Cantor Confusion
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From: Lester Zick on 24 Oct 2006 15:52 On Mon, 23 Oct 2006 22:06:05 -0400, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >Han de Bruijn wrote: >> And no, binary trees will not be found in Halmos' "Naive Set Theory". >> Because it's too naive, I suppose .. > >I don't see how you would know it is not there since you said that while >you own the book, you've never read it. Funny how books aren't much use >if you don't actually read them. > >Anyway, I didn't say binary trees were in the book. I said that if >Muckenheim wants to talk using the mathematical concepts of sets, >functions, and relations, he should use standard terminology, where >"standard" means the same as in some mathematics book. It is like saying >that if he wants to speak in English he should use words according to >the meanings as given in an English dictionary. So you're suggesting we should argue against a paradigm in terms set by the paradigm? So perhaps I should argue my definition of "infinity" in standard set analytic terms even though cast in Newtonian terms of the calculus? Curious to say the least. >Or, he could continue to do as Humpty Dumpty did and use his own >meanings for words without telling people what they are. I should have thought even modern math standard set analysts would be conversant with the calculus. Oh well. ~v~~
From: Lester Zick on 24 Oct 2006 16:04 On Mon, 23 Oct 2006 22:16:36 -0400, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >Dik T. Winter wrote: >> In article <1161377915.999210.39660(a)m7g2000cwm.googlegroups.com> "MoeBlee" <jazzmobe(a)hotmail.com> writes: >> > mueck...(a)rz.fh-augsburg.de wrote: >> ... >> > Okay, now that I asked for a definition of the relation you mentioned, >> > you're not giving that definition, but instead giving a >> > combinatorical/numerical argument with more terminology. What is a >> > "load of edges"? What is the definition of "a path carries a load of >> > edges"? If this is standard terminology in graph theory, then please >> > forgive my ignorance and supply me with the standard definition. If it >> > is not standard terminology, then please give me your own definition. >> >> By this time you ought to know that Mueckenheim *never* gives definitions. >> Or actually states that he is not able to give a definition for a >> particular term. Asking for definitions from Mueckenheim is as useful as >> talking to an eel. > >This does seem to be true. I suspect he doesn't know what the word >"definition" means in mathematics. "Definition" in mathematics would appear to mean pretty much whatever people who claim to do mathematics say it means. Ex: Cardinality is card(x)= least ordinal(x) equinumerous(x). In other words mathematical definition is when definition(x) defines mathematical definition(x).Or mathematics is when mathematician(x) and a mathematician is when mathematics(x). I don't know whether M. gives mathematical definitions or not.However I know a great many mathematikers who definitely don't. ~v~~
From: Virgil on 24 Oct 2006 16:26 In article <1161687405.722976.229350(a)f16g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > cbrown(a)cbrownsystems.com schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > cbrown(a)cbrownsystems.com schrieb: > > > > > > > > Of course we cannot get to mega by counting. According to set theory > > > > > we > > > > > get to omega by the limit: > > > > > > > > No, according to most set theories, we "get to" omega by simply > > > > assuming it exists. > > > > > > We get omega by assuming it exists. We get to omega by limits. > > > > No, I mean we get to omega by assuming omega exists. We do not get to > > omega by limits; unless in some bizzare way, you think the phrase "We > > get to omega by limits" is logically equivalent to the statement "We > > get to omega by assuming that omega exists". > > Idle discussion. > > > > > > > > > > lim [n-->oo] {1 + 1/2^2 + 1/3^2 + ... + 1/n^2} = (pi^2)/6. > > > > > > > > Here, one assumes you mean using the usual metric topology; > > > > > > No. This formula stems from Leonhard Euler. I works without any > > > topology. At that time topology did not exist. > > > > So, you claim that the above statement follows from no definitions at > > all? Instead it follows simply because Euler said it? > > No because one can prove it by using mathematics. Does that mathematics require a notion of the "distance" between two reals? If so then it is essentially provable because of metric topology, even if one does not call it that. > > > > > That observation shows but an inconsistency of set theory. > > > > No, it shows that if we are not specific about what we mean by "lim > > n->oo", it is possible for two people to have two entirely different > > definitions of "lim->oo", which result in two entirely incomatible > > results. > > No. When the direction is given (like here: t varying from 1 to oo), > then the limit is unique. WRONG! Since there are varying definitions of limits, not all of which give the same result, the result is, at least in part, dependent on which definition one is using. > > > > That's why it is generally wise to state what you mean explicitly by a > > statement such as "lim n->oo", in order to prevent confusion. > > > > > > > Cantor, in > > > his first proof, used the notation a_oo. Later he changed oo to omega. > > > > And from this, we can draw what conclusions? > > 1) First he did not distinguish betwenn potential and actual. > 2) Omega is what he meant by actual infinity. That may be your take on it, but need not be anyone else's. It might, for example, merely mean that Cantor later distinguished between omega and aleph_0 > > When Euler stated that (something) = pi^2/6, do you think he considered > > pi to be a number? Do you think he considered that pi could be divided > > by the number 6? Do you think that he felt that he was justified in the > > conclusion that pi^2/6 = pi*(pi/6)? > > He certainly considered it a number, because as far as I know he did > not know about the finite contents of the universe. To consider that Euler would have to think like "Mueckenh" if he were alive today is an insult to Euler. > > > > Conversely, do you think Euler would have considered that beauty could > > be divided by the number 6? Do you think he would have felt that it > > followed that beauty^2/6 = beauty*(beauty/6)? > > I don't know about Euler. But Pythagoas did so: Everything is number. And is every such "number" divisible by 6, according to Pythagoras? .... > That mathematics which is compatible with physics. Yes. No set theory. > It is a process of discovery where one is based upon the other and the > other is based upon the one. This is merely the common physicist claim that all mathematics is a subset of physics. From a mathematicians point of view, it makes more sense to claim that all physics is a subset of mathematics. > > > > In order to get out of bed in the morning you need beliefs and > > > > assumptions - you need to believe, for example, that the floor will > > > > support your weight, and that you will not instead plunge into a fiery > > > > hell. > > > > > > > > In order to have a reasonable mathematical discussion, we need to > > > > /agree/ on what we are talking about. Those agreements are codified by > > > > axioms; > > > > > > That is the present, deplorable opinion of the average mathematicians. > > > > What is the alternative? That we all blindly stumble about, > > misunderstanding each other's statements; because for example what you > > call a real number is different than what I call a real number? > > That is but a definition. > > > > I think instead you simply want all other mathematicians to naturally > > agree with your statements, whether those statements are logical or > > not. > > The statements must be realistic. If I hear that it needs proof that an > ever increasing positive function is larger than zero, then I see that > your kind og mathemaics is no mathematics. If I read the reference correctly, the function in question is not ever increasing, at least not as a function of time, it is constant almost everywhere, and as a real function, having a fixed jump discontinuity on a set of points having 0 (noon) as accumulation point and therefore necessarily discontinuous at 0. > Well, then let's take the set of al sets. How precisely do you claim that you can do that in ZF, with or without AoI? Certain sets are "takeable", like {}, but there are limits to what you are allowed to "take".
From: Virgil on 24 Oct 2006 16:30 In article <1161693049.753655.156520(a)m7g2000cwm.googlegroups.com>, "georgie" <geo_cant(a)yahoo.com> wrote: > David Marcus wrote: > > georgie wrote: > > > > > > David Marcus wrote: > > > > georgie wrote: > > > > > > > > The OP said that a definition was invalid because it was self- > > > > referential. However, there is no rule against self-reference in modern > > > > mathematics, so the OP's objection is not valid. > > > > > > So the set of sets containing themselves is ok by you. > > > > I don't see where I said that. I said there is no rule against self- > > reference. The rules are specified by the axioms of ZFC. > > So it's valid or it isn't and it's only valid when you like it. Is > that your > position. From your reactions it appears that way. If you don't > explain > why you think self-reference is ok in some cases and not others, you > lose credibility. "Self-reference" within ZFC is allowable when permitted by the axioms of ZFC. Anyone, like "georgie" who objects to that, is the one losing credibility. > > > > > Almost a century ago, Russell and Whitehead attempted to develop such > > > > rules as a way of avoiding the paradoxes, but their approach was too > > > > cumbersome. So, ZFC avoids the paradoxes in a different way. The > > > > diagonal argument follows the rules of ZFC. If you want more details on > > > > what the rules are, there are quite a few good books on the subject. > > > > > > This is a bunch of BS. There are no references to ZFC in Cantor's > > > proof. > > > > All the steps of Cantor's argument can be justified within ZFC. People > > don't normally trace their arguments all the way back to the axioms. The > > whole point of Mathematics is that we build definitions and theorems and > > then use those to build more. > > > > Which books on set theory have you read? > > All the ones you've read plus some more.
