An uncountable countable set
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From: Tony Orlow on 5 Sep 2006 22:13 Dik T. Winter wrote: > In article <44fdcaf1(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> writes: > > Dik T. Winter wrote: > > > In article <1157367096.604428.36330(a)i42g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > ... > > > > A number has only one of the following properties: It is larger than or > > > > smaller than or equal to any natural number. > > > > > > So omega and aleph-0 are numbers. They satisfy the definition. Thanks. > > > > So, they are numbers which are larger than any finite number? Why then > > do we not consider an inductive proof of the form E y e N A x>y P(x) not > > to prove P(aleph_0) or P(omega)? > > That would be a new axiom, and you may consider it, but it leads to > contradictions when you retain the current definitions of aleph_0 and > omega. Yes, I am well aware of that. That's why I have chosen to reject those concepts in favor of finding something better, based on this infinite-case induction. Noting personal. :) Tony
From: Dik T. Winter on 5 Sep 2006 22:15 In article <1157465239.847094.136310(a)e3g2000cwe.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1157367591.092721.79760(a)i42g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > ... > > > > > Because I have the German text available and because I do not want to > > > > > be blamed of mistranslating. > > > > > > > > So you can blame other persons of mistranslating? > > > > > > Those who cannot yet read German but are interested in the origins of > > > set theory should learn to do it. > > > > Yes, everybody who is not able to read German is not allowed to speak > > about set theory and the origins... > > She needs translations which might be erroneous. (Compare: law, axiom) Yes, they need translations, but law in this case is not a good translation > > > > > > Cantor: So while a changing quantity x that successively > > > > > > takes the various values of finite numbers 1, 2, 3, ..., > > > > > > v, ... , is a potential infinite, on the other hand, a > > > > > > through the axioms completely determined set (N) of all > > > > > > integral finite number is an example of an actually finite > > > > > > quantity. > > > > > > > > > > Not through the axioms, but through "a law" (ein Gesetz). > > > > > > > > What is the difference? > > > > > > A law is derived from the natural properties of arithmetics. > > > > Oh. What law derived from the natural properties of arithmetics is he > > talking about when he gets the completely determined set of all integral > > finite numbers? > > That is but his conviction. I ask you what law, but you are not willing to answer? Again, what law is he using (note that in English law generally refers to Theorem). -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: MoeBlee on 5 Sep 2006 22:20 Aatu Koskensilta wrote: > Similarly, to the question of the status of > definitions in ordinary mathematics it is quite unnecessary to bring to > bear the formal theory of definitions in logic. I find that the formal theory of definitions illumines understanding of informal definitions tremendously. At least it does for me. > Much easier, it seems > to me, would be offer certain informal observations about definitions. > If these fail to clarify the matter, further elucidations could be > offered, and if these too fail, it seems a foregone hope that going > formal is of any use. Of course informal remarks were first offered, and by different posters. And of course anything more will fail with a crank. It would be a big mistake to think that the purpose of posting to a crank is to convince him or her of anything at all. > It seems your motivation, if I have understood you correctly, is simply > a wish to present the formal treatment as a proof, against the idiocy of > the loons you nominally offer them as a reply, of the possibility of > handling these questions in a rigorous and precise mathematical way, to > whatever spectators these "debates" may have. Yes, a kind of "for the record" gesture. Also, I admit that posting to cranks is often irrational. I mean, I don't require of myself that I have some rational purpose in engaging in a discussion that is, as far as acheiving intellectual understanding, doomed to failure anyway. And, by the way, I sometimes like to offer the formal version even outside the context of posting to cranks, just because I think the formal versions do shed light on the informal questions (at least they do for me) and I like the idea that they stand, in "for the record" style, available in the discussion. > > Many people, including me, have tried to intellectual contact with > > Orlow with many different approaches, including keeping things less > > technical (I've tried it too). Nothing works, since Orlow is indeed a > > crank. > > Indeed. What puzzles me is, why then try at all? I don't require of myself that I have a rational anwser to that question. Sometimes it's just a matter of reading a remark and having a feeling (I stress, a feeling) or an urge even, to have one's own view posted in contrast, for whatever it is worth, even for the sheer sake of it. Then that often leads to a rounds and spirals of responses back and forth. > The sensible course > seems to give up. I very much agree. > > So, the objective is not always to make intellectual contact - > > either by avoiding technicalities or by adducing them - but rather I > > have my own objectives in posting, such as the satisfaction I mentioned > > of at least putting on the table the rigorous formulations that I > > consider to be an important part of the intellectual achievments of > > mathematics. > > A much more constructive way to go about that would be to write up clear > expositions of these achievements, answering common questions and > dissolving common sources of puzzlement, published in some other way > than as