An uncountable countable set
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From: mueckenh on 5 Sep 2006 11:50 Virgil schrieb: > In article <1157368118.735322.299590(a)74g2000cwt.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > David R Tribble schrieb: > > > > > > I was not aware that abstract mathematical concepts (e.g., sets) > > > had any physical constraints. > > > > Then you should learn it. It you are unable to physically (i.e. in > > written form or in your mind) distinguish all the elements of a set, > > then the set does not exist. > > While "in one's brain" may be considered physical, > "in one's mind" is distinctly not physical. The mind is like the soul a present from God? > > So that sets "exist in the mind" with no physical constraints whatsoever. What a fortune that nobody can know what your mind contains. Not even 10^20 different numbers. But you pretend it contained infinitely many. Regards, WM
From: Tony Orlow on 5 Sep 2006 11:58 Dik T. Winter wrote: > In article <44ef3da9(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> writes: > > Dik T. Winter wrote: > .... > > > > Why do you need an axiom for that? Why is it > > > > not derivable logically? > > > > > > Because without the axiom of infinity the set of naturals need not exist, > > > and indeed, you can build a completely logical system with the negation > > > of the axiom of infinity and with all other axioms remaining. It is > > > similar to the parallel axiom in geometry. > > > > But without an axiom of infinity, it is demonstrable that, given the > > axiom of internal infinity (continuity), x<z -> x<y<z, that any finite > > interval includes an infinite number of points. Start with the line, and > > identify points. There's infinity. > > 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. 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? > > > > You both do not have an linearisation of the reals. You both are doing > > > other things. > > > > Describe those "other things". How are they "other"? > > You are expanding the reals in some undefined matter. That's the first time I've heard the H-riffics described as "expanding" the reals. The usual complaint is they don't include ALL the reals, but neither do the digital number systems, without an infinite number of digits. > > > > > The second proof of the uncountability of the reals is not about reals? > > > > That's news. What was it about, then, in your opinion? > > > > > > Sorry, I already explained that in previous articles to which you have > > > replied. But if you do not read the articles you reply to there is > > > something seriously wrong in the discussion. > > > > It is about digital representation, which is the same as power set, even > > in other number bases, which means more than two states of inclusion per > > member. > > Darn. Try to read. Cantor's proof is not about reals, it is neither > about digital representations. It is about none of the things you are > mentioning. But nevertheless you maintain that it is news that it is > not about the reals, while you read what I wrote? I asked what you thought it was about, if not reals or symbolic systems, and you refuse to answer the question, which is an answer in itself. You're not even curious enough to ask what I mean by more than two states of set inclusion? Oh, well. :S > > > > > Larger than any finite. The set of naturals is as large as, but no > > > > larger than, every natural. > > > > > > That is not a definition, because it makes no sense. "The set of naturals > > > is as large as every natural"? > > > > It is no larger than all naturals > > That is something completely different again. It's not LARGER than every finite. > > > > From that: "The set of naturals is as > > > large as 1", "The set of naturals is as large as 2". What is the meaning > > > of these statements? > > > > That is when you substitute "every", meaning "each", for "all". Careful. > > Yes, you should be careful in what you mean, and not use a word that has > multiple meanings so that you can be misunderstood. So I will refrase: > > > > Larger than any finite. The set of naturals is as large as, but no > > > > larger than, all naturals. > Is that what you intended? In that case you just stated a tautology. If I say it's larger than all naturals, how do you read that? Why read it differently when I add a negative to the sentence? > > > > > 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. > > 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. > > > > > I thought it was clear that I was using a notion of infinite, like WM, > > > > from a quantitative standpoint, rather than set-theoretic. > > > > > > Without definition. > > > > Greater than any finite. Simple enough? > > 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. Nowhere has the set ever become infinite in count, as long as you only count finite units. In this Wolfgang is correct.
From: Tony Orlow on 5 Sep 2006 12:29 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). 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. 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. > >>> 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). 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? > > 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). No, but we can take as axiomatic that a=b = A P P(a)=P(b), and that P=Q = A a P(a)=Q(a). 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. > > MoeBlee > :) TOny
From: Tony Orlow on 5 Sep 2006 12:36 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? 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. But, that requires the universal quantification over properties and/or sets, and so cannot be stated in first order logic, right? In any case, the statement you refer to from set theory can be universally applied when properties are considered as defining objects. 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? Tony
From: Tony Orlow on 5 Sep 2006 12:43
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. > > MoeBlee > 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? Tony |