From: Dik T. Winter on 24 Oct 2006 19:08
In article <1161683860.442902.89560(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1161518008.776999.238550(a)k70g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: .... > > > Not I define omega as the limit ordinal number. That is a matter of set > > > theory. > > > > You use oo, which is *not* the same as omega. > > oo is not the same as omega. Correct. But if omega exists, then the > limit n --> oo is omega. Only when you have *defined* that limit. And I would like to define it only when all 'n' are ordinal numbers. But in that case you can not talk about oo... > > But when we ignore that > > difference, more problems prop up. X(t) was a function from R to R, I > > think, not a function from ordinals to something else. But you stated > > that X(t) was 9.t. Again lacking precision. > > Absolute precision. X(t) = 10*t would be a bit sloppy. No, no precision, because you did not state what t was. I think t was a real, but you do not state that at all. > > There is no continuity required to find such a limit. Limits are > > independent on continuity. The function: > > f(x) = 1 when x = 0 and = 0 when x != 0 > > is certainly not continuous at x = 0, nevertheless the limit for x -> 0 > > exists, and is 0. What continuity do you use when you calculate: > > lim{x -> 0} f(x) > > ? > > If f(0) is undefined, i.e., if 0 is not element of the domain of f, > then we can find f(0) = 0 by l'Hospital' s rule. Ever heard of? In your > example we have no limit because of lacking continuity. There *is* a limit, but the limit is not equal to the function value, due to the discontinuity. Consider: lim(x -> 0) f(x) = a if for all epsilon > 0 there is a delta > 0 such that for all x != 0 and |x| < delta, |f(x) - a| < epsilon check a book on analysis. You will see that the function value at x = 0 plays *no* role in the definition. For my function, lim{x -> 0} f(x) = 0. > For every > interval delta covering the value x = 0, we cannot find an epsilon < > 1/3. And as you may know, there is no x > 0 which is closest to 0. With that definition of limit a limit would only exist if the function is continuous at the limit point, not when it is discontinuous, neither when it is not defined at that point. BTW, I wonder how you would use l'H?pitals rule for the following function: f(x) = 1 for x != 0 pray explain it. > > Yes, so what? I asked for a mathematical definition, not for handwaving. > > And none of the usages of Cantor do in any way define the limit. What > > *is* your mathematical definition of that limit? > > Learn mathematics, then you will understand the mathematical definition > of the set of all natural numbers as imagined by Cantor and unchanged > until toda: lim {t --> oo} {1,2,3,...,t} = omega. "Unchanged until today"? Where do you get *that* from? Pray provide me with the title of a text book on set theory where the limit of a sequence of sets is even *defined*. > > > Wrong is wrong. The limit *is* the irrational number. You can use > > > these and only these numbers in a Cantor list, not the equivalence > > > classes of sequences. > > > > You really do not understand how the reals are defined. The limit is > > *not* the irrational number. The limit does not even exist. This was when we were working in the rational numbers, to define the real numbers. > Therefore you cannot use the reals in a Cantor list. They even don't > exist. Within the *real* numbers the limit does exist. And a decimal number is nothing more nor less than a representative of an equivalence classes. > > Pray re-read what I wrote. > > Why? You do not explain how the reals are used in a Cantor list. I did explain how the reals were defined. > > The real numbers are defined as equivalence > > classes of sequences of rational numbers. The sequences do not belong to > > Q. They are sequences of elements of Q. So when defining reals (according > > to this methodology) we start with sequences of rationals. We call two > > sequences equivalent if their difference goes to 0, the concept of limit > > has not yet even been defined. > > Without Cauchy convergence you cannot prove equivalence. There is convergence to 0, no other convergence is involved. And with that you can prove that it is an equivalence relation. > > > It is easily shown that that is an > > equivalence relation, so we can divide the sequences in equivalence > > classes. Each of those equivalence classes is a real number. So a > > real number is an equivalence class of rationals. > > Equivalence classes cannot be defined without Cauchy convergence, which > is an improved definition with respect to the simple limit. Equivalence > classes cannot be used in a Cantor list, because the terms of he > sequences differ from one another. What has this all to do with a list of reals? I did tell you how the reals are defined! And the only thing used in the definition is convergence to 0, not even limit. Once we have defined the reals we can state that every Cauchy sequence of reals is convergent and has a limit. > > It is *extremely* losely speaking. The real (not irrational) number is > > an equivalence class, it is not a limit. So initially, 1/2 is *not* > > an element of this new system. An element of this new system is an > > equivalence class of which the sequence: > > 1/2, 1/2, 1/2, ... > > is a representative. > > What about the Cantor's list in this case? I do not understand why you want to tie that list in with the definition of the reals? > > > You > > > can use this and only this number in a Cantor list, not the equivalence > > > class of sequences (because the due terms are not uniquely defined). > > I see. Of course you see, because you wrote that yourself. > But you should try to understand. In Cantor's list, there are > limits of sequences, not equivealence classes. The list is a list of equivalence classes. In the decimal case representatives of those equivalence classes are used. > > > Summarizing the original question: You need the limit omeg |