Cantor Confusion
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From: Dik T. Winter on 27 Oct 2006 20:34 In article <1hm4k29rffqgjnk1fjctkb338sa9kmfu62(a)4ax.com> Lester Zick <dontbother(a)nowhere.net> writes: > On Thu, 26 Oct 2006 23:42:41 GMT, "Dik T. Winter" <Dik.Winter(a)cwi.nl> > wrote: > >In article <1161856793.990116.183680(a)e3g2000cwe.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > > > Dik T. Winter schrieb: > > > > > > > Within the *real* numbers the limit does exist. And a decimal number is > > > > nothing more nor less than a representative of an equivalence classes. > > > > > > So we are agian at this point: The real numbers do exist. For the real > > > numbers we have LIM 10^(-n) = 0. Therefore, i ther limit n--> oo tere > > > is no application of Cantor's argument. > > > >And again we are back at this point. You do not comprehend what I am > >writing. The reason real numbers exist is that there are sequences of > >rationals that come arbitrarily close to each other. > > Technically the reason real numbers exist is that there are straight > line and curve segments. It has nothing to do with decimal expansions. I do not agree with the first, but I agree with the second. It is Wolfgang Mueckenheim who thinks that the reals have everything to do with decimal expansion. For the first, I can state that you do not need curves for sqrt(2), and that I have no idea how to use curves to define 'e'. > > The reason that > >a decimal expansion is a representative of a real number is because the > >sequence of finitely terminations of that number is a sequence of > >rationals that comes arbitrarily close to other such sequences and so > >falls in an equivalence class. *No* limit is involved in all of this. > > What is the "sequence of finitely terminations" (sic) for > transcendentals? I did not use the word "transcendental" I think? And I am so sorry that English is not my native language. > Are you suggesting the sequence itself is finite? Any decimal representation of a real number terminates if ond only if it is rational and has (in its simplest expression) a denominator that contains only the primes 2 and 5. With "sequence of finite terminations" I understand the sequence where the first element is the decimal expansion terminated after the first digit, etc. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 27 Oct 2006 20:36 In article <h4n4k25f4lnlsrdfqb6mku1v7eg6gj8v0b(a)4ax.com> Lester Zick <dontbother(a)nowhere.net> writes: > On Fri, 27 Oct 2006 00:36:24 GMT, "Dik T. Winter" <Dik.Winter(a)cwi.nl> > wrote: > >You need transfinity when you want to show that something that holds in > >the finite case also is valid in the infinite case. Induction will not > >show that 0.111... is rational, it can only show that all the finite > >initial parts are rational. And I again note that the notation 0.111... > >(in the decimals) has only meaning due to the definition of that notation. > > However one can certainly show the square root of 2 without > transfinity through rac construction even though its decimal expansion > is infinite. You need not tell that to me. You should tell that to Wolfgang Mueckenheim who insists that sqrt(2) does not exist because it is impossible to know all the decimals in its decimal expansion. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Lester Zick on 28 Oct 2006 13:50 On Fri, 27 Oct 2006 16:17:53 -0400, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >MoeBlee wrote: >> David Marcus wrote: >> > Not sure if he ever said precisely "within Z set theory", but he >> > certainly said things very similar. Below are a few messages that I >> > found. There are probably others. >> > >> > In the first, he says that "standard mathematics contains a >> > contradiction". In the next two, he states there are "internal >> > contradictions of set theory". In the next, I say that he says that >> > "standard mathematics contains a contradiction", and he does not >> > dispute this. >> >> Thanks. And at least a couple of times I said that I was reading his >> argument to see whether it does sustain his claim about set theory, as >> I mentioned specifically Z set theory. >> >> How rude. He tells you he's going to demonstrate something "IN" set >> theory, so, ON THAT BASIS, you take the time to ponder his argument, >> then he just pulls the rug out from under by saying that it's something >> "outside" of ZFC but that "covers" ZFC. > >I seriously doubt he understands the difference. He doesn't seem to >really understand that modern mathematics rests on an axiomatic >foundation. Assumptions of truth usually do. > And, that there are certain agreed upon rules of argument >(codified in the axioms) that people use. Since when are arbiter dicta arguments? > If someone wants to use some >other rule of argument, they should clearly state that they are doing >so. This is just a common sense prerequisite for communication. So you can clearly communicate your assumptions of truth? ~v~~
