An uncountable countable set
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From: Tony Orlow on 5 Sep 2006 12:53 MoeBlee wrote: > Tony Orlow wrote: >> MoeBlee wrote: >>> Tony Orlow wrote: >>>> Logic determines truth. Induction is more than just a form of proof. >>> Logic determines VALIDITY. And induciton is indeed more than just a >>> form of proof. But your concept of induction is uninformed, >>> superficial, and confused. >>> >>> MoeBlee >>> >> That second sentence refers to inductive logic, as opposed to deductive >> logic, of which "inductive" proof is a form. > > My second sentence referred to induction in deductive mathematics. If > you were referring to, inductive logic, then of course, my remarks > would have to be adjusted in that light. > >> When you say my concept of induction is "uninformed, superficial, and >> confused", I would disagree. In my opinion, those that think it just >> "springs from the axioms" without a sound logical basis are the confused >> and uninformed ones. > > That is again a clear demonstration of your ignorance. If you read just > the least bit on this subject, then you'd find how induction is sourced > in mathematical logic. > > MoeBlee > 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. 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? Tony
From: Tony Orlow on 5 Sep 2006 12:56 MoeBlee wrote: > Tony Orlow wrote: >> So, is it technically incorrect to say 1 = 3/3 = 1.000... = 0.999...? I >> never thought so. > > It just occurred to me that even with the naturals isomorphically > embedded and not a subset of the reals, your demand that the naturals > are a subset of the reals can be met just by renaming. Call the finite > ordinals 'first-stage-naturals', and each for step in the construction, > keep nicknaming the members of range of the embedding as > 'second-stage-naturals', etc. (the construction of each system is like > a "staging area or staging exercise" for the next constructed system). > Then nickname the particular subset of the reals as 'naturals'. > > It's all trivial and irrelevent anyway, since these are but nicknames > while the formal theory has only predicate and operation symbols to > assign. We assign, say, the 32nd 1-place operation symbol to the > particular set of concern here that is a subset of the set of reals. > Whether we nickname that as 'the naturals' or as 'the embedded set from > the naturals' is merely a question of informal nicknaming practice and > has little mathematical importance. > > You learned, as we all did, in about the fifth grade, that the naturals > are a subset of the integers are a subset of the rationals are a subset > of the reals. However, when we get into the real serious stuff, we see > that with certain constructions, that fifth grade notion of subset is > not precise and is sharpened into a rigorous formulation in set theory. > > MoeBlee > 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. Tony
From: MoeBlee on 5 Sep 2006 13:23 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." > 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. MoeBlee
From: MoeBlee on 5 Sep 2006 13:47 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)). > 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. > 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. > 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. > 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? > > 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 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. > > 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). In second order, it would be a=b <-> AP(P(a) <-> P(b)), and P=Q <-> Aa(P(a) <-> P(a)). But those don't ential that, for example, b = {P | P is a defining property of b}. > 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. Please just read a book on logic. MoeBlee
From: MoeBlee on 5 Sep 2006 13:57
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. > 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. > 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. > 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. MoeBlee |