An uncountable countable set
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From: MoeBlee on 5 Sep 2006 16:04 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. > 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. ..> >> 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. > >>>> 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. > > 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. > > 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. > 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. > 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. 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 difference between the subjects but rather only that I failed to discern of which you were addressing. I'm surprised that even you would resort to such intellectual dishonesty as to take insinuate that I don't know the difference in the subjects themselves only because I may have been mistaken at some point as to which of the subjects you were addressing. > > I thought a 'bit' is a 0 or 1. In that case, in set theory, it is not > > the case that each object is a bit. > > Sorry, a bit position. Now, if I were you, I'd say you don't know the difference merely from the fact that you misspoke. But I'm not you, so I don't resort to that kind of cheap tactic. > Number the objects in the universe starting from > 0. Every unique set is therefore a unique bit string representing which > elements are members. If that's your axiom, fine. But it contradicts set theory. So, again, we need to be clear what theory we're talking about. But you don't have a theory, so it's nugatory anyway. You just keep announcing disparate axioms without stating a logistic system, primitives, or definitions. > >> The set y IS the set of > >> objects which are members of y, no more, and no less. > > > > That's correct regarding set theory, and it conflicts with your notion > > that a set is the set of its defining P
From: Virgil on 5 Sep 2006 16:18 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? > 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. > > > > 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? > > 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. > 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. > > > 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. > > 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. > 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.
From: mueckenh on 5 Sep 2006 16:20 Virgil schrieb: > Cantor's "diagonal" proof did not even concern them. It was others who > later applied it to the set of reals. it was originally about the set of > all lists (functions with domain N) or strings from an "alphabet" of two > "letters". A set of elements E = (x1, x2, ..., xï?®, ...), which depend of infinitely many co-ordinates x1, x2, ... xï?®, ... where each co-ordinate is either m or w. > > If one considers the alphabet of {"L","R"} for left branch and right > branch, Cantor's original proof essentially proves that the number of > paths in an infinite binary tree is uncountable. > > > > Give my a tree of infinite paths consisting of 0's and 1's, and I show > > that there > > are not less edges than paths. > > Cantor shows less edges than paths. In a choice between a proof by > Cantor and a proof by "Mueckenh", I will choose Cantor every time even when Cantor's proof is wrong and WM's proof is correct. Regards, WM
From: Virgil on 5 Sep 2006 16:33 In article <44fda5fe(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > In any > case, I agree with Leibniz that the unique identity of an object is > defined by its unique set of properties, and that equivalence between > two objects is the SAME as equivalence between the entire set of > properties of each. So, it's not that a->b -> b->a, but that a=b <-> > b=a. The object IS the set of properties which defines it, 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. In Zf, ZFC and NBG, Sets are determined only by what are and are not members of them. Anything else is irrelevant. Axiom of extensionality: Two sets are the same if they have the same members. > And what is z besides one of the set of properties which defines the > sets x and y? The distinction between elements and properties is rather > tenuous. Is it not a property of y that z e y? But is it a property of z, necessary to the identification of z, that z e y? Not in ZF, ZFC, or NBG. > No, but we can take as axiomatic that a=b = A P P(a)=P(b), Not in ZF, ZFC or NBG. > and that P=Q = A a P(a)=Q(a). Not in in ZF, ZFC or NBG. > Thus properties are defined by the objects to which > they pertain, and objects are defined by the properties which pertain to > them. Despite the fact that this statement is not first-order, I see no > problem arising from it. Certain problems arise from the "property" of a set being, or especially not being, a member of itself. So unless allowable "properties" are somehow constrained to avoid those problems, TO's set theory will be self contradictory. In ZF, et all, such constraints are built in.
From: Virgil on 5 Sep 2006 16:38
In article <44fda79d(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > MoeBlee wrote: > > 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 > > > > That's assuming x and y are sets. If they are atomic objects, > urelements, then A z not(zex v zey). In this case, neither x nor y has > any elements, but does that mean they are the same? If x and y are sets, then it does mean there are the same. but if they are ur-elements which are not sets, it need not, but having different rules for sets and non-sets poses no serious problem. > > How did we get into this? I think it was distinguishing the staircase in > the limit from the diagonal line, no? If there's a distinction, then > they're not the same object, are they? If they are sets and have the same members, they are the same set. |