buried in thousand message threads celebrating crankishness, surely? Yes, of course, that would acheive much more. But I don't post as a public service. Of course, like those who are generous to help me with individual questions, I hope to give good answers for the material I do know for beginners even more beginning than me. But as to overall posting, especially in crank wars, I offer no justifications and feel that none is required. > > Ha! Let's see you do it with Orlow. Yes, simple, informal arguments > > would be nice. But they're in vain with him anyway. The point of > > posting is not always to get through to the cranks. > > That much is obvious. After all, it is blatantly clear that there is > practically no hope of getting through to Orlow, Zick, Finlayson, > Easterly, and the others in the current bunch of loons and cranks people > have chosen to "debate". Usually the futility of such an exercise > becomes apparent after just a few posts - which is why it is somewhat > baffling to me that otherwise seemingly reasonable people wish to engage > in endless and pointless threads to the inhumane extents they do. Lack of self-discipline; lack of restraint. Had I more of those qualities, then indeed I would waste much less time posting to cranks. MoeBlee
From: Tony Orlow on 5 Sep 2006 22:30 MoeBlee wrote: > Tony Orlow wrote: >> MoeBlee wrote: >>> Tony Orlow wrote: >>>> Indeed, or we may NOT allow such things, if we adopt a rule of the form >>>> x = S: A P P(x) = P e S. >>> We don't need to go to second order to disallow urelements. The plain >>> axiom of extenstionality disallows urelements. You don't even >>> understand the axiom of extensionality. You're overflowing with >>> opinions about set theory but don't even know what the axiom of >>> extensionality says. >> Yes, I do. You think you are such an expert on what I don't know, but >> that's only because you can't think beyond what you do know. > > I read what you write. > >> What I am talking about is not covered by the axiom of extensionality, > > No, your point about disallowing urelements IS covered by the axiom of > extensionality. You may have other things to say about identity and > extensionality, but on the point of urelements, I just gave the > information you didn't know. My point was that that axiom only applied to sets, but I went further to say that even urelements can be considered to be equal to the sets of all 1--place predicates which apply to them, and so the axiom may be taken to be generally true. The big point there was about properties defining objects and sets. > >> which does not address the notion of properties defining objects or sets >> whatsoever, now, does it? >> It simply says that two sets whoich contain >> the same objects are the same set. How does that address the >> relationship between sets, elements, and properties? > > Obviously, the axiom of extensionality doesn't tell us everything about > sets that we get from the rest of the axioms. No, it doesn't. The axiom which DOES mention properties states that, given a set S and a property P, we can form a subset of S which includes all elements x of S for which P(x) is true. Given the universe as a set (possible, depending on how you define the universe), this would imply exactly what I'm saying, that every property defines a set. > >>>> That is, every object is a set of values which >>>> each property has when applied to that object. The truth value of each >>>> statement that may be made about x is EQUAL to the truth value of that >>>> property's membership in x. While some restrictions need to be made >>>> regarding what constitutes a property, this general notion seems sound, no? >>> No. What I don't understand is why you think you wouldn't benefit from >>> understanding what other people (such as those who have studied the >>> subject and written books on it) have come up with so that you can >>> contribute or even dissent on an informed basis rather than flounder in >>> your ignorance. >> If all you can produce is insults, what's the point? > > Because I produce more than deserved insults. For over a year, I've > given you not just deserved insults, but I've give you technical > explanations and technical facts, profuse informal explanations, > discussions about your own formulas (such as they are), and have > engaged also certain philosophical sidebars with you, as well as I've > provided you with references. Yes, at times you've been very generous, and I appreciate your contributions. You just seem rather cranky lately. Did I tell you I like your shoes? They're very nice! :) > >> You answer "No" to >> the general notion that properties of the elements define the set. > > I never said any such thing. I said that it remains to be seen how you > could have an object BE the set of its defining properties. See above. I asked you if the idea was reasonable, you said no, and went on a diatribe about my ignorance. But, okay, whatever you say. > >> Can >> you give any coherent reason why, besides my not having read the same >> books as you? > > I can't because I didn't say what you said I say. As to how an object > IS its set of defining properties, I can only await for YOU to show a > theory in which that all takes place. Okay, I've only recently really regressed it to that point, and am thinking about how to avoid difficulties in that area. Sure, it's all been done before, but then, where has IFR been all these years? > >>>> That's nice. Is there a difference between an abstract object like a >>>> set, and its definition? >>> Yes. >> What? > > The set is an object in some domain of discourse. The definition is a > syntactical object, which is a member of the theory but almost never > (if ever) a member itself of the domain of discourse. So, there are members of the theory that transcend the domain of discourse? Somehow, I thought the theory and its objects WAS the domain of discourse, but I guess some parts of the theory are not discussed, and therefore are beyond the domain of discourse. Is that what you're saying? I hope not. > >>>> If you change the definition, does that also >>>> change the set? >>> That a different definition may be of a different set does not imply >>> that the definition IS the set. >> The set consists of the elements within it and nothing more. The >> elements are distinguished from all other objects in the universe by >> some set of properties. Do you have a counterexample? > > No, because I never claimed otherwise. I set the set is not the same > object as the definition of the set. Is the set the collection of all the members within it? If so, how does one describe that collection, if not by properties of those members? > > MoeBlee > ToeKnee
