An uncountable countable set
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From: Lester Zick on 29 Oct 2006 19:49 On Sun, 29 Oct 2006 15:11:48 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >Lester Zick wrote: >> On Sat, 28 Oct 2006 20:33:16 +0000 (UTC), stephen(a)nomail.com wrote: >> >> >Tony Orlow <tony(a)lightlink.com> wrote: >> >> imaginatorium(a)despammed.com wrote: >> >>> Tony Orlow wrote: >> > >> ><snip> >> > >> >>>> The formulaic relationship is lost in that statement. When you state the >> >>>> relationship given any n, then the answer is obvious. >> >>> >> >>> Do "state the relationship given any n"... I mean, what is it, exactly? >> >>> >> > >> >> Uh, here it is again. in(n)=10n. out(n)=n. contains(n)=in(n)-out(n)=9n. >> >> lim(n->oo: contains(n))=oo. Basta cosi? >> > >> > >> >What is in(n)? The sets I and everyone but you are talking about are >> > IN = { n | -1/2^(floor(n/10)) < 0 } >> > OUT = { n | -1/2^n < 0 } >> >Noone has ever mentioned or defined in(n) >> > >> >What is the definition of in(n)? Is is a set? >> >> According to MoeBlee mathematical definitions require a "domain of >> discourse" variable such as IN(x) and OUT(x). > >I think you've used this joke enough already. Why don't you come up with >a new one? Mainly because everyone seems to want to ignore the point Moe raised regarding mathematical definitions. It would seem either Moe is right or Stephen (I think) drew an improper mathematical definition or Moe is not right. The last time around it was Stephen who was telling me that dr is velocity. Who did what is unimportant. I've seen both types of mathematical definition and I don't know as either is absolutely correct to the exclusion of others. But it does seem curious in such a didactic domain of discourse as mathematical definition one should be unable to tell which if either is which. You bust my chops over every nickel and dime mathematical issue passing my lips with accusations of trolling for no better reason than you decline to pay close attention to any point I raise and demand I come up with new jokes. Maybe you should come up with a few new jokes besides the same old modern math. ~v~~
From: Lester Zick on 29 Oct 2006 20:06 On 29 Oct 2006 12:27:33 -0800, "Ross A. Finlayson" <raf(a)tiki-lounge.com> wrote: >Lester Zick wrote: >> On Fri, 27 Oct 2006 20:08:18 -0400, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >> >> > The trick here is to >> >declare, as Ross aludes, to a "universe", a complete range of values for >> >the set or sets, and measure according to that "range". >> >> The problem here is that with respect to sets there is no complete >> range of values which includes both finites and infinites. There is no >> closed set of naturals in this sense. Yet every value in naturals is >> finite. In other words there is no [1, 2, 3 . . . 00].Mathematikers >> seem to want to call the set of naturals infinite but that can only >> refer to application of the generating mechanism "1+1". It doesn't >> mean the contents generated are or can be infinite. And as far as I'm >> concerned there is no way to generate infinities except through >> infinitesimal subdivision because mathematikers can't define their >> idea of infinity through "1+1" because that only produces finites. >> > >Various considerations of the natural integers have there being a point >at infinity. That can be useful in a form of nonstandard analysis, to >say that, for example, an infinite sum with a limit actually equals in >evaluation that limit. Consider for example something along the lines >of > > 1 >s = ----- > 2^n > >for n from zero to infinity. The limit exists and is two and then >consider replacing the 2 in the denominator with s. When you can >actually say that the sum equals that limit, you get the same >expression for s. > >Otherwise, s could never equal 2. > >There are other reasons to consider the infinite naturals as containing >an infinite element, that N E N. > >In these arguments here, if there is no infinite value for n, then the >process never completes. Re Zeno, the arrow goes half and then half >and then half again ad infinitum to reach the mark, that it does is >well-known. That's similar to the above equation with the domain over >(1, 2, ... (infinitely many times). The distance is one, the arrow >travels the distance, the distance can be decomposed in that manner to >partial distances, thus the sum is the distance. > >Similarly for the real numbers there are considerations of points at >infinity, in the "projectively extended" real numbers for example. >Similarly as to how division by zero in the "complete" ordered field is >undefined, ie that every other number than zero has a multiplicative >inverse, there is some consideration and ready application of there >being points at infinity, and meaningfully that their multiplicative >inverses are defined in similar ways as zero's. > >There are applications in integral transforms and so on for points at >infinity, for which the "standard" real numbers are insufficient, and >they could only be real numbers. > >When you have lim n-> oo, that "oo" is an "infinity", it's right there >in the expression. With lim x-> 0+ or lim x-> 0-, those can differ >with the sign, and do, and in signed numeric formats the sign takes a >bit, and in some there are dual representations of zero, >mechanistically different. > >Ross ~v~~
