An uncountable countable set
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From: Virgil on 26 Aug 2006 16:47 In article <1156610544.196827.4250(a)74g2000cwt.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On the other hand, > the natural number consisting of the first 10^100 digits of pi (in case > it is a normal irrational) does not exist. In mathematics, there is are essential distinctions between (1) mere existence and (2) being constructable in theory and (3) being explicitely constructable in practice. The "natural number consisting of the first 10^100 digits of pi (in case it is a normal irrational)" exists in at least the first two of those three senses.
From: Virgil on 26 Aug 2006 17:00 In article <44f089fd(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > MoeBlee wrote: > > I said that IN A SYSTEM > > that constructs the reals as either Dedekind cuts or as equivalence > > classes of Cauchy sequences, then the naturals and rationals are > > isomorphically embedded but are not actual subsets of the reals. (That > > is BASIC undergraduate mathematics. I mean, even I know that, and I > > don't even have an education in mathematics! It's appalling how > > ignorant you are WHILE you so categorically opine.) > > How does the constructive treatment of the various number systems affect > the fact that the set of points representing the naturals is a subset of > the set of points representing the rationals or reals? Since representations are not identical to what they represent, such representations will have properties not shared by what they represent. To imply, as T does, that every property of the representation is necessarily a property of the represented is a deliberate falsehood. > If you claim to > defend set theory, then it's pointless of you to claim that the naturals > aren't a subset of those two sets. On the contrary, there is a very important point to it which is the point of general semantics: "The map is not the territory". http://en.wikipedia.org/wiki/General_semantics > > > > I don't define 'number' in a formal theory. But in informal discussion > > about mathematics I don't demur from using the word 'number' in its > > ordinary dictionary senses. Set theory, being a part of the foundations of mathematics. is all about formal theory, and those who reject "formal theory" in discussing set theory delude themselves.
From: MoeBlee on 26 Aug 2006 17:48 Tony Orlow wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> No, it all rests on the notions of identity and equality. As Leibniz > >> pointed out, when the properties of two objects are all exactly the > >> same, then they are the same object. So, when we say two numbers are > >> equal, that means all properties of the two are equal. > > > > Ha! The fallacy of reversing implication right there! An example of > > just about the most basic fallacy. > > When you say "a=b", you say "A a A b P(a)=P(b)". The two are equivlent > statements, and therefore imply each other. I explained to you a long time ago that in general, in first order logic we cannot state the identity of indiscernibles, even as a schema, let alone have it implied from something else in first order logic. However, as exception to the generalization just mentioned, in a language with only finitely many non-logical primitive symbols, we can state the identity of indiscernibles as a schema. And in very general terms not tied to any specific kind of system, we may say that the indiscernibility of identicals implies the identity of indiscernibles only in the sense that the identity of indiscernibles is a logical priniciple (or at least taken by many people to be a logical principle), thus a given. But what was incorrect in your original statement was the word 'so' in the sense that you were RELYING on one principle to infer the other. In that sense, you committed the fallacy of inferring B -> A from A -> B. > > No, the indiscernibility of identicals does NOT imply the identity of > > indiscernibles. You need both implications; you can't derive one from > > the other. And, in first order logic, one direction can be posited only > > in the semantics not in the axioms. > > You prove two quantities equal by showing there is no difference, do you > not? It depends on the specific theory. In a first order theory with infinitely many primitive predicate symbols, we have no theorem schema for doing what you suggest. But set theory has only two primitive predicates (one if you take equality as defined) so we can state such a theorem schema. However, we don't need to do that since the axiom of extensionality allows us to prove x=y merely by proving Az(zex <-> zey). Anyway, I don't disagree with the principle of the identity of indiscernibles; my point is that we don't infer it (or if we do, then a demonstration is required) from the principle of the indiscernibility of identicals (except in the trivial sense that we can infer a logical principle from anything at all). MoeBlee
From: MoeBlee on 26 Aug 2006 17:58 Tony Orlow wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> So, is it technically incorrect to say 1 = 3/3 = 1.000... = 0.999...? I > >> never thought so. > > > > Those are notations that can be understood precisely only in context of > > the particular treatment in which they occur. > > BLAM!! That's exactly what I'm saying. If a particular treatment cannot > distinguish between two elements, then they are the same, By a 'treatment' of a formal system, I don't mean the formal system itself, but rather an informal presentation of a formal system. For example, a treatment would include the English text in a textbook that discusses a formal system. So the treatment includes both the formal system and the informal presentation of that formal system. For example, a textbook, lecture, oral explanation, or even an Internet post may be considered treatments or at least fragments of some presumable overall treatment. And your "cannot distinguish" is too vague and broad. We prove equality of objects in a theory in very specific ways. We don't just say, "Well, no one seems to be able to distinguish them, so they must be the same." For example, in set theory, x=y <-> Az(zex <-> zey). MoeBlee
From: MoeBlee on 26 Aug 2006 18:22
Tony Orlow wrote: > > I said that IN A SYSTEM > > that constructs the reals as either Dedekind cuts or as equivalence > > classes of Cauchy sequences, then the naturals and rationals are > > isomorphically embedded but are not actual subsets of the reals. (That > > is BASIC undergraduate mathematics. I mean, even I know that, and I > > don't even have an education in mathematics! It's appalling how > > ignorant you are WHILE you so categorically opine.) > > How does the constructive treatment of the various number systems affect > the fact that the set of points representing the naturals is a subset of > the set of points representing the rationals or reals? If you claim to > defend set theory, then it's pointless of you to claim that the naturals > aren't a subset of those two sets. First, just as a terminological point, I don't know whether 'constructive' is a correct way of describing the constructions. That involves a lot more considerations, so I just want to say that by 'construction' I mean a general informal sense and not necessarily the more technical sense of 'constructive'. Anyway, your question reflects that you don't understand what I posted. In set theory that uses either the Dedekind cut approach or the equivalence class of Cauchy sequences approach, the system of natural numbers is isomorphically embedded in the system of real numbers. With these approaches, we do not necessarily claim that the thus constructed (defined) objects in the theory are some actual Platonic set and its superset. It would be much more productive to talk about this after you've read and understood the constructions. Right now, you're like me saying to an automobile mecanic, "But it's not possible for there not to be a radiator on this car! All cars have a radiator!" > > If the system and > > construction are of a different sort, then we'd have to evaluate upon > > the specifics of that system and construction. But, again, as I said, > > for contexts that do not need to be so pedantic, such notation is > > understood well enough that the equations you mentioned do of course > > hold. > > If that's the case, then the set of natural poitns on the real line is a > subset of those other sets, and there is nothing i what I said to object to. Again, you simply do not know what I'm saying. You would if you read a book, though. I don't think there is any other way I can get you to understand this matter in the vacuum of your knowledge of the subject. I can't PROPERLY compress four chapters of a set theory book into a few posts. MoeBlee |