Cantor Confusion
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From: Lester Zick on 2 Nov 2006 12:39 On 1 Nov 2006 16:22:37 -0800, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >Lester Zick wrote: >> Okay(x). So(x)exactly(x)which(x)are(x)mangled(x)versions(x)of(x) >> what(x)you(x)were(x)claiming(x)? > >I already pointed them out several times. Now, on the one hand you >indicate a sense of wanting to move on from the subject, but you also >continue to post about it, including asking me a question which would >prompt me yet again to post examples of a subject you indicate you want >to move on from. You're hopeless. Moe, do you have any idea of the difference between misrepresentation and outright mockery? ~v~~
From: Noehl on 2 Nov 2006 13:27 Lester Zick wrote: > On Thu, 2 Nov 2006 00:28:49 -0500, David Marcus > <DavidMarcus(a)alumdotmit.edu> wrote: > >> Dik T. Winter wrote: >>> In article <sjnfk256m5tau0bmk5u4vjdg53al1f2sc9(a)4ax.com> Lester Zick <dontbother(a)nowhere.net> writes: >>> > Yours are some of the very >>> > few that don't. And I've seen posts of other .nl correspondents which >>> > come through in color. >>> >>> It has nothing to do with the domain where you come from. It has everything >>> to do with the manner things are quoted and in the way *your* newsreader is >>> able to handle that. I know the reason your newsreader does not show colours. > > But my newsreader does show colors. > >>> It is because I prepend the '>' sign with a space when quoting. > > Ok. Mine reacts the same way. But when there is no preceeding space > quoted text is shown in a different color. > >> And I >>> have pretty good reasons to do that. (One of the reasons being that this >>> article would be rejected because there is more quotation than new text.) > > I've never experienced that. > >> My newsreader lets me post articles that have more quotation than new >> text. > > ~v~~ Thunderbird quotes nicely -- Noehl u144241 Games that I like to play Multiplayer Online Games <http://www.gamestotal.com/> Strategy Games <http://www.gamestotal.com/> Unification Wars <http://uc.gamestotal.com/> Massive Multiplayer Online Games <http://uc.gamestotal.com/> Galactic Conquest <http://gc.gamestotal.com/> Strategy Games <http://gc.gamestotal.com/> Runescape <http://www.stephenyong.com/runescape.htm> Kings of chaos <http://www.stephenyong.com/kingsofchaos.htm>
From: Virgil on 2 Nov 2006 14:59 In article <1162470378.676172.129570(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1162405520.008395.100850(a)e64g2000cwd.googlegroups.com> > > mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > ... > > > > Finally you give a definition. Why did it take so long? > > > > > > I thought that this was so clear that no explanation was required. > > > > I asked you for a definition quite a few times. I would have thought > > that that was enough indication that it was not clear at all. > > > > > > > > And in mathematics 1/oo is *not* defined. > > > > > > > > > > Not in mathematics. But in a theory which assumes omega to be a > > > > > whole > > > > > number. > > > > > > > > Which theory? > > > > > > Set theory. > > > Cantor invented omega and defined omega as a whole number. > > > Who changed this standard meaning? > > > Why do you think this meaning was changed? > > > When do you think the contrary meaning became standard? > > > What is the contrary meaning? > > > Do you agree that A n: n < omega is incorrect? > > > If not, why do you complain about on-standard meaning on Cantor's > > > definition? > > > > A nice rant. > > Why don't you answer? > > > Where did Cantor define 1/oo? A quote might be > > appropriate. > > see below > > > > > > Where in the above quote is 1/oo defined? > > > > > > It is defined that omega (which Cantor used later instead of oo) is a > > > number larger than any natural number n. Omega is the limit ordinal > > > number. Therefore 1/omega must be a number smaller than every fraction > > > 1/n. What theorem says that a limit ordinal must have a reciprocal ordinal? Or even that a natural must have a reciprocal natural? The arithmetic of ordinals is quite different from the arithmetic of rationals, to the extent that there is only one ordinal that has a reciprocal ordinal, and it is not omega. > > > > Why? As long as 1/omega is not defined you can not talk about it. You > > simply assume that 1/omega is a number. But that is not the case, it > > is not defined. > > If omega is a number > n What sort of number? The arithmetic allowable depends on which set of numbers is being considered. Omega is an ordinal, but not a natural nor a real nor a rational nor an integer. So the only arithmetic allowable on omega is that of the ordinals, and most of them, including omega, do not have ordinal reciprocals.
From: Virgil on 2 Nov 2006 15:10 In article <1162470537.261044.229200(a)h54g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1162405329.073198.286680(a)h48g2000cwc.googlegroups.com> > > mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > ... > > > > > Irrational numbers have no last digit. Therefore, with a sequence > > > > > of > > > > > digits like the diagonal number is, one can never have a completed > > > > > number but only come as close as possible to any number --- or > > > > > avoid to > > > > > do so. > > > > > > The problem is that: > > > For the diagonal number of Cantor's list it is not sufficient to come > > > arbitrarily close to a number which is different from any list number > > > --- or avoid to do so. . > > > > You lost me here. Numbers do not come arbitrarily close to each other. > > Numbers are fixed entities. Sequences can come arbitrarily close to > > each other. > > Irrational numbers are sequences. Or equivalence classes of sequences, or Dedekind cuts, or mere abstractions without any concrete representations in terms of "simpler" numbers. If real numbers are to be represented by sequences that any two sequences which are "arbitrarily close" represent the same number. For example, as when one represents the same number in different bases. > > > And, representatives are > > *not* limits. When considering the equivalence classes, most sequences > > have one limit: the equivalence class it is sitting in. > > > > I think that you are still thinking that *some* represenation defines > > a real number; but that is not the case. > > In Cantor's list there are those unique representations required. Not so. Even in decimal, Cantor's diagonal rule allows for certain rationals having dual representation. > > > That is especially not the > > case when you consider only representations to some integral base. > > There are other methods to define numbers. You do not like to call > > them numbers, but ideas. But you can not prevent me to call something > > like sqrt(2) a (real) number. And in common mathematics that is just > > what it is. > > > > By definition, every sequence (use any definition, I know Cantor, > > Dedekind, Baudet and Weierstrass, they all lead to the same): > > {sum{k = 1...n} a_k/10^k} > > is a representative of a "number". It is just a sequence of rationals. > > Therefore I do not understand why you say "Numbers are fixed entities". > They are merely defined by sequences. The sequence 1 + 1/2 + 1/4 + 1/8 + ...+ 1/2^n + ... "defines" a fixed number. That that number has other representations does not make that number into a variable quantity.
From: MoeBlee on 2 Nov 2006 15:12
mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > MoeBlee schrieb: > > > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > All entries of the list have a finite number of letters. An infinite > > > > > sequence is larger than any finite sequence. The diagonal of a list > > > > > cannot have more letters than the lines. > > > > > > > > > > According to your logic the list can have infinitely many lines. But > > > > > even if that was correct it would not facilitate an infinte diagonal. > > > > > > > > > > The number of diagonal elements is the minimum of columns and lines. > > > > > > > > 0 > > > > 1 2 > > > > 3 4 5 > > > > 6 7 8 9 > > > > ............... > > > > ................. > > > > ................... > > > > > > > > infinitely downward for an infinite list of finite lists. > > > > > > > > The diagonal is 0 2 5 9 14 ... infinitely across. > > > > > > > > All entries in the infinite list are finite lists. > > > > > > better say finite sequences or numbers or entries > > > > 'sequence' and 'list' are synonymous here. > > a list is an injective sequence Then that is your definition. Usually, injection is not required. But for this particular discussion it doesn't matter, since all the sequences I've used are injections. > > I'm pefectly happy to use just 'sequence'. Doing so does not at all > > harm my argument. > > > > > > The infinite list is > > > > longer than any finite list. > > > > > > The entries surpass every finite entry. Nevertheless you call all of > > > them finite. > > > > I don't know what you're trying to say. > > Because you did not read what I wrote. I defined it above: "better say > finite sequences or numbers or entries" No, I read it over a few times. When I say I don't understand something, you can take me at my word that I mean just that - I read it a few times, thought about it, and don't understand it. Thus, you can save yourself the wasted typing of saying false things such as that I didn't read what you wrote. I don't know what you mean by entries SURPASSING every finite entries. What entries surpass which other entries? What does 'surpass' mean? If you give me ordinary discourse, then I'll have a better chance of understanding you, just as I defined each of my terms, 'sequence', 'entry', etc. in my own remarks. > >Even using just the word > > 'sequence', my point is correct. > > > > We have an infinite sequence S of finite sequences. Being an infinite > > sequence, the length of S is longer than the length of any finite > > sequence. > > Maybe, if you say so. But omega is not the maximum of all finite > sequences. Yes, since omega is not a sequence at all, let alone being a finite sequence, let alone being the maximum of all finite sequences. > Therefore the width of the list is less than omega. My argument does not mention 'width of the list'. If YOU want to refer to 'width of the list', then YOU need to define it. And that means first proving that there exists a unique object that meets the description. > > > > The diagonal of the list is infinite. > > > > > > That is your assertion. But obviously the diagonal elements are > > > simultaneously elements of the entries. > > > > No, we trivially PROVE the diagonal sequence is infinite. > > You may also prove that the maximum of numbers less than 5 is 5. > Nevertheless it is false. No, I can't prove that. > The diagonal of a list of sequences with less than 5 terms is less than > 5. > The diagonal of a list of sequences with less than omega terms is less > than omega. > > This simple truth should convince you that ZFC is not acceptable. You claim it is a simple truth without proving it. And your claim is not even compatible with the simple intuitive picture that uses ellipses. So not only do you not have a mathematical proof of your claim, you don't have an intuitive explanation, except an argument by ANALOGY in which you analogize between the finite and infinite, only assuming, as a form of question begging, that what holds for a finite sequence must hold for an infinite sequence. > > > The diagonal elements are simultaneously elements of the entries. > > > Therefore the diagonal elements cannot sum up to a number which is > > > larger than any natural number unless also the elements of list entries > > > sum up to a number which is larger than any natural. > > > > In my example, I said nothing about summing up. And I said nothing > > about anything in S being larger than any natural number. > > You said the domain is omega. You said "we trivially PROVE the diagonal > sequence is infinite". omega is larger than any natural number. > "Infinite" means "larger than any natural number". The common definition of 'is infinite' I use is: x is infinite <-> ~En(n is a natural number & x is equinumerous with n) which in turn reduces to: x is infinite <-> ~Enf(n is a natural number & f is a bijection between x and n) No mention there of "larger". > > > Or put it so: Every segment of the diagonal is covered by an entry. > > > > Which 'entries'? > > > > > There is no segment which is not covered. > > > > What is the initial segment {<0 2>}, of the diagonal, covered by? And > > what does it matter? > > > > > If all entries are finite, > > > > Yes, all entries of S are finite sequences. > > Without a maximum. Yes, if you mean that there is no entry has a greater length (notice, by the way, that 'greater' here is just the usual 'greater than' relation among natural numbers; i.e., finite) than all other entires. > Without a sequence of infinite length. Correct. No entry of S has infinite length. > > > then the diagonal cannot be infinite (if infinite omega is larger than > > > any finite n). > > > > In this post I PROVED that the diagonal of S is infinite. The diagonal > > of S is an infinite set. It is an infinite sequence. It has an infinite > > domain. It is an infinite set of ordered paris. (And, by the way, it > > has an infinite range.) And you've not shown that that contradicts any > > theorem of any Z set theory. That it may contradict your own confused > > word jumbles is not of concern to me. > > You have derived a nice contradiction. No, I haven't. Please state the sentence P in the language of set theory such that you think I derived P and the negation of P. |