An uncountable countable set
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From: MoeBlee on 5 Sep 2006 20:36 Tony Orlow wrote: > A better way to view the universe of sets is as a relation between the > objects within it and 1-place predicates concerning objects, where each > object has a truth value associated with each predicate, between 0 and 1 > inclusive. Thus, these predicates define subsets of the universe, eh? I bet you could find that there's is work that's been done on model theories with a continuum of values. I might be barking up the wrong tree, but I think there might be something in the work of C.C. Chang (not included in the book) that might at least suggest some points of departure, if not directly, then at least through the bibligographic trails. MoeBlee
From: Tony Orlow on 5 Sep 2006 20:49 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. What I am talking about is not covered by the axiom of extensionality, 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? > >> 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? You answer "No" to the general notion that properties of the elements define the set. Can you give any coherent reason why, besides my not having read the same books as you? > >> That's nice. Is there a difference between an abstract object like a >> set, and its definition? > > Yes. What? > >> 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? > >>>> But, that requires the >>>> universal quantification over properties and/or sets, and so cannot be >>>> stated in first order logic, right? >>> Right. But it's worse. >> Worse than what? > > Worse in the sense that there are even more problems with you mixed up > idea than that of moving to second order. Says you. > >> That's nice. You're playing Grandpa trying to show the six year old how >> to program the VCR like it's a Victrola. > > No, in this latest matter, it's more like you can't distinguish the > picture on the screen from the buttons on the remote control that > define which picture will be on the screen. The set defined is not its > own definition any more than the picture of the cowboy on the screen is > the sequence of buttons on the remote that brought that picture on. Yet another meaningless analogical insult. How enlightening. > > MoeBlee >
From: Aatu Koskensilta on 5 Sep 2006 20:54 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, 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. 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. 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. 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 - but seems, to me, ultimately counterproductive; if the objective is to actually establish intellectual contact with the crank, 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? -- Aatu Koskensilta (aatu.koskensilta(a)xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: MoeBlee on 5 Sep 2006 20:55 Tony Orlow wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> And yet, you are saying it's not technically correct. So, 1 is a natural > >> but not a rational or real? If rigorous formulations come to that > >> conclusion, then rigor does not ensure correctness. > > > > You're hopeless. You didn't understand a thing I said, which may be my > > fault for not providing adequate explanation in the context of your > > ignorance of the subject; but it is no my fault that you won't even > > look at a book on mathematics, such as even a introductory text in real > > analysis. > > > > MoeBlee > > > > Your whole point here is ludicrous. No, my whole point just happens to conflict with the oversimplification we were all taught in the fifth grade. > You are arguing that the naturals > are not a subset of the rationals, which are not a subset of the reals. That is correct, given, as I said, that we're working with either the Dedekind cut or equivalence class of Cauchy sequences method. A natural number is a finite ordinal. An integer is a certain kind of equivalence class of naturals. A rational is certain kind of equivalence class of integers. A Dedekind cut is a certain kind of infinite proper subset of the set of rationals. An equivalence class of Cauchy sequences is a certain kind of infinite set of infinite sequences of rationals. And, one can see that a finite ordinal is neither an infinite proper subset of the set of rationals nor an infinite set of infinite sequences of rationals. > While each superset may require a more complex construction than the > subset, all elements of the subset are covered by the superconstruction. How in the world would you know? You know NOTHING about these constructions. Refer to any set theory text or to any introductory analysis text that mentions the construction of a complete ordered field. > As points on the real line, they are all real numbers. No, there is an isomorphic embedding into the real number system. > What you're > saying amounts to what I said above, which is indeed absurd. So, no, I > don't understand a thing you're saying, when you say naturals aren't > reals. Sorry. You don't understand it because you don't know the meaning of a construction, an embedding, or even the axiom of extensionality. But I could even consider mollifying you this way: We could call the finite ordinals not 'natural numbers' (we could call them 'pre-naturals' or 'staging area naturals' or whatever) but instead call the set defined from the embedding the 'natural numbers'. Then you'd have the natural numbers, AS SO DEFINED, to be an actual subset of the set of real numbers. Though, of course, this requires you understanding that what we CALL these things is not essential since definitions are eliminable and the CONTENT of the theory resides ONLY in the axioms and primitives. That is to say that you still don't know how the principle of 'a rose by any other name is still a rose' is upheld in mathematics. MoeBlee
From: Tony Orlow on 5 Sep 2006 20:58
MoeBlee wrote: > Tony Orlow wrote: >> MoeBlee wrote: >>> Tony Orlow wrote: >>>> I am NOT accepting the axioms. The axioms are artificial statements >>>> which can be made to work together, but which do not follow from >>>> elementary logic the way they should. Every axiom should be justifiable >>>> based on fundamental concepts. I don't see that here. >>> You keep avoiding what I've told you several times. To get an adequate >>> amount of mathematics, you have to adopt axioms that are not derivable >>> from pure logic alone. So whatever axioms you produce will be what you >>> call "artificial statements" just as you call the axioms of set theory >>> "artificial statements." >> Perhaps I ignore that statement because you have not given any >> justification for it. I am not convinced that it's necessary to invent >> rules which do not follow from more elementary fundamentals. > > You ignore instead of asking me for justification. The justification is > in the incompleteness theorem, which is even STRONGER than what I > mentioned. Aside from incompleteness, if you knew even just a little > bit about the subject you'd understand the sense in which you can't get > adequate mathematics (such as, say, enough to do calculus) from just > logical axioms. Listen, in retrospect, I agree with the statetement, "To get an adequate amount of mathematics, you have to adopt axioms that are not derivable from pure logic alone." However, that does not mean they should not have ANY justification. As I said below, IFR is justified geometrically given the graph of a function, very intuitively. N=S^L is based on combinatorics. Is that pure logic? Somewhere in the statement of an axiom should be included the intent, as should be the case with governmental laws. > >>>> Yes, I am working on that, or was trying to until I got stuck with child >>>> care duty the last several weeks. It's not far off, but it's not at all >>>> like ZFC. I have tried to present several axioms, such as N=S^L and IFR, >>>> and the axioms of internal and external infinity. >>> Those are not statements of pure logic, hence they're "artificial >>> statements". Or, if they are pure logic, then they're inadequate. >> Inadequate why? N=S^L is inductively provable, and inductive proof is >> derived directly from logic by forming an infinite loop. > > Oooh boy. You don't even have a logicistic system, logical axioms, nor > rules of inference, and yet you're telling me that you can prove your > mathematical axioms, let alone your notion of inductive proof as > provable by logic by forming an infinite loop shows you really have no > idea what induction is. If you say so. What is it, then, in a nutshell? I mean, you can't define "mathematics", but maybe something a little more restrictive could be a good place to start. So, why don't you explain induction? > >> That's all very >> elementary. IFR is justified primarily geometrically, and is also very >> straightforward. Together they provide means for measuring the relative >> sizes of infinite sets of the symbolic and the quantitative variety, >> respectively. The axioms of internal and external infinity are >> constructive axioms like Peano's, and therefore somewhat "artificial", >> but justified as a construction. Were Peano's axioms inadequate? > > First order PA doesn't give you real analysis. And the PA axioms are > non-logical axioms. Like I said, my axioms of infinity are constructive axioms, rather than deductive. There are rules for creating the system and rules which follow from those regarding how it behaves. > > MoeBlee > Tony |