From: Dik T. Winter on 5 Sep 2006 22:42
In article <44fd9eba(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> writes: > Dik T. Winter wrote: > > In article <44ef3da9(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> writes: .... > > > But without an axiom of infinity, it is demonstrable that, given the > > > axiom of internal infinity (continuity), x<z -> x<y<z, that any finite > > > interval includes an infinite number of points. Start with the line, > > > and points. There's infinity. > > > > Your axiom uses things that are not defined. What is the *meaning* of > > "x<z"? > > Geometrically I thought we were talking about arithmetic? > it means that x is left of z on the number line. What is "left" on the number line? Where are you looking from, and how are you looking? But whatever you are trying to do, arithmetic is defined without any reference to a number line. > It means > A y (y<x ^ y<z) v (x<y ^ y<z) v (x<y ^ z<y). That says about all it > needs to, wouldn't you say? No. > > > Describe those "other things". How are they "other"? > > > > You are expanding the reals in some undefined matter. > > That's the first time I've heard the H-riffics described as "expanding" > the reals. The usual complaint is they don't include ALL the reals, but > neither do the digital number systems, without an infinite number of digits. Makes no sense to me. > > Darn. Try to read. Cantor's proof is not about reals, it is neither > > about digital representations. It is about none of the things you are > > mentioning. But nevertheless you maintain that it is news that it is > > not about the reals, while you read what I wrote? > > I asked what you thought it was about, if not reals or symbolic systems, > and you refuse to answer the question, which is an answer in itself. I have explained it, but apparently you were to lazy to read. It is about sequences of symbols from a set of two elements. The symbols he used were 'm' and 'w', I do not know why. And that is not what I think it is about, but really what it is about. I have read his article. So it was not about reals. Nor was it about digital representations, those sequences represented nothing more that just the sequences they were. > You're not even curious enough to ask what I mean by more than two > states of set inclusion? Oh, well. :S This is the first time I read this term. I have no idea what it means, but probably something unrelated to the things we are discussing. > > > > > Larger than any finite. The set of naturals is as large as, but no > > > > > larger than, every natural. > > > > > > > > That is not a definition, because it makes no sense. "The set of > > > > naturals is as large as every natural"? > > > > > > It is no larger than all naturals > > > > That is something completely different again. > > It's not LARGER than every finite. Again something different. So it is smaller or equal to some finite? Can you come back when you have a definition that makes sense? > > > > From that: "The set of naturals is as > > > > large as 1", "The set of naturals is as large as 2". What is the meaning > > > > of these statements? > > > > > > That is when you substitute "every", meaning "each", for "all". Careful. > > > > Yes, you should be careful in what you mean, and not use a word that has > > multiple meanings so that you can be misunderstood. So I will refrase: > > > > > Larger than any finite. The set of naturals is as large as, but no > > > > > larger than, all naturals. > > Is that what you intended? In that case you just stated a tautology. > > If I say it's larger than all naturals, how do you read that? Why read > it differently when I add a negative to the sentence? Probably because it is a different sentence? But if you state that it is larger than all naturals I think I have no difficulty reading it. And that is the case. > > > > But they are wrong. The proof was *not* about the uncountability > > > > of the reals. The diagonal proof Cantor provided was not about > > > > that. It was a proof about the things I outlined just above. > > > > > > It was about power set and digital representation, which are identical. > > > It was about symbolic sets. > > > > You finally did read it? If so, you really should improve your German. > > Huh? Are you agreeing with my statement? I don't have to know German to > discuss mathematics. Not entirely. Power sets and digital representations are not identical. But apparently you need to know German to discuss with Wolfgang Mueckenheim. > > > > > I thought it was clear that I was using a notion of infinite, > > > > > like WM, from a quantitative standpoint, rather than > > > > > set-theoretic. > > > > > > > > Without definition. > > > > > > Greater than any finite. Simple enough? > > > > So the cardinality of the naturuals is infinite? > > Cardinality schmardinality. There is no natural with an index in the set > larger than any natural, and all there are in the set are naturals. > Nowhere has the set ever become infinite in count, as long as you only > count finite units. In this Wolfgang is correct. But I do not think WM is stating that. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |