Cantor Confusion
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From: David R Tribble on 24 Oct 2006 19:11 Randy Poe wrote: >> Since when can definitions not be wrong? >> >> A definition is a statement by a person that a certain >> symbol will be used by them to stand for some concept. >> >> What is there in such a statement that can be wrong? >> >> Example: I will use the term "gleeb" to refer to an integer >> which is divisible by 2. >> >> How can that statement be wrong? How would you >> define "wrong" for such a statement? > Lester Zick wrote: >> The concept? >> >> Self contradictory predicates defining the concept in the definition. >> Ex: "squircles are square circles" "x is an even, odd" "gleeb is a >> finite integer divisible by 0". > Randy Poe wrote: > All of those are perfectly valid definitions. Just because it > doesn't exist doesn't mean the concept can't have a name. Indeed. Perhaps Lester can tell us if these definitions are "wrong" or not: 1. A "Fermat exponent" is a positive integer n > 2 such that x^n + y^n = z^n for some integers x,y,z. 2. A "prime pair limit" is an integer n such that no relative primes p and p+2 exist for any p > n. 3. A "Riemann tooth" is a non-trivial complex root of the zeta function such that its real component is not 1/2.
From: Dik T. Winter on 24 Oct 2006 19:16 In article <1161684094.162258.28800(a)h48g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > Eh? Using the definition Virgil gave leads to my conclusion without > > further ado. And I just stated that it was not desirable. Obviously > > wrong is something else. If Virgils definition is the definition, my > > conclusion is obviously right. So there should be a search for a > > definition that allows what is wanted. Virgils definition, although > > not wrong, definitions can not be wrong, > > False. But idle dscussion. On what grounds can a definition be wrong? > > > lim [n-->oo] {-1,0,1,2,3,...,n} = N > > > > > > is obviuously wrong too. > > > > Depends on how you define the limit. You did not state why that one was obviously wrong. > > > Therefore lim [n-->oo] {1,2,3,...,n} = N. > > > > Depends on how you define the limit. > > That *is* the definition of the this limit. It is a wrong definition > only in case N does not actually exist. You state "therefore", in general definitions are not conclusions from earlier arguments. But N obviously exist; it is simply a letter. > > > So it must be wrong and needs no further attention. Why? > > > > But you never give a definition of the limit of sets. You only state that > > definitions are *wrong* (although I fail to see why a definition can be > > wrong). But you never state a proper definition. > > Look here: lim [n-->oo] {1,2,3,...,n} := N. Finally. A definition. So you define the notation lim for a particular sequence of sets. But can you, in that case, use that definition in your arguments? This is not the only sequence of sets for which you did use the limit notation. > > > Because the contents of the vase increases on and on. Such a process > > > cannot lead to emptiness in any consistent system - independent of any > > > "intuition". > > > > That requires proof. > > LOL. Idle discussion. Perhaps, but you use it as argument, and I want to know whether it is a valid argument, and so I want to see a proof. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Virgil on 24 Oct 2006 19:34 > In article <1161683860.442902.89560(a)b28g2000cwb.googlegroups.com> > mueckenh(a)rz.fh-augsburg.de writes: > > If f(0) is undefined, i.e., if 0 is not element of the domain of f, > > then we can find f(0) = 0 by l'Hospital' s rule. Ever heard of? L'Hopital's rule has very restricted application in the finding of limits, and is in no way applicable in anything cited here. L'Hopital's rule is also often a refuge of the mathematically incompetent in many of the rare cases where it is applicable.
From: Dik T. Winter on 24 Oct 2006 19:39 In article <1161683454.085728.22020(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1161517636.934369.301190(a)i42g2000cwa.googlegroups.com> muecke= > nh(a)rz.fh-augsburg.de writes: > > > > > > Dik T. Winter schrieb: > > > > > > > > lim{n --> oo} 1/n = 0 proclaims "omega reached". > > > > > > > > It proclaims nothing of the sort. You want to read more in > > > > notation than is present. > > > > > > You want to read less than present and necessary. > > > lim{k --> oo} a_k = pi means omega reached. Or would you state that > > > pi belongs to Q like every a_k for natural k? > > > > Of course not, and of course I do not state that oo is reached. Stating > > (for the first limit above) that omega is reached is tantamount to stating > > that there is an n such that 1/n = 0. > > Your logic is strange. If there were an n with 1/n =3D 0, then we would > not need to write "limit". What else would "omega reached" mean? > > There *is* no such n. > > Therefore we write the limit. Yes, because the limit point is not reached and can not be reached. > The limit of all n is omega. That is by your definition, as you wrote in another of your articles. > > second statement it would mean that there is a k such that a_k = pi. > > Both are nonsense. > > Because there is not such an n, we use the limit. Indeed. Omega is not reached because we can not reach it. > > With straight-edge and compass. Or else, please solve the problems of > > 'doubling the cube', 'squaring the circle' and 'trisecting an angle', > > using straight-edge and compass. Apparently you are back to 15-th > > century mathematics. > > It is ridiculous to see how you willful try