Cantor Confusion
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From: Lester Zick on 29 Oct 2006 19:54 On Sun, 29 Oct 2006 14:58:37 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >Virgil wrote: >> In article <1162139168.281239.183260(a)b28g2000cwb.googlegroups.com>, >> mueckenh(a)rz.fh-augsburg.de wrote: >> >> > > 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. >> > >> > It does not exist as a number. It has no b-adic representation. It >> > exists as a geometric entity. It is an idea. >> >> Being a "number" is nowhere defined as having to have a complete decimal >> expansion, except possibly by WM himself, but everyone knows how screwed >> up he is. >> >> Real numbers have two standard definitions generally accepted: >> (1) as Dedekind "cuts" >> (2) as equivalence classes of Cauchy sequences of rationalsmodulo the >> null sequences. > >You can also construct the real numbers as infinite decimals. The problem is you can construct integers and irrationals using rac techniques but not transcendentals. >> WM has no generally accepted definition, so they are particular to him >> alone, and are of no weight elsewhere. ~v~~
From: Lester Zick on 29 Oct 2006 20:03 On Sun, 29 Oct 2006 15:06:08 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >Lester Zick wrote: >> On Sun, 29 Oct 2006 13:03:00 -0500, David Marcus >> <DavidMarcus(a)alumdotmit.edu> wrote: >> >> >mueckenh(a)rz.fh-augsburg.de wrote: >> >> >> >> David Marcus schrieb: >> >> >> >> > mueckenh(a)rz.fh-augsburg.de wrote: >> >> > > William Hughes schrieb: >> >> > > >> >> > > > And once again WM deliberately confuses, "all positions are >> >> > > > finite", with "there are a finite number of positions". >> >> > > >> >> > > There is no confusing! Every finite position belongs to a finite >> >> > > segment of positions (indexes). If you don't believe that and assert >> >> > > the contrary, then try to find a finite position which dos not belong >> >> > > to a finite segment (of indexes). >> >> > > >> >> > > It is simply purest nonsense, to believe that "all positions are >> >> > > finite" if "there are an infinite number of positions". >> >> > >> >> > Before any of us wastes more time on this, please pick one: >> >>> >> >> > 1. You are making a statement that you say is provable within some >> >> > standard mathematical system, e.g., ZFC. >> >> > >> >> > 2. You are making a statement that is true within your own system. >> >> >> >> My above statement is true within any system which is free of self >> >> contradictions. >> > >> >That's nice, but it isn't what I asked. I'll try again with a more >> >specific question: >> > >> >1. Is your statement provable within ZFC, i.e., using the axioms and >> >rules of inference of ZFC? >> >> Is ZFC free of self contradictions? > >In W. Muckenheim's opinion? I didn't ask him; I asked you for the simple reason he's already indicated agreement for any system free of self contradiction. If you say and can demonstrate ZFC is free of self contradiction WM should accede to your request. I know of no axiomatic system which can be proven free of self contradiction because the axioms themselves are problematic and only undemonstrated assumptions of truth. > I couldn't say. Some days he seems to say >no, other days he seems to say yes. He is impossible to pin down to a >clear statement about anything. Whereas it's certainly been easy enough to pin you down on arbitrary assumptions of truth and character assassination. >Anyway, my #1 asked about "provability" and W. Muckenheim replied about >"truth". So, it didn't seem worthwhile to worry about the rest of his >sentence. Best to go one step at a time. You might as well if you're interested in getting to the truth of a contentious situation. I'm not quite sure what your interest in "provability" might be if you're not interested in demonstrating the truth of WM's contentions. I'm not a supporter of WM's contentions but neither am I a fan of modern mathematical assumptionss of truth and self consistency. ~v~~
From: Dik T. Winter on 29 Oct 2006 20:09 In article <1162135300.256879.37750(a)m7g2000cwm.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1161860460.122685.294590(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > > > > > > 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. > > > > > > You think it must be proved that -1 is not a natural number? As it must > > > be proved that an always increasing positive function does not converge > > > to zero? > > > > No, you must tell me what that limit is, until now you have not provided > > any definition of a limit of sets so I can verify whether the above is > > true or false. Until now you have defined the limit of exactly *one* > > sequence of sets, but that definition is not sufficiently general to > > determine whether the above statement is true or false. > > The definition is sufficiently special (= non-general) to determine > that lim [n-->oo] {1,2,3,...,n} = N is correct. Yes, of course, because that was the definition you provided. So because it is so defined, the definition is correct. > > But this is again different, you state here that N is the set of natural > > numbers, and you define the limit of a particular sequence of sets as > > being that set N. But it does *not* cover arbitrary sequences of sets. > > What, according to that definition, is: > > lim {n -> oo} {-1, 0, 1, ..., n} > > the definition does not make that clear. So the definition is not about > > the "operator" but about a particular sequence of sets. > > The operator "lim [n-->oo]" defines