > >>> David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > >>>> Tony Orlow wrote: > >>>>> David Marcus wrote: > >>>>>> Your question "Is there a smallest infinite number?" lacks context. > >>>>>> You > >>>>>> need to state what "numbers" you are considering. Lots of things can > >>>>>> be > >>>>>> constructed/defined that people refer to as "numbers". However, these > >>>>>> "numbers" differ in many details. If you assume that all subjects that > >>>>>> use the word "number" are talking about the same thing, then it is > >>>>>> hardly surprising that you would become confused. > >>>>> I don't consider transfinite "numbers" to be real numbers at all. I'm > >>>>> not interested in that nonsense, to be honest. I see it as a dead end. > >>>>> > >>>>> If there is a definition for "number" in general, and for "infinite", > >>>>> then there cannot both be a smallest infinite number and not be. > >>>> A moot point, since there is no definition for "'number' in general", as > >>>> I just said. > >>>> -- > >>>> David Marcus > >>> 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. If someone were to ask "does there exist a smallest > >>> positive non-zero number?", the answer depends on what sort > >>> of "numbers" you are talking about. > >>> > >>> Stephen > > > >> Like, perhaps, the Finlayson Numbers? :) > > > > If they were sensibly defined then sure you could talk about them. > > Nothing Ross has ever said has made any sense to me, and > > I severely doubt there is any sense to it, but I could be wrong. > > The point is, there are different types of numbers, and statements > > that are true of one type of number need not be true of other > > types of numbers. > > > > Stephen > > Well, then, you must be of the opinion that set theory is NOT the > foundation for all mathematics. Non-sequitur. > but only some particular system of > numbers and ideas: a theory. Set theory is a valid basis for all sorts of systems numbers. The TO is unable to keep them separate in his mind is only a weakness of his mind, not of those systems that he cannot keep straight in his mind.
From: Tony Orlow on 28 Oct 2006 15:54
stephen(a)nomail.com wrote: > Tony Orlow <tony(a)lightlink.com> wrote: >> stephen(a)nomail.com wrote: > > <snip> > >>> What does that have to do with the sets IN and OUT? IN and OUT are >>> the same set. You claimed I was losing the "formulaic relationship" >>> between the sets. So I still do not know what you meant by that >>> statement. Once again >>> IN = { n | -1/(2^(floor(n/10))) < 0 } >>> OUT = { n | -1/(2^n) < 0 } >>> > >> I mean the formula relating the number In to the number OUT for any n. >> That is given by out(in) = in/10. > > What number IN? There is one set named IN, and one set named OUT. > There is no number IN. I have no idea what you think out(in) is > supposed to be. OUT and IN are sets, not functions. > OH. So, sets don't have sizes which are numbers, at least at particular moments. I see.... >>> Given that for every positive integer -1/(2^(floor(n/10))) < 0 >>> and -1/(2^n) < 0, both sets are in fact the same set, namely N. >>> >>> Do you agree, or not? Or is it the case that the >>> "formulaic between the sets is lost." >>> ? >>> >>> Stephen > >> The formulaic relationship is lost in that statement. When you state the >> relationship given any n, then the answer is obvious. > > What relationship? For a given n, -1/(2^(floor(n/10))) < 0 > if and only if -1/(2^n) < 0. The two conditions are logically > equivalent for positive integers. If n is a member of IN, > n is a member of OUT, and vice versa. > > What other relationship do you think there is between > -1/(2^(floor(n/10))) < 0 > and > -1/(2^n) < 0 > ?? Like, wow, Man, at, like, each moment, there's, like, 10 going in, and, like, Man, only 1 coming out. Seems kinda weird. There's, like, a rate thing going on.... :D > > Do you think there exists a positive integer n such that > -1/(2^(floor(n/10))) < 0 > and > -1/(2^n) >= 0 > > Stephen Hell no! |