An uncountable countable set
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From: MoeBlee on 5 Sep 2006 14:04 Tony Orlow wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> MoeBlee wrote: > >>> Tony Orlow wrote: > >>>> Yes, and the universe is consistent by definition, so math should be > >>>> consistently overall as well. > >>> Unless the universe is a set of sentences, the notion of consistency > >>> does not even apply. > >> The universe is governed by the properties of the elements within it, > >> which properties are statements true about those elements. > > > > You are taking properties to BE statements. That's fine if you have > > some coherent philosophy to support it. > Actually, I am taking properties to be truth values applied to > statements about a given object. "It is red" is a statement with a truth > value close to 1 for most fire engines, and a truth value near 0 for > most newspapers (except in old jokes). Fire engines have the property > that that statement is usually true for them. So, a property is a truth > value associated with a statement. Better? So, by directly plugging in your definition of 'properties', your previous statement becomes: "The universe is governed by the truth values applied to statements about given objects of the elements within it, which truth values applied to statements about given objects are statements true about those elements. Whatever that means, it is difficult to see that it implies that consistency is a property of the universe as opposed to a property of sets of sentences. MoeBlee
From: MoeBlee on 5 Sep 2006 14:14 Tony Orlow wrote: > I find this last sentence vague because of the word "induction". If you > refer to inductive proof, then I am familiar with the Peano axioms and > the underlying logical construction which makes inductive proof valid. You have only a faint idea. If you actually read a book on the subject, you'd find how much deeper, richer, and rigorous this is. > If you refer to inductive logic, the formulation of rules from instances > of fact, then there are statistical methods coupled with feedback that > make it possible. Which were you talking about? Mathematical induction. Inductive sets, et al. As in mathematical logic and set theory, which is deductive. Not inductive logic as in the other sense of inductive - empirical based inference (or however you want to define it). MoeBlee
From: MoeBlee on 5 Sep 2006 14:17 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
From: Tony Orlow on 5 Sep 2006 14:18 MoeBlee wrote: > Tony Orlow 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). > >> 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? > > In such a theory, we have a primitive predicate for 'is a set'. Then > the axiom of extensionality is: > > (x is a set & y is a set) -> (x=y <-> Az(zex <-> zey)) and we would > have some kind of axiom saying there exists a unique set x such that Az > ~zex. > > That allows that there may be distinct urelements. 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. 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? > >> Well, if every >> object is defined by the set of properties which pertains to it, then no >> object is not a set, and this statement holds. > > You're musings are already starting out with vagueness. One time you > say an object IS its set of defining properties and then you say an > object is DEFINED by its set of defining properties. And all of that is > without regard to how formal theories work or even what a formal theory > is. You are STILL operating in the vacuum of your own ignorance about > any of this. That's nice. Is there a difference between an abstract object like a set, and its definition? If you change the definition, does that also change the set? Do equivalent definitions correspond to equivalent sets? > >> 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? > >> In any case, the statement you refer >> to from set theory can be universally applied when properties are >> considered as defining objects. > > Again, you're playing mathematics like a six year old pretends to drive > his Daddy's car. That's nice. You're playing Grandpa trying to show the six year old how to program the VCR like it's a Victrola. > > MoeBlee >
From: Tony Orlow on 5 Sep 2006 14:39
MoeBlee wrote: > Tony Orlow wrote: >> MoeBlee wrote: >>> 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. >> Looking at what I wrote now (I've beeen tied up for a bit) it's not >> correct. I should have said a=b = A P P(a)=P(b). > > Better would be a=b <-> AP(P(a) <-> P(b)). 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. > >> Since one cannot >> quantify over sets, or the properties that define them, in first order >> logic, this is not a first order statement, but second order. > > Right. > >> 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. > > '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. > >> So, it's not that a->b -> b->a, but that a=b <-> >> b=a. > > That part seems messed up. a=b <-> b=a is just the symmetry of > identity. 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. :) > >> The object IS the set of properties which defines it, > > That's going well beyond the principles of the identity of > indiscernibles and the indiscernibility of identicals. I think you're > going to run into some big problems if you try to have an object equal > to its set of defining properties, beginning with defining a 'defining > property' and then vicious circles that I suspect will arise from your > postulate. I am aware that there are difficulties defining what constitutes a valid property in this sense, as Russell's Paradox demonstrates, but I think the kind of statement that produce such issues can be identified. That would be an interesting discussion.... > >> 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? If there is no universal system of cohesive mathematics, then this is surely the case. In the example of the staircase in the limit vs. the diagonal line, point-set topology cannot discern the two objects because it looks only at proximity of corresponding points. When one defines the objects using a segment-sequence topology, as I suggested, there is a very discernible difference between the two objects, namely, as I intuited from the beginning of that discussion, in direction of the curve. Thus, from the "subjective" perspective of sets of points, they are equal, but from the "subjective" perspective of sequences of segments, they are not. So, are they equal? Two objects are equal only if there exists no way to distinguish them. How do we know if this is the case? By enumerating all possible properties of each. Can we do that? No. We can only say that, given the set of properties under discussion in any given theory, the two are not distinguishable, within that theory. We cannot say that they are absolutely the same object. > >>> 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). >> And what is z besides one of the set of properties which defines the >> sets x and y? > > That's a confused view of the axiom of extensionality and the role of > variables. The perception of confusion would appear to be a subjective and rather relative phenomenon. > >> The distinction between elements and properties is rather >> tenuous. Is it not a property of y that z e y? > > For any PARTICULAR z, it's a property of y that z is or is not a member > of y. That doesn't entail that y IS the set of properties that y has. Consider each object in the universe to be a bit. Does each unique set correspond to a unique bit string, where each object's bit posiiton is a 1 if the object is a member, and 0 if it is not? The set y IS the set of objects which are members of y, no more, and no less. > >>> Anyway, I don't disagree with the princi |