An uncountable countable set
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From: Lester Zick on 29 Oct 2006 14:32 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). ~v~~
From: Lester Zick on 29 Oct 2006 14:34 On 28 Oct 2006 16:44:48 -0700, cbrown(a)cbrownsystems.com wrote: >Tony Orlow wrote: >> stephen(a)nomail.com wrote: > ><snip> > >> > 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 >> > >You may or may not realize it, but this is /exactly/ how your arguments >come across in a mathematical context. Spark up that bowl! Party on, >Garth! Whereas yours come across as self righteous and reactionary, Oh Sacred Cow! ~v~~
From: David Marcus on 29 Oct 2006 15:10 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. -- David Marcus
From: David Marcus on 29 Oct 2006 15:11 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? -- David Marcus
From: Ross A. Finlayson on 29 Oct 2006 15:27
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 |