Cantor Confusion
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From: Virgil on 2 Nov 2006 15:17 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. His claim is trivially and obviously false, but he keeps repeating it ad nauseam, as if by sufficient repetition of that lie , he can make the truth go away.
From: Virgil on 2 Nov 2006 15:38 In article <1162471212.373690.109320(a)m73g2000cwd.googlegroups.com>, 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. > > > > 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? WM conflates "whole" with "natural". > > > 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"? > > > 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. Note that in ZF, such universal quantifiers, and existential quantifiers, are required to be limited to members of some pre-existing set, which must be specified before a meaning can be assessed. Until such specification is made, "A n: n < omega" is meaningless in ZF. > > > If not, why do you complain about non-standard meaning on Cantor's > > > definition of omega as a whole number? What we complain about is that it has no meaning until "whole number" is defined. > > > > 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. In terms of ZF, your position does not even exist. > > > 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? If by N you mean a set containing all of the finite ordinals, and if by ">" you mean either contains as a subset or contains as a member, then yes. Otherwise, probably not. I know that > modern set theory says so. If something can be larger than a number, > then it must be a number. In the sequence of ordinals, starting with {} and minimally closed under x -> x union {x}, then s = {{}, {{{}}} } is "larger than " {} but is not an ordinal. And not every set of ordinals is an ordinal. > 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".
From: Virgil on 2 Nov 2006 15:45 In article <1162471594.046811.299910(a)f16g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > 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. There has been no compatible with ZF version posted. And absent any we will continue to view your version as incompatible with ZF. > > > > > > 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. Your time would be better spent learning to do so. > > 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. Except that with nothing assumed nothing can be proved. Until we have a clear list of what assumptions WM makes to justify his alleged proofs, he is merely blowing hot air. > > > > > > 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. But everything smaller does. That means that for every n in N there is an nth digit in the diagonal. > > Regards, WM
From: Virgil on 2 Nov 2006 15:53 In article <1162471702.093869.290570(a)h54g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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. The above in not correct. Without an axiom of infinity one cannot prove the existence of infinite sets, but, equally, without an axiom of finiteness, one cannot prove that all sets are finite. Absent both axioms, the issue is incapable of being settled either way. And any claim that one can prove all sets finite without any assumptions is absolutely false. > > In modern set theory you cannot say for all sets Ax x is finite. In ZF you cannot even say "for all x" unless you have a set known to exist from which the x's are to be taken.
From: MoeBlee on 2 Nov 2006 15:59
mueckenh(a)rz.fh-augsburg.de wrote: > 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. Wrong. If we're talking about the axiom of infinity, then we're talking about a set theory or some other theory you are welcome to specify. As to set theory, for the tenth time: Without the axiom of infinity it is UNDETERMINED whether 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. In Z set theory: ~Ax x is finite -> Ex x is infinite. I don't know what point you're trying to make. Whatever your point, you won't be able to show that merely dropping the axiom of infinity from the Z axioms entails that there are only finite sets. > Modern set theory simply cannot describe developing sets as > it apparently cannot describe sets with limited contents of > information. Whatever your definition of "developing sets" and 'limited contents of information", the fact remains that dropping the axiom of infinity does NOT entail that there are no infinite sets. > 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. Oh please, I've read more in that book than you have. But what is more important, at least I know what that book is ABOUT, as opposed to your unfamiliarity with the rudiments of the subject. And Fraenkel, Bar-Hillel, nor Levy would never agree with you nonsense that merely dropping the axiom of infinity entails that there are only finite sets. > 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) Especially notice the scare quotes in that passage around "growing" and "produce" (if those are scare quotes in the original quote). I highly doubt that in context that is meant to be a claim that literally there are sets in set theory that grows bigger and bigger (as opposed to our option to adopt different definitions of a universe for set theory). I highly doubt that the full contex of that quote permits it to be taken literally that any given definition of a universe for a theory is not of a fixed set or class. (I will look at the book again, and that passage, next time I'm at the library.) > [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) And there has been progress made even with proposals for intuitionistic set theories. That doesn't alter that in Z set theories, dropping the axiom of infinity does not entail that only finite sets exist. MoeBlee |