Cantor Confusion
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From: mueckenh on 2 Nov 2006 07:57 Virgil schrieb: > > > > It is unbounded but always finite. > > > > > > Unbounded implies not finite, as finite implies bounded. > > > > That is the definition of potential infinity. But for this definition > > we never have a completet set and never can be sure whether a set is > > Dedeking infinite. > > A set which f(x) = x+1 maps to a proper subset is Dedekind infinite. If a set is not actually completed, then it cannot be decided whether it maps to a proper subset. > > > > Sorry, to say, but you always mix up these two very different things. > > A difference which makes no difference is no difference. > > The only difference in whether the set of finite naturals is potentially > or actually infinite is whether one is looking at its members or at the > set as a whole. Interesting that you mean to have better insights than Cantor who mainly developed this field. > The difference is in what the viewer chooses to see not > in what is there to be seen. That is a true word! In set theory one sees what one wants to see, not what is really there. > > To those of us who choose to see both the members and the set, the > artificial distinction that Mueckenheim chooses to impose does not exist. It is not my distinction. Actual infinity is mainly due to Cantor. With regard to MatheRealism it does not exist. Therefore there is no distinction necessary on my behalf. > All sorts of people have tried to show that this assumption leads to > contradictions within those theories, so far without success. Of course. As long as the believers believe that a matrix of finite lines can have an infinite line (which is the projection of the diagonal) there can't be any success. Astonishing is only that so many people decided to join this club. But perhaps they were not aware of these things when deciding to study set theory. Therefore I'll try to inform them. > WM deliberately blinds himself by assuming that if every string of characters in an infinite list of strings is finite that there must be a UNIFORM limit on their lengths. If actual infinity like omega does exist and if it is larger than any natural number, then omega is a uniform limit of all natural numbers. A limit which is not assumed. It is not a maximum but only a supremum. This is a very small but nevertheless decisive difference. > But a trivial example proves him wrong: For each n in the infinite completed set of naturals Net f(n) be a string of length at least n, then there is no maximally long string in the image of f, as for each m in N there is n in N with n > Length(f(m)). Correct. There is no maximum. But the supremum omega is *not* taken. omega is by definition the number of numbers, in a matrix containing all natural numbers as lines, omega is the number of line. But there is no line omega. > Note that the "diagonal" in my example above MUST be greater than any finite n, so that either there is a number greater than every finite n and simultaneously less than omega or WM is wrong again. Or the assumption of actual infinity is inconsistent. > WM is under the mistaken impression that there must be a longest line in any list of lines. No, there is no maximum. But every line is shorter (has less than omega elements). The diagonal can be projected on the line lines. > Any "diagonal" for this function will have to be of length greater than every n in N. This shows the inconsistency of ZF and NBG. > Does WM choose to claim that a set of naturals which has no maximum must still have a finite upper bound? You can see the inconsistency of actual infinity best by the following list 1 1 2 1,2 3 1,2,3 .... ... n 1,2,3,...,n .... ... omega 1,2,3,... omega + 1 1,2,3,...,omega This is a contradiction. omega cannot count the natural numbers, because they count themselves: There are n numbers up to number n. omega is not their maximum (which does not exist) but their supremum which is not reached. > Or is it true or such a set S that for every n in S there is an m in S with m > n. Of course that is true. And it is true that every such m is finite. > > It is limited by the number of letters in *one line* of the list. > Which line? It does not matter. We cannot find it. We know that it is finite. It's existence is very similar to that of a well ordering of R. > You keep requiring a finite maximum when there is none. The length of the diagonal, like the set of line lengths and the set of line numbers, need not have any finite maximum. Neither they have an infinite maximum. > Since that "proof" requires assumptions which contradict the axioms of ZFC and other standard set theories, Which assumptions do you have in mind? > If one wishes to be limited to speaking of only some of, but never all of, the naturals, finite sets may work well enough, but many of us wish to speak about all of them collectively or of those not in some finite set, for which we need infinite sets. Which are not in some finite sets? I put just this question. > Then WM should be able to give us the position number of that last digit, or else the number of digits. > If he can do neither then he lies. First you should give the position number of the first number not in a finite set. Regards, WM