N. In your example lim {n -> oo} > {-1, 0, 1, ..., n} we have N too but in addition the numbers -1 and 0. "lim {n -> oo}" is an operator that works on sequences, apparently. And you have defined that operator for precisely one sequence of sets. But if you now state that it defines N, my next question is "what is N"? And if you state: "the set of natural numbers", my next question would be: can you prove that your definition defines the same set of natural number as the standard method? And how do I know that the second limit is N plus some additional elements if you did not *define* your notation? > > > In the first > > > case, we have 1/n < epsilon for every positive epsilon and we may > > > *define or put* > > > lim [n-->oo] 1/n = 0. > > > In the second case we have without further ado > > > lim [n-->oo] 1/n = 0. > > > That is the difference between potential and actual infinity. > > > > Well, in mathematics the first form is valid, the second is not valid. > > If actual infinity is assumed to exist, then the second case is valid. No. The second is never valid, because the limit notation is defined in such a way that the limit point is *never* reached (all definitions of limits in mathematics are formulated in such a way that the limit point itself, or possible function values at the limit point, are not used). And in mathematics 1/oo is *not* defined. > > But now you are talking in analysis where limits are properly defined > > (because there is a topology and a metric). > > Set theory was discovered and defined as being based on analysis. Cp. > Cantor's first proof. Yes, so what? Set theory is more basic than analysis. In set theory limits are in general not defined. > > You do not know that (and can not prove it), until you have *defined* the > > ordering relations. Once you *have* the ordinal numbers, you can define > > the ordering relations, and a valid definition would be the following > > (assuming the letters are ordinal numbers): > > (1) a = b if there is an order preserving bijection between a set with > > ordinal a and a set with ordinal b > > (2) a <= b if there is an order preserving injection from a set with > > ordinal a and a set with ordinal b > > from these you can define: > > (3) a != b iff not a = b > > (4) a < b iff a <= b and a != b > > (5) a > b iff b < a > > (6) a >= b iff b <= a > > Now you have an ordering. You have to show that it defines an ordering > > relation indeed (not so very difficult). And as there is an injection > > from {} to N, we know that {} <= N (from 2), and as there is no > > bijection from {} to N, we know that {} != N (from 1 and 3), and so by > > (4), 0 < omega. > > So apply this knowledge to the case of the balls and the vase. it can not be applied to it. -- 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 29 Oct 2006 20:29 In article <1162135345.044588.103450(a)k70g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1161856793.990116.183680(a)e3g2000cwe.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > 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. 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. > > For the diagonal number of Cantor's list it is nnot sufficient to come > arbitrarily close to a number which is different from any list number. Sorry, but numbers are fixed and not variable. The sentence "number ... to come arbitrarily close" is nonsense. Numbers do not come arbitrarily close to each other, it is sequences that can come arbitrarily close to each other. > > > > I do not understand why you want to tie that list in with the > > > > definition of the reals? > > > > > > > Because the real numbers are used in that list. > > > > Representatives of the equivalence classes that actually are the real > > numbers. > > How many different representatives can be chosen in Cantor's list for > one equivalence class? As many as you want. But if you want to make the diagonal argument to work it makes sense to select particular representatives for each equivalence class. So selecting for some a base pi notation and for others a base (1 + i) notation is not likely to work. But again (and we have discussed this before), in my opinion, Cantor's original diagonal proof was not about the reals at all. It was about the theorem he states in the first part of the paragraph for which you wish only to quote the second part: "there are sets that have larger cardinality than the naturals" and uses the set of infinite sequences of two objects, called 'm' and 'w'. > > > > > But you should try to understand. In Cantor's list, there are > > > > > limits of sequences, not equivealence classes. > > > > > > > > The list is a list of equivalence classes. In the decimal case > > > > representatives of those equivalence classes are used. > > > > > > And that are limits. > > > > No, they are not, they are sequences of rational numbers. > > In Cantor's list there are omega irrational numbers. Yes, and so? > > > > > > Where in the construction above did I use the limit omega? > > > > > > In using any infinite sequence already > > > > Oh. I did not know that I did use the limit omega there. > > Cantor uses it. Otherwise everything was finite. Perhaps Cantor uses it, I did not. I simply used the axiom of infinity, which implies that there are infinite sequences. Omaga is defined *based* on the axiom of infinity, I did not use it. > > > Why then do you think omega is needed at all in mathematics? > > > > Because it comes in handy on many occasions. What *is* needed is the > > axiom of infinity, because that guarantees that you can even talk about > > infinite sequences. > > Omega is but a convenient name of the set which exists according to the > axiom of infinity. 