Cantor Confusion
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From: mueckenh on 2 Nov 2006 07:34 William Hughes schrieb: > Each diagonal digit is formed by a line. and by a column! > The number of diagonal digits is the number of lines. > The number of lines is a supremum. Clearly the number > of lines does not yield a diagonal digit. > > Let A be the list of all finite strings. > Let D be a diagonal formed from this list. > > The question is "How many digits does D have". > > It is not enough for you to say that this question does not have a > defined > answer. You must also show that the usual answer (omega) > leads to a contradiction. It does, because the number of columns of each line is finite, i.e., less than omega. > > Let the number of digits in D be X. We don't know if X exists. > However, > it is clear that if X does exist it must be greater than any natural > number. > Hence: > > - X does not exist > or > - X is at least as great as omega > Correct. > > > > > > Why do you say incorrect? Given any integer N, we certainly have more > > > letters > > > than N. > > > > But the smallest number which is larger than any integer is omega. And > > we have excluded a line with omega letters. > > But excluding a line with omega letters does not mean that D cannot > have omega letters. D can be projected or mapped into a line. Then it is a line longer than any line. > The number of letters in D is less than the number > of letters in a line only when this line has more letters than any > other line. > There is no line with more letters than any other line, > so no line we can use to limit D. > We only know it must be finite. > > Correct. If it exists, it must be infinite (or omega). Only then the > > number of lines can be omega. But we have excluded that the number of > > letters is omega. > > > > No, we have not excluded omega. You wish to introduce a line with omega letters? That yields what I always say: There is no infinite number of natural numbers without an infinite number omega. > > > If it does not exist you cannot use it to limit the > > > diagonal. > > > > If it does not exist, then you cannot build a diagonal of that length. > > In which case you cannot use the fact that it does exist to show > a contradiction. Yes, you can assume the diagonal does not exist. > No, you do not get a contradiction if you assume it does exist. You don't get a contradiction if you assume that a supremum is taken which, by definition, is not taken? OK. That's st theory. Regards, WM
From: mueckenh on 2 Nov 2006 07:40 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. > > 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". > > > > 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. > 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? 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.# Regards, WM
From: mueckenh on 2 Nov 2006 07:46 David Marcus schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > David Marcus schrieb: > > > > > > I know of a really good logician who told me that he is unable to > > > > translate my proof into ZFC. > > > > > > Hardly surprising. If he is truly a good logician, then he didn't want > > > to tell you that you were talking gibberish. > > > > No, he (and he is not the only one to do so) does understand the > > arguing but does not know how to translate it into ZF language. That is > > all. > > If your "really good logician" understands the argument, have him post > his version of it. I have posed for several times a version which can be understood and has been understood by him and by other mathematicians. > > > > Fine. Then stop claiming that you can prove ZFC is inconsistent. If you > > > don't like ZFC, that's your concern. But, that is completely different > > > from saying that you have a proof within ZFC of a contradiction. > > > > I have a proof which shows that the real numbers have less elements > > than a countable set. That is all. ZFC and what there can be formalized > > does not bother me at all. > > If the proof cannot be given in ZFC, then it is irrelevant to whether > ZFC is inconsistent. I cannot construct a horoscope. Nevertheless I can find out whether a horoscope tells the future correctly, at least when the future has become present time. > So, please desist from making such a spurious > claim. > > If you can prove that the real numbers are both countable and > uncountable, then that simply shows that the system that you are using > for your proofs is inconsistent. It says nothing about the real numbers > and countability as defined in ZFC. The countability of R is not proven in "my system" but in general mathematics which is valid prior to ZFC and will emain so after ZFC has gone. The proof of uncountability of R therefore shows ZFC is false. > > > The maximum of a set of finite numbers which has no maximum is simply > > not present. It is *not* an infinite number which is larger than any > > finite number, because a maximum must belong to the set. > > If by "maximum" you mean the largest element and by "finite numbers" you > mean natural numbers, then it is true (in ZFC and standard mathematics) > that an unbounded set of natural numbers does not have a maximum. Correct. > > > And a supremum not belonging to the elements of the set does not yield > > a diagonal digit. > > No clue what this is supposed to mean. Feel free to rephrase using > common mathematical terminology and/or define what a "diagonal digit" > is. A supremum is a least upper bound of a set (like the set of columns of a list). If there is no maximum, then the supremum is not taken, i.e., it does not belong to the set. Regards, WM
From: mueckenh on 2 Nov 2006 07:48 MoeBlee schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > Sorry, to say, but you always mix up these two very different things. > > It should be comprehensible that potential infinity is possible without > > the axiom of infinity. But that is not what set theory requires. > > Therefore it is correct to say that in set theory theory there is no > > infinity present or detectable without the axiom of infinity. > > "Detectable", informally, okay. Yes, without the axiom of infinity, in > set theory (throughout all these discussions, by 'set theory' I mean, > in any given instance, some given Z or NBG variant) we cannot prove > there exists an infinite set. So, in that sense, we cannot "detect" the > existence of an infinite set. But we still are not permitted to claim > that infinite sets do not exist. We can only say that we cannot ever > detect whether they exist or not. You use these words such as 'is > present', 'detecable', etc., which are your words for your OWN way of > comprehending. But set theory is NOT in those terms that you use to try > to comprehend set theory. > > Instead of your own METAPHORICAL language, let's look at the formulas > (or English renderings of actual formulas) of set theory: > > Without the axiom of infinity, we cannot prove either of these two > formulas ('finite' and 'infinite' I define, respectively as > 'equinumerous with a natural number', 'not equinumerous with a natural > number'): > > Ex x is infinite > > ~Ex x is infinite > > So without the axiom of infinity (but with no added axioms to the ZFC > axioms), we cannot prove: > > Ax x is finite > > To prove that formula, we need to adopt an axiom that entails it. Just > dropping the axiom of infinity does NOT entitle us to conclude Ax x is > finite. > > Would you please just say whether you understand this point. The following is correct: There does not exist any actual infinity (i.e. what in modern set theory is just called "an infinite set") neither in reality nor in mathematics, unless you introduce the axiom of infinity. Only then an actually infinite set exists in mathematics. Therefore, without this axiom, everything and every set is finite. This includes sets without bound which are called potentially infinite but which do not have an infinite cardinal number. In modern set theory you cannot say for all sets Ax x is finite. But the negation of this statement is not what modern set theory calls infinite. Modern set theory simply cannot describe developing sets as it apparently cannot describe sets with limited contents of information. These things are unknown to the slaves of formalism. Read a good book like Fraenkel, Abraham A., Bar-Hillel, Yehoshua, Levy, Azriel: "Foundations of Set Theory", North Holland, Amsterdam (1984). There you will find more about that topic. to look at the universe of all sets not as a fixed entity but as an entity capable of "growing", i.e., we are able to "produce" bigger and bigger sets. (p. 118) [Brouwer] maintains that a veritable continuum which is not denumerable can be obtained as a medium of free development; that is to say, besides the points which exist (are ready) on account of their definition by laws, such as e, pi, etc. other points of the continuum are not ready but develop as so-called choice sequences. (p. 255) Regards, WM
From: William Hughes on 2 Nov 2006 07:49
mueck...(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > Each diagonal digit is formed by a line. > > and by a column! Yes, and the number of columns is equal to the number of lines. The fact that no single line contains all the columns is not important. > > > The number of diagonal digits is the number of lines. > > The number of lines is a supremum. Clearly the number > > of lines does not yield a diagonal digit. > > > > Let A be the list of all finite strings. > > Let D be a diagonal formed from this list. > > > > The question is "How many digits does D have". > > > > It is not enough for you to say that this question does not have a > > defined > > answer. You must also show that the usual answer (omega) > > leads to a contradiction. > > It does, because the number of columns of each line is finite, i.e., > less than omega. > > > > Let the number of digits in D be X. We don't know if X exists. > > However, > > it is clear that if X does exist it must be greater than any natural > > number. > > Hence: > > > > - X does not exist > > or > > - X is at least as great as omega > > > Correct. > > > > > > > > > Why do you say incorrect? Given any integer N, we certainly have more > > > > letters > > > > than N. > > > > > > But the smallest number which is larger than any integer is omega. And > > > we have excluded a line with omega letters. > > > > But excluding a line with omega letters does not mean that D cannot > > have omega letters. > > D can be projected or mapped into a line. Then it is a line longer than > any line. > D cannot be projected or mapped into any line. > > The number of letters in D is less than the number > > of letters in a line only when this line has more letters than any > > other line. > > There is no line with more letters than any other line, > > so no line we can use to limit D. > > > We only know it must be finite. > > > > Correct. If it exists, it must be infinite (or omega). Only then the > > > number of lines can be omega. But we have excluded that the number of > > > letters is omega. > > > > > > > No, we have not excluded omega. > > You wish to introduce a line with omega letters? > That yields what I always say: There is no infinite number of natural > numbers without an infinite number omega. All you are doing is calling the infinite number of natural numbers omega. Then you say, there is no omega without omega. You do not want to merely assume that omega doesn't exist. You want to show that assuming its existence leads to a contradiction. > > > > > If it does not exist you cannot use it to limit the > > > > diagonal. > > > > > > If it does not exist, then you cannot build a diagonal of that length. > > > > In which case you cannot use the fact that it does exist to show > > a contradiction. Yes, you can assume the diagonal does not exist. > > No, you do not get a contradiction if you assume it does exist. > > You don't get a contradiction if you assume that a supremum is taken > which, by definition, is not taken? > What definition says that a supremum is not taken? - William Hughes |