From: mueckenh on 2 Nov 2006 08:06 David Marcus schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > The diagonal is an infinite sequence. So the diagonal is longer than > > any of the finite sequences. But the diagonal consists of elements of > > the finite sequences. So it cannot be longer than the maximum of the > > finite sequences. If this maximum does not exist, you cannot take the > > supremum omega for it, because the supremum is not a member of the > > sequences and does not supply elements of the diagonal. > > Let's try a simpler problem. Consider the following list. I would hesitate to call it a list, because injectivity is lacking. But that is not important. > > 1 > 1 > 1 > ... > > In other words, consider the sequence x where x(n) = 1 for n a natural > number. How long is this sequence? You expect the answer omega. No problem. There may be omega elements, the first, the second, and so on. The diagonal of that list is very short. It is 1. Regards, WM > > -- > David Marcus
From: Han de Bruijn on 2 Nov 2006 08:23 mueckenh(a)rz.fh-augsburg.de wrote: > Karel Hrbacek and Thomas Jech: "Introduction to set theory" Marcel > Dekker Inc., New York, 1984, 2nd edition, p. 2: "So the only objects > with which we are concerned from now on are sets." Karl Marx, "Das Kapital": because capitalism is only interested in its "ungeheure Waren-sammlung". And nothing else matters. Han de Bruijn
From: David Marcus on 2 Nov 2006 08:33 mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > 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. OK, I'll bite: How does defining a number as a sequence contradict that a number is a fixed entity? -- David Marcus
From: David Marcus on 2 Nov 2006 08:40
mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > mueckenh(a)rz.fh-augsburg.de wrote: > > > David Marcus schrieb: > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > David Marcus schrieb: > > > > > > > > > > > The set of natural numbers is an infinite set that contains only finite > > > > > > numbers. > > > > > > > > > > Please do not assert over and over again this unsubstantiated nonsense > > > > > (this word means exactly what you think) but give an example, please, > > > > > of a natural number which does not belong to a finite sequence. If you > > > > > cannot do so, then it is obviously unnecessary to consider N as an > > > > > infinite sequence, because all its members belong to finite sequences. > > > > > > > > I didn't say anything about sequences, finite or otherwise. So, your > > > > request is irrelevant to my statement. > > > > > > The sequence of natural numbers is not comprehensible in ZFC? Neither > > > is the sequence of partial sums of a converging series? Nor are the > > > finite sequences which are called (initial) segments of sequences which > > > are ordered sets. Also the expression "extended sequence" for an > > > uncountable ordered set is new to you? > > > > Non sequitor. > > ? > I did not yet conclude anything but asked some questions. Your questions seem to have no relationship to anything I wrote, but OK, I'll see if I can answer your questions. See below. > > Let's make it simple. I'll give a statement and you say whether you > > think it is provable in ZFC. Is > > > > The set of natural numbers is infinite > > > > provable in ZFC? Please answer "Yes" or "No". I see you didn't answer my question. > > > > Don't you think that you should label all your posts as > > > > "NON-STANDARD MATHEMATICS"? > > > > > > 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 non-standard meaning on Cantor's > > > definition of omega as a whole number? > > > > Since Cantor predates axiomatic set theory, if you write anything that > > uses Cantor's definitions without checking whether the definitions are > > still standard is "Non-Standard Mathematics". > > Therefore I put above list of questions in order to find out what your > understanding of the standard is. If you say: My position is standard, > that is fine for you, but it is not sufficient to show anything but > orthodoxy. > > > If you want to discuss > > history, that is fine, but you should label your posts as such. This is > > simple courtesy. If you use words without defining them, readers assume > > you are using them in their current meanings. If you are using > > historical meanings, then either say so or use a different word. > > > > As for the current definition of omega, Kunen's book is a good > > reference. According to Kunen, omega is not a natural number. > > That is out of any question. Sorry, but I don't understand. Please rephrase. > > I'm > > guessing that by "whole number" you mean natural number, but I really > > don't know, since you seem to have your own language for everything and > > you never give definitions for any of the words that you use. > > "Whole number" is Cantor's name for his creation. > > My question is : Do you maintain omega > n for all n e N? Before I can answer the question, I need to know what you mean by the words/terms. So, please define "omega", "N", and ">". Also, by "maintain" do you mean that ZFC proves it? > I know that > modern set theory says so. If something can be larger than a number, > then it must be a number. If it cannot be a fraction because ZF does > not yet know how to divide elements, then it can only be a whole > number, I would guess.# -- David Marcus |