Cantor Confusion
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From: Franziska Neugebauer on 18 Nov 2006 15:46 David Marcus wrote: > Franziska Neugebauer wrote: >> mueckenh(a)rz.fh-augsburg.de wrote: >> >> > I used this suitable word because it allows to speak of lines, >> > columns and diagonal. >> >> _Define_ what you mean! > > That would make it too simple! LOL. Indeed. >> I suspect you are speaking of matrices, lines, >> columns and the like in default of a reasonable point of view. >> >> > If you don't like it, say triangle or structure. I >> > propose to use "infinite triangle" in order to be clear and to show >> > that your commentary below fails to show anything. Instead of >> > "square" we should speak of "equilateral". So we have an >> > Equilateral Infinite Triangle: EIT. >> >> Another misnomer. Cf. my posting in reply to David Marcus. >> <45550ba7$0$97245$892e7fe2(a)authen.yellow.readfreenews.net> > > At least I defined what I meant. And, I never said "equilateral"! I prefered "monangle" or simply "angle". F. N. -- xyz
From: Franziska Neugebauer on 18 Nov 2006 15:48 Virgil wrote: > In article <1163856355.707119.306700(a)m7g2000cwm.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > >> Virgil schrieb: [...] >> And there are lines as long as the diagonal is. > > Name one. The names "can" be found in the set of finished emptiness. F. N. -- xyz
From: David Marcus on 18 Nov 2006 15:49 Virgil wrote: > WM again has cart-before-horse-itis. He thinks he doesn't need a horse. -- David Marcus
From: Franziska Neugebauer on 18 Nov 2006 16:01 David Marcus wrote: > Franziska Neugebauer wrote: >> mueckenh(a)rz.fh-augsburg.de wrote: >> > Franziska Neugebauer schrieb: >> >> mueckenh(a)rz.fh-augsburg.de wrote: >> >> > What *counts* is this: It is asserted that the number of natural >> >> > numbers is omega >> >> >> >> "It" is asserted that the *set* of natural numbers is (does exist) >> >> and is named omega. The *cardinality* of omega is aleph_0. >> > >> > The cardinality of omega is omega. >> >> The cardinality of omega is |omega| not omega. > > Kunen's "Set Theory" defines |A| to be the least ordinal that can be > bijected with A. So, with this definition, |omega| = omega. You are absolutely right. But we debate the wrong thing instead of ,----[ WM in <1163782643.189768.296460(a)e3g2000cwe.googlegroups.com> ] | What counts is this: It is asserted that the number of natural | numbers is omega (the elements of the first column = number of lines). | By adding 1 element to the first column we see that, if this assertion | is correct, the number increases to omega + 1. But are there enough | lines? No. That means: The finite natural numbers are not sufficient | to stand in bijection with all omega elements of the column, but only | with the finite ones. Therefore the unavoidable conclusion is: There | are not omega finite numbers. `---- F. N. -- xyz
From: Virgil on 18 Nov 2006 16:09
In article <1163874845.525985.180420(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > >> omega "is" not "the first column". What you may write is, that in > > >> your sketch the first column represents omega. > > > > > > That is a matter of taste. > > > > You are in error. Set theory under discussion does not deal with > > "columns". > > This column contains all natural numbers in their natural order. It is > N, it is omega, written horizontally. A column as a set of elements may biject with N as a set of elements, but that does not make that column and N into the same set. And columns are only horizontal when they have fallen over. > > > > On the other hand omega represents what before Cantor was commonly > > > abbreviated by oo. > > > > Irrelevant to contemporary set theory. > > which you, unfortunately, don't know any better than its history. And which WM, by comparison, does not know at all. > > > > > Omega is the first transfinite number. > > > > Omega is the first transfinite ordinal number. > > > Please look it up in any modern text book. You will find there that > omega is a cardinal number too. Depends on the book. Omega has the smallest non-finite cardinality, but in some books it is not itself a cardinal. > > > > You need not interpret n as a set (though you can do it). > > > > In contemporary set theory almost everything is a set. > > In ZFC everything is a set. Not in every set theory. Which set theory does WM use in which it is otherwise? > > > > A treatise in which variables ("n") and number symbols ("0", "1", ...) > > do not refer to sets is not a treatise on set theory but > > a treatise of application of set theory, if ever. > > Don't mistake set theory with ZF or ZFC. Then which set theory does WM use? > > > > It is said to be the number of natural numbers. > > > > In contemporary set theory it is said that omega is the set of > > natural numbers [as Virgil pointed out the ordered set]. The number > > (cardinality) of omega is named aleph 0. So it is said that the number > > (cardinality) of the set of natural numbers is aleph 0. > > You are completely in error. The number (Anzahl) of a set is its > ordinal number. The cardinal number is something different > (M?chtigkeit). But even the cardinal number can be denoted by the > least ordinal which can be put in bijection with a set of that number > class. But need not be. And even so, there is a difference between "can be denoted by" and "is". Such distinctions are important in mathematics, and are ignored at one's peril. > > > > > > If you say that the diagonal can or even must be longer than > > > every line, > > > > I never said that a diagonal (d ii) i e omega "can" or "must" be longer > > than every (finite) line occupancy. What every clear-thinking person > > agrees upon is that the cardinality of the sequence (d ii) i e omega is > > greater than the cardinality of (the occupancy of) every single line of > > that "list" or "matrix". This is due to the fact that there are only > > finitely many occupied memebers in each line-sequence. > > I agree. But on the other hand, the cardinality of the sequence (d nn), > i. e. omega, cannot be > greater than the cardinality of every single line of that "list" or > "matrix". WM seems to be conflating the set of lines with individual lines. The "number" of terms in the diagonal equals the "number" of lines, but note that the number of lines exceeds the length of any one line, at least unless there is a last line and therefore only a finite number of lines. > If omega is a number which can be completed and even surpassed, then > there must be at least one line with omega units. False.Trivially false. Stupidly false. > > It is politeness. I could have said: Everybody whose intelligence is > not too far below the average level will understand this > interpretation. But I didn't. Wm certainly didn't understand any correct interpretation. > > > Knowledge of literature. Cantor. > > > > As you have been told quite some time before: Mathematics is no > > Zitierwissenschaft (quotation(s)/citation(s) science). > > Sometimes it is necessary to quote. In particular if you are uninformed > but nevertheless refuse to take advice from me. Those who decline taking advice from you, particularly on matters on which you are so obviously wrong, are just using their native good judgment. > In ZFC the cardinality > of a set S is the least ordinal alpha such that there is a bijection > from alpha to S. > > > You claim that the diagonal of a matrix can be longer than > > > every line. > > > > This is your wording not mine. > > It is not your wording, because you like to veil your inconsistencies, > but it is your opinion. Our claim is that the diagonal of the infinite list we have been discussing is necessarily longer than any single line, because it is at least as long as the next line, and there always is a next line in an infinite list. > > > > > As the diagonal is defined to consist of the ends of terms > > > of lines, this claim is easy to conradict. The diagonal entries are certainly determined by the end of line entries in a list in which each line is of length equal to its line index. Does WM dispute this? > > > > "ends of terms of lines" is your wording not mine. > > It is not your wording, because you prefer to veil your > inconsistencies, but it is your opinion. WRONG as usual. It is WM's claim and no one else's. > > > > > In order to defend your claim that there are infinitely many finite > > > numbers, > > > > In the lack of an effective offense I don't have to defend but to inform > > you of facts. > > Why then do you complain about the wording "the diagonal of a matrix > can be longer than every line"? "Every" is misleading when "any" is intended. > > > > > you could simply say: > > > > > > "There are more natural numbers d nn than natural numbers n." > > Your wording not mine. > > Can you understand mathematical symbols? Can you understand d nn <--> > n? We can understand why you want that misinterpretation, but it is still a misinterpretation. What we mean is that there is no line as long as the diagonal. Our justification is that each line is of finite l |