Cantor Confusion
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From: mueckenh on 3 Nov 2006 09:01 Virgil schrieb: > If real numbers are to be represented by sequences that any two > sequences which are "arbitrarily close" represent the same number. Therefore Cantor's proof is invalid. With increasing length of the list, the difference introduced by exchanging the diagonal becomes smaller and smaller. For an infinite list it vanishes at all. > > 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. Which two numbers could that be? > > > > > > 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. The sequence 1 - 1/3 + 1/5 - 1/7 +-... does not define a fixed number. Regards, WM
From: mueckenh on 3 Nov 2006 09:15 MoeBlee schrieb: > > > > > > > > > > 0 > > > > > 1 2 > > > > > 3 4 5 > > > > > 6 7 8 9 > > > > > > > > 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. Entries are 1 2 and 3 4 5 and so on. The numbers written in a line. > > 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. The digits of the numbers written down in your list (above) are not bounded by a finite number. > > > > 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 width of the list is the number of digits of the number with most digits. As such a number does not exist, he width is the supremum, namely omega. > > > > > > 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. I did not introduce a number omega which is larger than every natural number. But IF such a number is introduced, THEN we should be allowed to use the inequality omega > n for every natural number n, i.e. for the n digits of the n-th list entry. > > > > > 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". So you do not man that omega > n holds fo every n e N? Then we have no dissent. > > > > > 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. But the list has a diagonal which, if mapped on a line, has infinite length. > > 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. > > There is NO "maximum of the finite sequences". If you want to use "the > maximum of the finite sequences" in your argument, then you need to > prove that there exists an object that meets that description. > > > 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. > > I said nothing about taking a supremum. > > Please address the proof I gave; not a strawman of my proof. Omega is used in se theory. It is the supremum of the squence of > > If your next response is just more strawman and use of descriptions not > proven to properly refer, then I may very well just note that rather > than waste my time yet again explaining your own errors to you. > > / > > For reference, here is my proof: > > A sequence is a function such that the domain of the function is an > ordinal. A finite sequence has a natural number as its domain. A > denumerable (countably infinite) sequence has omega as its domain. An > uncountable sequence has an uncountable ordinal as its domain. > > The entries in a seque
From: mueckenh on 3 Nov 2006 09:19 Virgil schrieb: > In article <1162470874.593282.36250(a)b28g2000cwb.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > WM merely repeated his automatic error several more times here. > > WM claims that a list in which the nth listed element is a string of > length at least n characters cannot produce a diagonal of length > greater that any finite number of characters. > The diagonal needs an element from every line, the n-th element from the n-th line. Therefore it cannot be longer than every line. Map the diagonal (d_kk) on some line (d_nk). You should see it. Regards, WM
From: mueckenh on 3 Nov 2006 09:26 Virgil schrieb: > > > > Cantor invented omega and defined omega as a whole number. > > > > Who changed this standard meaning? > > WM conflates "whole" with "natural". No, there is a difference. Neither Cantor nor I do conflate it. > > > > > Why do you think this meaning was changed? > > What change? "Whole number" today has a variety of meanings which may > well still include whatever Cantor meant by it. > > > > > When do you think the contrary meaning became standard? > > What "contrary meaning"? Contrary to "whole" is "not whole". > > > > > What is the contrary meaning? > > What "contrary meaning"? > > > > > Do you agree that A n: n < omega is incorrect? > > It is ambiguous in ZF, thus effectively meaningless in ZF until a > context for the universal quantifier is made. A n eps N. > > 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.# > > Except that ZF does not know what whole numbers are. There is no > definition within ZF for "whole number". ZF doesn't seem to know much, not even the main property of the only species of numbers which it deals with. Reagrds, WM
From: Dik T. Winter on 3 Nov 2006 09:25
In article <1162557178.206693.187650(a)e3g2000cwe.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > I think we have obtained agreement, that we disagree in this point and > that further discussion will not lead to a result acceptable for both > parties. The final state is: > > 1) The number of columns is equal to the number of lines. > 2) Both are infinite, according to set theory this means omega or > aleph_0. > > But you believe: omega or aleph_0 is the maximum of the set of lines > and the supremum of the set of columns, which, however, is not taken by > any column. You believe the different meaning is not important. Again you misinterprete what I am saying. It is clear that there will never be agreement if you continue misinterpreting what is written. Omega (or aleph_0) is *not* the maxium of the set of lines. It is the supremum of the cardinalities of the initial finite segments of the set of lines. There is *no* position omega taken by any line. > I believe that a taken maximum and a not taken supremum are so > different (although the difference is very tiny) that set theory, > necessarily assuming this, is wrong. There is not a taken maximum. > The difference between maximum und supremum is, by the way, the reason > why I insist that an actually infinite set of natural numbers must > contain a non-natural number. Yes, you keep insisting that, but as there is no maximum, that insistance is futile. But let's take the actually infinite set of natural numbers. Suppose it contains your non-natural number. Now remove that non-natural number from that set. Isn't the set still actually infinite? What you are confused about is that the cardinality (and ordinality) of the set of natural number is indeed actually infinite; but the *contents* are only potentially infinite. > > > 0.1 > > > 0.11 > > > 0.111 > > > ... > > > > > > you remember? Without an infinite number in the list there is no > > > infinite diagonal defined. > > > > I state (you know that) that the diagonal: 0.111... > > Yes, again mixing up maximum and supremum. No, not mixing at all. Supremum only. > > > Fine. But the width is finite by definition. (We do not put finite > > > segments together, but we have only finite seqments.) > > > > Nope. The width is unlimited, as is the length. And so is not finite. > > But the length is realized as a whole number by a set of elements > (lines) while the width is not realized by a set of columns. You are confused between two concepts. The length is not realised in the sense you think it is. > > > > That is not contradicted. Width and length are equal. > > > > > > This amounts to say that there are infinite natural numbers or that the > > > diagonal is longer than any line. > > > > Wrong. > > If you map the diagonal on a line by d_kk -> d_k, then the line (d_k) > is longer than any line (a_mk) for any m e N. Therefore the diagonal > has more elements than any line (if omega > m for all m e N). Right. The diagonal has more elements than any line. That is obviously true by its definition. Suppose it had equally many elements as some line, than it would be shorter than the next line, and that next line would not contribute to the diagonal. But your mapping above (d_kk -> d_k) is not a mapping on a line, because there is no such line. > > > The maximum of a set of finite numbers which has no maximum is simply > > > not present. > > > > Right. > > > > > It is *not* an infinite number which is larger than any > > > finite number, because a maximum must belong to the set. > > > > Right. > > > > > And a supremum > > > not belonging to the elements of the set does not yield a diagonal > > > digit. > > > > And again right. And still your conclusion is wrong. Obviously the > > diagonal has no finite length, because if it had it would have the > > same length as one of the numbers, and then the next number would be > > longer, which is a contradiction. So the diagonal is infinite in length. > > And nevertheless its length is assumed to be described by a > (non-natural) number. (One cannot compare numbers with non-numbers.) Yes? What is the problem with that? The length of each and every line is described by a natural number, and each and every natural number describes a line. So if we want to describe the length of the diagonal, it can not be a natural number. That length is called omega (if we want to also maintain the ordering information) or aleph-0. > > > Let us stick to Euclidean geometry. But that is unimportant. I see it > > > is impossible to convince you of the existence of reality. > > > > What is the "existence of reality" in this context? > > That what exists, that was really is, contrary to a straight line which > crosses itself. What do you mean with "that what exists, that what really is"? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |