Cantor Confusion
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From: Virgil on 16 Nov 2006 16:34 In article <1163705561.843253.9860(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > > > Indeed. If people *object* to an axiom, that is philosophy. > > > > > > But if people choose a set of axioms, that is what? > > > > > > > Everyone is welcome to choose their own axioms. > > > > > > That's mathematics? > > > > Of course. > > And if people not decide to use an axiom, that is what? A different axiom system, as in ZF being a bit different from ZFC. > But if people decide not to use an axiom, that is philosophy? Merely a different axiom system. > > > > Would like to do. Please le me know which words are available in your > > > universe of discourse. > > > > I told you several times that the terminology in any modern textbook is > > fine. For some reason you do not like this answer. > > I told you the terminology used in a modern textbook to show that > finished infinity is used there. For some reason you do not like to > understand it. > > "Some mathematicians object to the Axiom of Infinity on the grounds > that a collection of objects produced by an infinite process (such as > N) should not be treated as a finished entity." > No one is forced to use any particular axiom system unless they want results that only that axiom system can provide. If one wants, for example, any of the common versions of set theory, one is constrained to choose among those axiom systems that provide such a system or go off and try to invent a new axiom system on their own that will provide a theory which satisfies them. > > > > If an actually infinite set of numbers existed, and if neighbouring > > > elements had a fixed distance from each other, then the set must > > > contain an infinite number. > > > Is that a "no" or a "yes"? > > Read again, simplified: If neighbouring elements have a fixed distance, > the answer is yes. > If neighbouring elements have not a fixed distance like the rational > numbers: the answer is no > > Regards, WM Then, according to WM, one can have a set containing NO infinite element, such as the rationals, and a proper subset of it which MUST contain an infinite element, such as the integral rationals. I find such self-contradictory systems quite unsatisfactory. As do, I suspect, all those who are mathematically at least minimally competent.
From: Virgil on 16 Nov 2006 16:46 In article <1163705807.229775.45500(a)k70g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > This sequence does not have a last element: > > > > 1 2 3 4 5 ... > > > > This last sequence has three dots on the right. None of the other > > sequences do. So, this last sequence is clearly different in some way > > from all the other sequences. > > Yes. We do not know its last element. We do not even know that it has a "last" element. > Perhaps it can change its size. A sequence, being a function, a sort of set, is of fixed "size", in the sense that its membership does not vary with time or with any other "variable". > But mathematics does not obey commands. Neither interpreted as a > command nor as a magic formula the "..." can create infinity. It is not "infinity" so much as "infiniteness" that we are interested in. > > > > "Crank" means someone who makes up pejorative names for people whose > > language they do not understand and thinks that definitions which are > > explained in many books are "private". > > So you are a crank? Look in the nearest mirror to find one. > > How has it been done over 4000 years? With axioms, often implicit, but axioms, nevertheless. And over the course of that 4000 years mathematicians have discovered the dangers of implicit axioms, so that the most secure mathematics requires all axioms to be explicitly stated. > > > I can verify I'm using words with the same meanings as other people by > > asking them what definitions they are using, then seeing if they are the > > same as mine. > > Why do you say "we" if you talk about mathematicians? Because you are not one of us.
From: mueckenh on 17 Nov 2006 07:49 David Marcus schrieb: > > > > I see. But recently you used the word "completed infinity". > > > > > > I don't think I ever said that. Do you have a quote? > > > > Here it is: > > In the below post, I was just trying to paraphrase what you are saying. Please don't try to paraphrase what I said, because I don't believe that you understand it sufficiently. So in most cases you will fail to repeat my ideas. Please quote only full sentences. > I didn't say I would say that or that I understood what you were trying > to say. The question was whether you "ever said that" it. I hope this question as been settled now. > In fact, I don't know what you you mean by the phrase. Did you > really misunderstand what I wrote? There is no misunderstanding possible. You refuted Lester's interpretation, by proposing to have a better one: "That doesn't seem to be what WM is saying. He seems to be saying that the notion of a completed infinity leads to either absurdities or contradictions. Perhaps he thinks the way to avoid these absurdities is to only consider things that can be physically produced." I now that cranks never admit having made an error. But do you think that obvious lies like this are a way to reach your aim?> > > I don't know what you mean by "completed anywhere". The completed initial segment contains every natural number. Another segment contains not every natural number. Regards, WM
From: mueckenh on 17 Nov 2006 07:55 Franziska Neugebauer schrieb: > > In my approach, the lines contain unary representations of natural > > numbers. > > What the lines contain (occupancy) does not effect the number of > sequence menbers. It does, if the members enumerate themselves. This is the case in the present EIT: 1 12 123 .... > >> Equivocation: "Adding one element" names two different things > >> (changing the occupancy vs. changing the domain). It names two different things only if there are two different things in the initial bijection. But that cannot be the case because these things are identical (except that one is noted vertically and the other one horizontally.) Hence, any difference can only be that the nunmber of these things is different. That is what I proved. > > The question is whether there is a bijection between the > > initial segments of the first column and the lines like > > > > 1 > > 2 > > 3 > > ... > > n > > and 1,2,3,...n. > > ISOTFC := { {1}, {1, 2}, {1, 2, 3}, ... } > L := { 1, 2, 3, ... } > > B := { <{1}, 1>, <{1, 2}, 2>, <{1, 2, 3}, 3>, ... } > > The bijection between the initial segments of the first column (ISOTFC) > and the lines (L) is explicitly contructed. I have named it "B". I > cannot see any reason to debate about the existence of B. This bijection contains only finite sets. If you add one element to a finite set, then you get a finite set. If you add an element to an infinite set you get a set of ordinal number omega + 1. Therefore a set of ordinal number omega must exist (it is the first column) but it does not appear in your bijection. The ISOTFC is no part of your B. > > > If this is possible, and if the ordinal of the first column is omega, > > What does "possible" mean? Sets are or are not. The opposite of impossible. > There is no set > theoretical notion of "possibility of sets". But an explanaion: The set of all sets is an impossible set. > > Sets (including bijections) are neither possible nor impossible. They > exist or do not exist. If they exist, then they are possible. If not, then they are impossible. Regards, WM
From: mueckenh on 17 Nov 2006 07:57
Franziska Neugebauer schrieb: > > The natural numbers count themselves. Bijection of initial segments of > > column and lines > > > > 1 > > 2 > > 3 > > ... > > n <--> 1,2,3,...n > > A n e omega. > > > If there is no infinite number then there are not infinitely many > > numbers. > > 1. If *where* is no infinite number? There is no infinite number among the lines 1,2,3,...,n. There can be no bijection with the completed first column. > 2. Define "infinitely many" [, please.] Here I mean with "infinitely many" a number which, like the column, by adding 1 element yields a transfinite number omega + 1. Regards, WM |