From: Lester Zick on 29 Oct 2006 20:32 Oops. My bad. Hit the wrong button on the duplicate reply. - LZ On 29 Oct 2006 12:27:33 -0800, "Ross A. Finlayson" <raf(a)tiki-lounge.com> wrote: >Lester Zick wrote: >> On Fri, 27 Oct 2006 20:08:18 -0400, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >> >> > The trick here is to >> >declare, as Ross aludes, to a "universe", a complete range of values for >> >the set or sets, and measure according to that "range". >> >> The problem here is that with respect to sets there is no complete >> range of values which includes both finites and infinites. There is no >> closed set of naturals in this sense. Yet every value in naturals is >> finite. In other words there is no [1, 2, 3 . . . 00].Mathematikers >> seem to want to call the set of naturals infinite but that can only >> refer to application of the generating mechanism "1+1". It doesn't >> mean the contents generated are or can be infinite. And as far as I'm >> concerned there is no way to generate infinities except through >> infinitesimal subdivision because mathematikers can't define their >> idea of infinity through "1+1" because that only produces finites. >> > >Various considerations of the natural integers have there being a point >at infinity. That can be useful in a form of nonstandard analysis, to >say that, for example, an infinite sum with a limit actually equals in >evaluation that limit. Consider for example something along the lines >of > > 1 >s = ----- > 2^n > >for n from zero to infinity. The limit exists and is two and then >consider replacing the 2 in the denominator with s. When you can >actually say that the sum equals that limit, you get the same >expression for s. > >Otherwise, s could never equal 2. Okay. I take the point, Ross. But what rule is there requiring 00 to be part of the same set as finites raised to a power of infinity? I think the power of infinity could be defined using the "number of infinitesimals" which is reciprocally defined with differentiation. >There are other reasons to consider the infinite naturals as containing >an infinite element, that N E N. N E N? >In these arguments here, if there is no infinite value for n, then the >process never completes. Well infinitesimal subdivision certainly never completes. > Re Zeno, the arrow goes half and then half >and then half again ad infinitum to reach the mark, that it does is >well-known. That's similar to the above equation with the domain over >(1, 2, ... (infinitely many times). The distance is one, the arrow >travels the distance, the distance can be decomposed in that manner to >partial distances, thus the sum is the distance. But the problem with Zeno's paradoxes is that what is established primarily is that the arrow or whatever goes the distance first. In other words the paradox establishes unity and attempts to subdivide the unity infinitesimally then claims the unity cannot exist unless the process of infinitesimal subdivision can complete and be finite. Infinitesimal subdivision is not a finite process. It's only used as a method of drawing tangents for the purpose of comparing otherwise infinites. >Similarly for the real numbers there are considerations of points at >infinity, in the "projectively extended" real numbers for example. >Similarly as to how division by zero in the "complete" ordered field is >undefined, ie that every other number than zero has a multiplicative >inverse, there is some consideration and ready application of there >being points at infinity, and meaningfully that their multiplicative >inverses are defined in similar ways as zero's. In other words people are just trying to make zero a natural number. It isn't. It has very specialized restrictions when it comes to use in arithmetic contexts. >There are applications in integral transforms and so on for points at >infinity, for which the "standard" real numbers are insufficient, and >they could only be real numbers. Well the so called "points at infinity" don't have to be defined as part of the same set as natural numbers. Natural numbers are finite. Infinites are not. They don't require combination in a common set. Everything supposedly true of infinitesimals is reciprocally true of infinitesimals and not the set of naturals. >When you have lim n-> oo, that "oo" is an "infinity", it's right there >in the expression. With lim x-> 0+ or lim x-> 0-, those can differ >with the sign, and do, and in signed numeric formats the sign takes a >bit, and in some there are dual representations of zero, >mechanistically different. I'm not quite sure yet what the mechanical implications of zero are. But basically if you're going to integrate something that something has been infinitesimally differentiated and if not you aren't doing definite integration. Either way the zero and the infinity defined refer to the limits of infinitesimal integration and not to natural numbers. But I certainly appreciate your considered comments. ~v~~
From: David Marcus on 29 Oct 2006 20:45 Lester Zick wrote: > On Sun, 29 Oct 2006 15:11:48 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > >Lester Zick wrote: > >> According to MoeBlee mathematical definitions require a "domain of > >> discourse" variable such as IN(x) and OUT(x). > > > >I think you've used this joke enough already. Why don't you come up with > >a new one? > > Mainly because everyone seems to want to ignore the point Moe raised > regarding mathematical definitions. It would seem either Moe is right > or Stephen (I think) drew an improper mathematical definition or Moe > is not right. Or, you misunderstood what Moe said. I would think that would be the heavy favorite. > The last time around it was Stephen who was telling me > that dr is velocity. Who did what is unimportant. I've seen both types > of mathematical definition and I don't know as either is absolutely > correct to the exclusion of others. But it does seem curious in such a > didactic domain of discourse as mathematical definition one should be > unable to tell which if either is which. You bust my chops over every > nickel and dime mathematical issue passing my lips with accusations of > trolling for no better reason than you decline to pay close attention > to any point I raise and demand I come up with new jokes. Maybe you > should come up with a few new jokes besides the same old modern math. I never said you were trolling. I asked someone else whether they thought you were trolling. In truth, I don't think you are trolling. So, I guess you are going to stick with this joke for a while. -- David Marcus
From: Ross A. Finlayson on 29 Oct 2006 21:15
Lester Zick wrote: > Oops. My bad. Hit the wrong button on the duplicate reply. - LZ > De nada. >> ... > > > >Various considerations of the natural integers have there being a point > >at infinity. That can be useful in a form of nonstandard analysis, to > >say that, for example, an infinite sum with a limit actually equals in > >evaluation that limit. Consider for example something along the lines > >of > > > > 1 > >s = ----- > > 2^n > > > >for n from zero to infinity. The limit exists and is two and then > >consider replacing the 2 in the denominator with s. When you can > >actually say that the sum equals that limit, you get the same > >expression for s. > > > >Otherwise, s could never equal 2. > > Okay. I take the point, Ross. But what rule is there requiring 00 to > be part of the same set as finites raised to a power of infinity? I > think the power of infinity could be defined using the "number of > infinitesimals" which is reciprocally defined with differentiation. > > >There are other reasons to consider the infinite naturals as containing > >an infinite element, that N E N. > > N E N? > > >In these arguments here, if there is no infinite value for n, then the > >process never completes. > > Well infinitesimal subdivision certainly never completes. > Les, Lester, there is some consideration that it does. Is not the differential intuitively the atomic subdivision of one? In an interesting way, variously on what you consider interesting, the notion of subatomic particles in physics is a similar one as the consideration of sub-iota reals in mathematics. > > Re Zeno, the arrow goes half and then half > >and then half again ad infinitum to reach the mark, that it does is > >well-known. That's similar to the above equation with the domain over > >(1, 2, ... (infinitely many times). The distance is one, the arrow > >travels the distance, the distance can be decomposed in that manner to > >partial distances, thus the sum is the distance. > > But the problem with Zeno's paradoxes is that what is established > primarily is that the arrow or whatever goes the distance first. In > other words the paradox establishes unity and attempts to subdivide > the unity infinitesimally then claims the unity cannot exist unless > the process of infinitesimal subdivision can complete and be finite. > Infinitesimal subdivision is not a finite process. It's only used as a > method of drawing tangents for the purpose of comparing otherwise > infinites. > Right, it's not a "finite" process in the sense that there are finitely many integers, but I think you would agree that it is a "finite" process to the extent that the above description is self-contained. > >Similarly for the real numbers there are considerations of points at > >infinity, in the "projectively extended" real numbers for example. > >Similarly as to how division by zero in the "complete" ordered field is > >undefined, ie that every other number than zero has a multiplicative > >inverse, there is some consideration and ready application of there > >being points at infinity, and meaningfully that their multiplicative > >inverses are defined in similar ways as zero's. > > In other words people are just trying to make zero a natural number. > It isn't. It has very specialized restrictions when it comes to use in > arithmetic contexts. > When I say "naturals" it includes zero and its successors. Zero and its successors is the naturals. Notice the verb is conjugated "is", not "are." Do you see how that subtle play on words, representing mathematical concepts, leads to N E N? > >There are applications in integral transforms and so on for points at > >infinity, for which the "standard" real numbers are insufficient, and > >they could only be real numbers. > > Well the so called "points at infinity" don't have to be defined as > part of the same set as natural numbers. Natural numbers are finite. > Infinites are not. They don't require combination in a common set. > Everything supposedly true of infinitesimals is reciprocally true of > infinitesimals and not the set of naturals. > In number theory there is some consideration of a "prime at infinity" or for that matter "composite at infinity." That's similar to this notion of passing the bar whether "infinity" would have properties of being prime and/or composite. Here I was talking about "points at infinity" in the reals instead of the integers, but the discussion surrounds both. > >When you have lim n-> oo, that "oo" is an "infinity", it's right there > >in the expression. With lim x-> 0+ or lim x-> 0-, those can differ > >with the sign, and do, and in signed numeric formats the sign takes a > >bit, and in some there are dual representations of zero, > >mechanistically different. > > I'm not quite sure yet what the mechanical implications of zero are. > But basically if you're going to integrate something that something > has been infinitesimally differentiated and if not you aren't doing > definite integration. Either way the zero and the infinity defined > refer to the limits of infinitesimal integration and not to natural > numbers. But I certainly appreciate your considered comments. > > ~v~~ Yes, Les, here I was referring to integers, then reals, interchangeably with these notions, where obviously the integers are reals are not the same thing. While that be so, you might be interested in the notion of naturals as continuum, per Spinoza as I heard from I believe Dean Buckner some time ago. Consider zero in sign-magnitude, with zero's dual representation, although negative zero can be reserved for INT_MIN, as it were, preserving the range seen in 2's complement for the same fixed width. The dx's and dy's and dt's and etc that litter multivariate analysis are algebraic values, in the sense that dx/dx = 1. And warm regards, Ross |