Cantor Confusion
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From: David Marcus on 2 Dec 2006 23:52 Eckard Blumschein wrote: > On 11/29/2006 7:41 PM, MoeBlee wrote: > > Eckard Blumschein wrote: > >> On 11/29/2006 5:24 PM, mueckenh(a)rz.fh-augsburg.de wrote: > >> > Then we have finite sets without > >> > a largest element. > >> > >> Why not with a growing largest element? > > > > Because Fraenkel, Bar-Hillel, and Levy said nothing that implies any > > such thing as that there are finite sets without a largest element. WM > > is just latching onto a particular quote(that even uses scare quotes) > > out of context and with utter disregard to what the quote is meant to > > actually summarize. > > I will look into the book myself. I think it is enough that WM is misinterpreting the book. I don't think you need to do so as well. You've already misinterpreted the writings of enough historical figures. > Everyone of us my be wrong in details. Some people are wrong in all details. > I guess, we cannot be cautious enough with respect to possibly changing > meaning of words. Some people aren't cautious at all, and think they understand things that they do not. -- David Marcus
From: David Marcus on 3 Dec 2006 22:19 mueckenh(a)rz.fh-augsburg.de wrote: > Randy Poe schrieb: > > MoeBlee wrote: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > MoeBlee schrieb: > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > That is not the question any longer. The question is: Can a set > > > > > > theorist admit that she is in error? It seems impossible. They all are > > > > > > too well trained in defending ZFC. > > > > > > > > > > In error as to what? Set theorists admit mistakes. It is not uncommon > > > > > for books to have errata sheets attached. > > > > > > > > Would you see a contradiction in these two statements? > > > > 1) "The cardinality of omega is |omega| not omega." > > > > 2) "The cardinality of omega, also written as |omega|, is omega". > > > > > > If you give me the context of those remarks, I might have something to > > > say about them. > > > > In its original context, statement #1 was about the NOTATION for > > "cardinality of omega". > > No. The first statement was from a set theorist who was not aware that > it is usual to denote the cardinal number aleph_0 by omega. This is not true. It is not usual to denote aleph_0 by omega. People who don't understand mathematics (like you) should carefully distinguish between cardinals and ordinals. They are not the same. > > In its original context, statement #2 was about a theorem > > that |omega| = omega. > > No. The second statement was from a set theorist who meanwhle had > learnt that it is usual to denote the cardinal number aleph_0 by omega. Nope. Speaking of admitting errors, when will you admit your myriad errors? -- David Marcus
From: David Marcus on 3 Dec 2006 22:29 Eckard Blumschein wrote: > On 11/22/2006 12:16 AM, David Marcus wrote: > > > It is hard to be always wrong. > > In what, according to your opinion, WM is not wrong? I believe I was referring to the fact that you agreed with Virgil, and hence might be right about something. However, WM does, now and then, say something that is correct, but generally immediately says something that is absurd. -- David Marcus
From: mueckenh on 4 Dec 2006 05:38 Dik T. Winter schrieb: > > Therefore we can denote a set by X and we can say that the set X grows. > > Wrong, again. Consider X a variable that stands for an unspecified > integral number. Can you state that the integral number X grows? Of course the unspecified number grows by taking the values of specified numbers. > No. You can state that X grows, but not that the integral numbers > grow. In the above X and Y are variables that denote sets in every > instance. But X and Y are *not* sets. > > > That is nothing else than to say that the number of states in the EC > > grows. Of course the number 6 has not gown to 25. But it is simply a > > matter of definition, how one interprets "to grow" and "number". > > In that case, please provide a definition. Everybody knows what the number of ther EC states is. > > > > That is not "the set of states". You can talk about "the current set of > > > states" or about "the set of states in 1957" or whatever. At least > > > mathematically. In mathematics, by definition, a set can not grow. > > > > Wrong. Read my explanation above. > > Wrong. Read my explanation above. I am talking mathematics. Do you really think so? > > > > You are, of course, entitled to use another definition, but that will > > > not clarify the discussion at all (and you are not using standard set > > > theory). > > > > Hrbacek and Jech teach standard set theory including the fact that in > > ZF everything is a set. > > Yes? I remember that you opposed. Now you agree because you have learnt? >Do they define sets as allowed to grow? Not in the quote you supply. > There they talk about set valued variables that can grow. No. X and Y do not grow, they remain "X" and "Y". The set they denote does grow. The number of EC states may be n. "n" does not grow. The number denoted by "n" does grow. > > > > > It is simply a matter of definition. > > > > > > Yes, with your definition a set can grow, but you put yourself outside > > > set theory, and you must at first consider all results from set theory > > > unproven theorems in your theorem, and you need to prove them (if > > > possible). > > > > Hrbacek and Jech teach standard set theory including the fact that in > > ZF everything is a set. > > But I see nothing that states that a set can grow. Pray reread. If you do not yet understand it reread again. Regards, WM
From: mueckenh on 4 Dec 2006 08:03
William Hughes schrieb: > > > Informally we have that a potentially infinite set is a set > which is always finite, but to which we can add an element > whenever we want. We say that x is an element of > the potentially infinite set if we can add enough elements > to get to x. Yes. In particular this method of adding elements guarantees that such a set can never be uncountable. > > > > A potentially infinite set like N can be produced by induction. > > Every set "produced by induction" has a largest element. > The set N does not have a largest element. The set N produced by induction is potentially infinite. It does not have a largest element because it is not a fixed set but a set which can grow. It has at most a temporarily largest element. > > >An > > actually infinite set cannot be produced by induction nor can it be > > produced by other means because it is simply nonsense. The only way of > > getting it is to postulate its existence by an axiom, although the > > degree of nonsense is not reduced by this method. > > > > > > Please do not mix up potential and actual infinity. Actual infinity is > > > > required, e.g., for the real numbers. > > > > > > > > > Piffle. > > > > > > The rational numbers form a potentially infinite set A. > > > > Try to learn the commonly accepted meaning of "potentially infinite". I > > am not wiling to discuss your personal definitions. > > > How does the definition I am using differ from the > 'commonly accepted meaning of "potentially infinite"'? You consider complete sets. Potential infinity is an unending process. > > The set of numbers is potentially infinite. So the real numbers > are potentially infinite. So you can construct the set of all real numbers (of the interval [0, 1] in binary representation) by: 0.0 0.1 0.01 0.11 .... This set is countable. Regards, WM |