An uncountable countable set
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From: Dik T. Winter on 5 Sep 2006 20:59 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. -- 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 21:15 Aatu Koskensilta wrote: > Tony Orlow wrote: > > So, no, I don't understand a thing you're saying, when you say naturals > > aren't reals. Sorry. > > MoeBlee's referring to a technicality about the usual set theoretic > construction of the various number systems. In that construction it will > be true that, strictly speaking, the natural number 1 is not the real 1, > although of course the naturals are embedded in the obvious fashion in > the reals. Such technicalities are usually entirely irrelevant, I don't think they're entirely irrelevent to the foundations. But I couldn't have belabored the point more in my original posts that at the stage of working with the real numbers, the distinction between the naturals as embedded and not a subset is pedantic and that for all practical purposes, at the stage of working in analysis, we may as well regard the naturals as a subset since we are within isomorphism anyway. > and one > can note that there are alternative constructions under which the (set > theoretical representation of the) natural 1 is the (set theoretical > representation of the) real 1. Yes, and I made clear that I was referring to the Dedekind cut or equivalence class of Cauchy sequences methods only. > One might also note that addition, > multiplication and so forth as functions of reals have quite different > properties than as functions of naturals, which might also make it > worthwhile to be extra-pedantic in some situations. And I think the constructions are worthwhile onto themselves as well. At least I find worth in them. > As a general observation, it seems that when arguing with cranks people > often have a habit of preferring overtly formal expositions and > arguments, stressing somewhat irrelevant technicalities. You'd have to go back to the original context of my remarks to see if the technicality is irrelevent. In any case, relevent or not to dissuading cranks, the point I made is correct and deserves to be upheld against ignorance on that basis. > This is > certainly understandable - and some people probably just enjoy going > through the formal details, possibly working out some of the details for > themselves for the first time That might be why some people mention such details, but it is not my reason. My purpose is to bring to the table the rigorous alternatives to the incorrect and vague formulations of cranks. Where a particular matters are under question, I feel that the rigorous formulations that have already been made to deal with such a matters deserve to be mentioned as a contrast to the nonsense of cranks. >- but seems, to me, ultimately > counterproductive; if the objective is to actually establish > intellectual contact with the crank, 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. 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. > and have him see the error of his > ways, surely most simple, informal, conceptual arguments and > explanations are to be preferred, setting aside all technicalities that > really are irrelevant - cutting through the crud, focusing solely on the > actual issue or confusion? 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. MoeBlee
From: Tony Orlow on 5 Sep 2006 21:26 MoeBlee wrote: > Tony Orlow wrote: >> MoeBlee wrote: >>> Tony Orlow wrote: >>>> The difference between = and <-> disappears when logical truth values >>>> are quantities from 0 through 1, so I don't see that as any better, but >>>> equivalent. >>> You say, in the absence of having specified a syntax for a language in >>> which this all happens. >>> >>>>>> 'equality' would be a better word than 'equivalence' here, I think. >>>> I suppose, though the same applies to "equivalence classes" doesn't it? >>>> No matter. >>> No, that is the point. There is a difference between members of an >>> equivalence class and the equivalence class itself. >> I didn't say that all objects within a class are EQUAL, > > I didn't say that you said that all objects in a class are equal. In > fact that is why it is imporatant to keep clear the difference between > equality and equivalence. Ugh. You keep missing the point. It must be deliberate. > >> but given some >> criterion for distinguishing objects, one can form CLASSES where a given >> property is the same for all members of any given class, ignoring all >> other properties. > > That's pretty much what we do in set theory. Except we don't "ignore" > other properties; rather, we just have only a finite number of > primitives. And we don't need to refer to classes; rather sets x and y > are the same set. Yep. You're doing it on purpose. There is no other explanation for you ignoring that I have already said that finite theories with a finite number of primitives can only address a finite number of properties. That's why I was saying they are "subjective" in a sense. But you didn't let on that that made any sense either. Very perplexing, now that you say the same thing back. > > .> >> Yes, it's that simple. If the object IS the unique set of logical > values >>>> applied to all properties, then each unique set of logical values for >>>> each statement about an object IS a unique object. :) >>> Whatever that means, I doubt it is the principle of the symmetry of >>> identity, which is that a=b <-> b=a, which makes no mentions whatsoever >>> of "logical values" or "properties". >> All I was saying is that if the set of property values IS the object, >> then the object IS the set of property values. Is that so difficult to >> understand? > > I understand that it is your postulate. But it is not the same > statement as the symmetry of identity. And apparently that IS difficult > for you to understand. Ugh again. Is is = = ???? Is IS =. = = is. Got it? x IS the set S of property values defining it. x=S <-> S=x. Comprende? Dios mio! > >>>>>> and the >>>>>> inability to discern two objects by their properties makes them equal, >>>>>> at least until some property is discovered which can discriminate >>>>>> between the two. >>>>> That's going to make the theory subjective - depending on discoveries. >>>>> Why don't you look at how different mathematical theories handle >>>>> identity? >>>> Ummm.... Isn't each isolated theory "subjective" in terms of the >>>> properties that it explores? >>> A theory is a set of sentences closed under entailment. Theories are >>> not made subjective for our reasons for interest in them. The >>> subjectivity is in our deciding to study one theory and not another, >>> but as a set of sentences closed under entailment, the theory itself is >>> not affected by whether we are interested in it or not or by our >>> reasons for interest or disinterest in it. >> You must need another cup of tea. I am not talking about psychological >> subjectivity, but the fact that any normal theory only addresses certain >> properties of the objects is discusses, and therefore may not have >> distinctions that are available in other theories. > > Sorry for not reading your mind when you mentioned, in your own scare > quotes, "subjective". As to your point, I don't want to comment at this > time since I see some philosophical complexities here I couldn't > address properly within only a paragraph. Anyway, my original point is > that a theory such as set theory avoids having the identity of objects > depend upon DISCOVERY (which is epistemological) of properties. > Of course not. It IGNORES any other properties besides those explicitly defined within the theory. There is no room for other possibilities within the theory. That's what I'm saying. >>> See, that is what is subjective (or epistemological). We don't define >>> equality by "way to distinguish" but rather by FORMULAS. >> Formulas are a fine way to distinguish objects. For instance, I >> distinguish a vastly greater number of different infinities than >> cardinality simply by ordering formulas on a unit infinity. Good suggestion. > > As usual, you seem not to recognize my point. I believe I do, but if you would like to clarify your point, here's your chance. > >>> No, we may do better than that in theories in which there are only >>> finitely many primitive predicate symbols, such as set theory. I told >>> you all about that already. >> If there are only finitely many primitive predicate symbols, then there >> are only finitely many properties being addressed by the theory. > > No, because properties are not just primitive but are also compound. Unless you compound them infinitely, you still have a finite set of compound properties. Perhaps yuo could give an example you think contradicts that statement. > >> For >> instance, set theory only uses 'e' and '=', and misses most properties >> of sets. > > The reason is that we are interested in those properties we need to > formulate mathematics. Well, then, one would think you would want to incorporate measure as early on as possible so as to have your mathematics as integrated as possible. > >> And you might not be confused over the nature of objects and properties, >> over inductive logic vs. inductive proof, > > I may be confused about ontology, but not because I have taken some > firm stance about it while doing everything I can to not understand the > many philosophies that have been formed. Ummm, are you sure about that? > > As to induction, I've never shown any confusion between inductive logic > and inductive proof. If at some time I did not realize which of the two > subjects you had in mind, then that doesn't entail that I dont' > understand the
From: MoeBlee on 5 Sep 2006 21:31 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. > 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. > >> 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. > 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. > 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. > >> 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. > >> 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. MoeBlee
From: Tony Orlow on 5 Sep 2006 21:37
Virgil wrote: > In article <44fd9eba(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Dik T. Winter wrote: >>> In article <44ef3da9(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> >>> writes: > >>> Your axiom uses things that are not defined. What is the *meaning* of >>> "x<z"? >> Geometrically it means that x is left of z on the number line. > > And for someone standing on the other side of the number line would x be > on the right of z? > > And does the line stay horizontal as one moves around earth? Which way > is larger if the line ever goes vertical. And how does the "larger" work > at antipodes? > Silly questions. > > >> 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? > > Not hardly. > A y (y = x) v (y = z) v (y<x ^ y<z) v (x<y ^ y<z) v (x<y ^ z<y) > is a bit better but still insufficient. True, I should have specified y<>x and y<>z. I guess it's usually done using <= for this reason, eh? > >>> > > That is not a definition, because it makes no sense. "The set of >>> > > naturals >>> > > is as large as every natural"? >>> > >>> > It is not larger than all naturals >>> >>> That is something completely different again. >> It's not LARGER than every finite. > > Which natural(s) is it "not larger" than", in the sense of not being a > proper superset of that natural or having that natural as a member? .....11111 binary (all bit positions finite) > >> If I say it's larger than all naturals, how do you read that? > > As its being a proper super set of that natural and containing that > natural as a member. Which natural? The "all" natural? > >> Why read >> it differently when I add a negative to the sentence? > > Who does? >>> > > > Then I don't know what proof you are talking about. When people say >>> > > > "Cantor's second", they are generally referring to his second proof >>> > > > of >>> > > > the uncountablility of the reals based on the diagonal argument, as >>> > > > opposed to the first, based on an unreachable intermediate value. >>> > > >>> > > 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. > > They are not the same for reals, as some reals have dual digital > representations in any base representation scheme. Only in systems which regard those two representations as referencing the exact same point, and not neighboring points. In any case, they are different digital representations, different strings. > >>> > 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. > > Then you should learn to read the English translations of Cantor's work > more carefully as I have done. Cantors original "diagonal" proof only > showed that a set of all infinite sequences of two symbols, such as the > set of all infinite binary strings, is uncountable. His followers then > modified the proof to apply to sets of decimal strings, and the set of > real numbers. Fine. There are various forms of the proof. It is generally referred to as the 2nd proof of the uncountability of the reals. All along I've said it's not about reals but about digital, and more generally symbolic, representations. So? Now you say I was right. Big surprise. > >>> 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. > > If there is a set of all naturals, and there are sets with which the set > of all naturals can biject then there is a cardinality for the set of > naturals. That TO doesn't like it is TO's failure. I failed to like it... :( > >> Nowhere has the set ever become infinite in count, as long as you only >> count finite units. > > For the definition of cardinality, that is not an issue. The only issue > is the possibility of bijection with other sets. Yes, cardinality ignores many issues. |