From: Lester Zick on 28 Oct 2006 15:52 On 27 Oct 2006 16:56:30 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >Lester Zick wrote: >> >> You mean that mathematical definitions can't have different "domains >> >> of discourse" and mathematical definitions in different domains of >> >> discourse can't borrow from one another? >> > >> >No, that's not what I said. >> >> Then why exactly are you complaining about what I said? Frankly, Moe, >> you don't seem to have said much of anything that I can make out. If >> mathematical definitions can have different domains of discourse then >> what I wrote should be perfectly acceptable according to your own >> definition of mathematical definitions and domains of discourse.. > >Since I never said anything that can be paraphrased as the jumble of >nonsense you just mentioned, nothing I did write entails that the >jumble of nonsense you wrote needs to be acceptable to me. What kind of jumble of nonsense do you prefer then, Moe? >> >No, you posted utter nonsense ("Cardinality(x)=least ordinal(y) with >> >equinumerosity(z)") as if it is something that I had said. >> >> I never said you had said that. > >You're absurd. You quoted me asking you what I said that justified a >certain statement you made. Gee it's sure too bad the relevant citations appears to have gone with the wind. No doubt my fault as well. > You directly replied to that quote with >"Cardinality(x)=least ordinal(y) with >> >equinumerosity(z)". Quoted you? I hardly ever quote anyone when I can paraphrase the substance of what they say actually means instead. >> Says who exactly, Moe? Who died and made you arbiter of the universe? > >Apparently a rival of the authority that died and made you the arbiter >as to what is "perfectly acceptable forensic modality". Clever devil that I am! >P.S. My doubleposts and duplicate passages (from posts I thought were >not previously accepted by the interface) are unintentional. But not your doubletalk apparently. That seems to remain pretty consistent from post to post. ~v~~
From: Lester Zick on 28 Oct 2006 15:59
On Sat, 28 Oct 2006 00:34:01 GMT, "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote: >In article <1hm4k29rffqgjnk1fjctkb338sa9kmfu62(a)4ax.com> Lester Zick <dontbother(a)nowhere.net> writes: > > On Thu, 26 Oct 2006 23:42:41 GMT, "Dik T. Winter" <Dik.Winter(a)cwi.nl> > > wrote: > > >In article <1161856793.990116.183680(a)e3g2000cwe.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > > > > > Dik T. Winter schrieb: > > > > > > > > > Within the *real* numbers the limit does exist. And a decimal number is > > > > > nothing more nor less than a representative of an equivalence classes. > > > > > > > > So we are agian at this point: The real numbers do exist. For the real > > > > numbers we have LIM 10^(-n) = 0. Therefore, i ther limit n--> oo tere > > > > is no application of Cantor's argument. > > > > > >And again we are back at this point. You do not comprehend what I am > > >writing. The reason real numbers exist is that there are sequences of > > >rationals that come arbitrarily close to each other. > > > > Technically the reason real numbers exist is that there are straight > > line and curve segments. It has nothing to do with decimal expansions. > >I do not agree with the first, but I agree with the second. It is >Wolfgang Mueckenheim who thinks that the reals have everything to do >with decimal expansion. For the first, I can state that you do not >need curves for sqrt(2), and that I have no idea how to use curves to >define 'e'. Never suggested one needs curves for sqrt(2). Maybe there's mis understanding And I'm not sure how to relate a curve to the definition of e either. But personally I'm convinced it's there. > > > The reason that > > >a decimal expansion is a representative of a real number is because the > > >sequence of finitely terminations of that number is a sequence of > > >rationals that comes arbitrarily close to other such sequences and so > > >falls in an equivalence class. *No* limit is involved in all of this. > > > > What is the "sequence of finitely terminations" (sic) for > > transcendentals? > >I did not use the word "transcendental" I think? And I am so sorry that >English is not my native language. No I understand and that isn't an issue since you seem pretty fluent. I was just wondering whether you meant there could be a finite sequence for transcendentals. Not that I think you did mean that. It was just a little hard to determine what you were trying to say with that expression. > > Are you suggesting the sequence itself is finite? > >Any decimal representation of a real number terminates if ond only if >it is rational and has (in its simplest expression) a denominator that >contains only the primes 2 and 5. With "sequence of finite terminations" >I understand the sequence where the first element is the decimal expansion >terminated after the first digit, etc. Okay. I think I understand what you mean now. ~v~~ |