to misunderstand me. Not wilfully. But if you start using terms with a meaning different from the meaning in common mathematics it is easy to get misunderstanding. In mathematics there is a precise meaning of "constructable numbers". You used that word and used theorems of them, but you actually did mean something else. > > > Of couse 0.110001000... and all its sums with rational numbers are > > > constructible numbers in my sense. > > > > But the constructable numbers in your sense are not countable. > > They are, because there can be not more than countably many > constructions. As you allow constructions by lists, and as there are uncountably many lists, there are also uncountably many WM-constructible numbers. > K?nig and Cantor did not yet know the constructible numbers in modern > sense. Nevertheless they were convinced that only such numbers which > can be constructed in my sense are meaningful. The reason is the > countable set of finite constructions. A real number with no definable > law cannot be constructed because an infinite amount of information > would be needed. Every construction is finitely defined. And as I have stated already a few times. You mean 'computable' numbers. The above is approximately the definition of 'computable' numbers in current mathematics. But you stated elsewhere that you did not mean that. Please rehearse modern theory. > K?nig used this argument to show a fault in set theory: Man zeigt sehr > leicht, da? die endlich definierten Elemente des Kontinuums eine > Teilmenge des Kontinuums von der M?chtigkeit aleph_0 bestimmen, ... > J. K?nig: Math. Ann. 61 (1905) 156 - 160 > ?ber die Grundlagen der Mengenlehre und des Kontinuumproblems. Yes, that is precisely what I wrote, quite sometime ago already, about the computable numbers. It is easy to show that the computable numbers are countable. But the theory has been developed a bit since then. The problem with what K?nig wrote is that not every finite definition also gives a number. > Cantor recognized the truth of K?nigs conclusion but doubted that the > finitely defined numbers were countable: W?re K?nigs Satz, da? alle > "endlich definierbaren" reellen Zahlen einen Inbegriff von der > M?chtigkeit aleph_0 ausmachen, richtig, so hie?e dies, das ganze > Zahlenkontinuum sei abz?hlbar, was doch sicherlich falsch ist. Cantor > to Hilbert. > > I think today there is no question that the constructed numbers (in my > sense) are countable. And of course there is no question that K?nig > was right. When you substitute "computable" for "constructable", both are indeed right. But Cantor was wrong. It is only after Turing that this was solved. The finite definitions are countable, but not every finite definition gives a number. So there exists a list of finite definitions (this has been formalised using Turing machines), but this is not a list of computable numbers, because there will be non-halting Turing machines in the list, and you do not know how to separate them from the rest. Look up the halting problem. There is no computable list of computable numbers. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Randy Poe on 24 Oct 2006 20:58
Lester Zick wrote: > On 23 Oct 2006 21:22:10 -0700, "Randy Poe" <poespam-trap(a)yahoo.com> > wrote: > > > > >Lester Zick wrote: > >> On 23 Oct 2006 08:48:07 -0700, "Randy Poe" <poespam-trap(a)yahoo.com> > >> wrote: > >> > >> > > >> >Lester Zick wrote: > >> >> On Mon, 23 Oct 2006 02:00:06 GMT, "Dik T. Winter" <Dik.Winter(a)cwi.nl> > >> >> wrote: > >> >> > >> >> >In article <1161518242.756958.103660(a)h48g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > >> >> > >> >> [. . .] > >> >> > >> >> > Virgils definition, although not wrong, definitions > >> >> >can not be wrong, does not lead to desirable results. > >> >> > >> >> Since when can definitions not be wrong? > >> > > >> >A definition is a statement by a person that a certain > >> >symbol will be used by them to stand for some concept. > >> > > >> >What is there in such a statement that can be wrong? > >> > >> The concept? > >> > >> >Example: I will use the term "gleeb" to refer to an integer > >> >which is divisible by 2. > >> > > >> >How can that statement be wrong? How would you > >> >define "wrong" for such a statement? > >> > >> Self contradictory predicates defining the concept in the definition. > >> Ex: "squircles are square circles" "x is an even, odd" "gleeb is a > >> finite integer divisible by 0". > > > >All of those are perfectly valid definitions. Just because it > >doesn't exist doesn't mean the concept can't have a name. > > Self contradictions can have names. They're just false definitions There's that weird phrase again. It's like saying that if you're defining an alphabet, some of your new squiggles can be "wrong". > unless of course you want to argue that self contradictions can be > true. No, giving it a name makes it neither true nor false. > As for validity I'd appreciate it if you could explain the difference > between "valid" "correct" and "true". "Valid" usually means correctly formed according to some rules. It doesn't imply either "true" or "false". For instance, x < 7 is a valid inequality, but obviously depending on the value of x it can be either true or false. "Correct" would seem to me to be usually synonymous with "true". > To me it would appear to be a > distinction without a difference and the various terms would only be > used to sanctify different kinds of subjects, definitions, concepts, Well, obviously I disagree, and I have no idea what you might mean by "sanctifying" a definition. - Randy |