'the set' -> 'a set'. > > (And ultimately about limits.) Without that axiom > > there is no way to prove that infinite sets do or do not exist. > > Correct. And with this axiom the first infinite entity which is proven > is omega. And then the digits of any irrational are enumerated. Their > number is omega. Yes, the "number" of digits is omega. But this does *not* imply that there is an "omega-th" digit, there is not. > > > Real > > > numbers without the axiom of infinity would be a nice construction. > > > > I would not know how to do that. Perhaps that is possible, but in that > > case you would need to reformulate the limit concept. > > Why then do you try to dispute that omega does play a role in > constructing he reals? The axiom of infinity essentially says: There is > the set omega. There are some models of set theory that state that the set of naturals equals omega. There are other models where that is not stated. > > > Omega is introduced only by the axiom of infinity. > > > > No. It is *defined* using properties obtained through the use of that > > axiom. In some fields of mathematics you do not need omega at all, but > > only the axiom of infinity (analysis, and I think also algebra, number > > theory, and a host of other fields). > > > The axiom of infinity states that there is a set with such and such > properties... And this set is omega. I think this is NBG. -- 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 29 Oct 2006 20:43
In article <1162135573.031943.175480(a)h48g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > In article <1161861368.657796.161130(a)k70g2000cwa.googlegroups.com> muecke= > nh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > > > > > > As you allow constructions by lists, and as there are uncountably ma= > ny > > > > lists, there are also uncountably many WM-constructible numbers. > > > > > > Where are those list? How are uncountably many different lists are > > > constructed by countably many words? > > > > Any infinite sequence of decimal digits is a list. > > Correct. And this sequence has to be defined somehow. But there are > only countably many definitions. Again correct. But there are uncountably many sequences of decimal digits that can not be defined. > > > If you mean that there are more finite definitions than numbers, you > > > are right. Therefore we have a countable set of constructable numbers. > > > > Pray, refrain from using non-standard terminology. WM-construcatble = > > computable. > > Definiert man die reellen Zahlen in einem streng formalen System, in > dem nur endliche Herleitungen und festgelegte Grundzeichen zugelassen > werden, so lassen sich diese reellen Zahlen gewi? abz?hlen, weil ja > die Formeln und die Herleitungen auf Grund ihrer konstruktiven > Erkl?rungen abz?hlbar sind. (Kurt Sch?tte, Hilbert's last pupil) > > Finite definitions! Call this set the set of computable numbers. Yes, that is what they are called. Turing even provided a precise formulation of what a finite definition actually was. > > > Why do you insist on this obvious fact? It does not support your > > > position. There is a list of all finite constructions or definitions > > > (encoded by numbers, G?del). This is the definition of countablity. > > > > I thought that G?del encoded theorems? > > Encode theorems as well as formulas for constructing numbers. Encode > everything which can be expressed by a finite sequence of letters. Then > build the diagonal number. This is also a finite sequence of letters, > because it cannot be longer than the finite list numbers. Again, this statement is completely in contradiction with the axiom of infinity. If you want to talk about mathematics without that axiom, please do so, but do not suggest that you are talking about mathematics with that axio. > > But it is trivial (you do not > > need G?del for that) to find that the number of finite definitions is > > countable. > > All finite sequences are countable. They yield another finite sequence. > Hence they are uncountable. Contradiction. The statement "they yield another finite sequence" is wrong in the context of the axiom of infinity. > > > The diagonal number of this list shows the listed numbers are > > > uncountable. Contradiction. > > > > But indeed, there exist such a list of finite definitions, but not every > > finite definition yields a computable number. Some of those definitions > > do not yield a number at all, so you can not even take the diagonal of > > the numbers defined with such a list. Getting back to the definition > > of computable number: a computable number is encoded by a Turing machine > > that, when it is given a natural number n, responds with the n-th digit > > of the number encoded by it. Taking the diagonal means that you can ask > > the n-th Turing machine the n-th digit of the number it represents. But > > how can you find that digit if the n-th Turing machine does not halt when > > asked for the n-th digit? > > Forget the Turing machines. The diagonal of the list of all finite > sequences (words) of a finite alphabet is a finite sequence because the > diagonal cannot have more places than the words in the list. Forget that arguing with negation of the axiom of infinity. I am talking in the context of that axiom. Due to that axiom there is a set of natural numbers (all finite), that is itself not finite. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |