The Definition of Points
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From: Mike Kelly on 19 Apr 2007 11:04 On 19 Apr, 15:01, Tony Orlow <t...(a)lightlink.com> wrote: > Mike Kelly wrote: > > On Apr 18, 6:55 pm, Tony Orlow <t...(a)lightlink.com> wrote: > >> Virgil wrote: > >>> In article <4625a...(a)news2.lightlink.com>, > >>> Tony Orlow <t...(a)lightlink.com> wrote: > >>>> Mike Kelly wrote: > >>>> < snippery > > >>>>> You've lost me again. A bad analogy is like a diagonal frog. > >>>> Transfinite cardinality makes very nice equivalence classes based almost > >>>> solely on 'e', but in my opinion doesn't produce believable results. > >>> What's not to believe? > >>> Cardinality defines an equivalence relation based on whether two sets > >>> can be bijected, and a partial order based on injection of one set into > >>> another. > >>> Both the equivalence relation and the partial order behave as > >>> equivalence relatins and partial orders are expected to behave in > >>> mathematics, so what's not to believe? > >> What I have trouble with is applying the results to infinite sets and > >> considering it a workable definition of "size". Mike's right. If you > >> don't insist it's the "size" of the set, you are free to do with > >> transfinite cardinalities whatever your heart desires. What I object to > >> are statements like, "there are AS MANY reals in [0,1] as in [0,2]", > >> and, "the naturals are EQUINUMEROUS with the even naturals." If you say > >> they are both members of an equivalence class defined by bijection, then > >> I have absolutely no objection. > > > Then you have absolutely no objection. Good that you recognise it. > > >> If you say in the same breath, "there > >> are infinitely many rationals for each natural and there are as many > >> naturals overall as there are rationals", > > > And infinitely many naturals for each rational. > > How do you figure? In each 1-unit real interval, there is exactly one > natural, and an infinite number of rationals. Which interval has one > rational and an infinite number of naturals? No interval in the ordering on the real line. But there are orderings where there are infinitely many naturals between each pair of rationals. You seem hung up on the idea of orderings on reals/ rationals/naturals that aren't the standard one. Maybe because you think about everything "geometrically"? > >> without feeling a twinge of > >> inconsistency there, then that can only be the result of education which > >> has overridden natural intuition. That's my feeling. > > > Or, maybe, other people don't share the same intuitions as you!? Do > > you really find this so hard to believe? > > Most people find transfinite cardinality "counterintuitive". Surely, you > don't dispute that. I imagine most people have never heard of it. From what I can remember of freshman analysis some people had problems with it, some people didn't. I certainly don't think there was a clear majority who did. And I don't remember anyone taking 2 years to understand that all cardinality is really about is bijections. > >> I'd rather acknowledge that omega is a phantom quantity, > > > By which you mean "does not behave like those finite numbers I am used > > to dealing with". > > Or, does not fit into what I understand quantities to be. OK, whatever. You don't think Omega is a "quantity". You haven't defined "quantity" in any unambiguous way but I'm perfectly happy to believe you don't think the set of all finite ordinals is a "quantity". So what? I don't think omega sounds much like a "quantity", either. So? This doesn't change anything about my understanding of set theory. It doesn't make me say silly things like "aleph_0/omega is a phantom". Who cares if they're called "numbers"? > >> and preserve basic notions like x>0 <-> x+y>y, and extend measure to the infinite scale. > > > Well, maybe you'd like to do that. But you have made no progress > > whatsoever in two years. Mainly, I think, because you have devoted > > rather too much time to very silly critiques of current stuff and > > rather too little to humbling yourself and actually learning > > something. > > That may be your assessment, but you really don't pay attention to my > points anyway, except to defend the status quo, so I don't take that too > seriously. Great. Don't let me discourage you from your noble quest. I'll continue to glance at your posts when I see them and see if there's ever going to be anything of yours that actually interests me. And, yes, when you make what I consider amusingly egregious errors about set theory or other areas of mathematics I will "defend the status quo" by pointing out that you're being silly. > >>>> I'm working on a better theory, bit by bit. I think trying to base > >>>> everything on 'e' is a mistake, since no infinite set can be defined > >>>> without some form of '<'. I think the two need to be introduced together. > >>> Since any set theory definition of '<' is ultimately defined in terms of > >>> 'e', why multiply root causes? > >> It's based on the subset relation, which is a form of '<'. > > > Get this through your head : every relation between objects in set > > theory is based on 'e'. It's really pathetic to keep mindlessly > > denying this. Set theory doesn't just "try to base everything on 'e'". > > It succeeds. > > > > If you say so. I do say so. Does this mean you're going to stop claiming that relations in set theory aren't based solely on 'e'? -- mike.
From: Tony Orlow on 19 Apr 2007 11:22 MoeBlee wrote: > On Apr 17, 9:56 pm, Tony Orlow <t...(a)lightlink.com> wrote: >> MoeBlee wrote: >>> On Apr 17, 11:33 am, Tony Orlow <t...(a)lightlink.com> wrote: >>>> But, I have a question. What, exactly, is the difference between >>>> "equality" and "equivalence"? >>> As I told you, in first order, we cannot generalize (I'll say about >>> exceptions in a moment) that identity can be "captured" by axioms. >>> Thus in the general case, we require stipulating a fixed semantics for >>> the identity symbol. Meanwhile, equivalence often pertains to an >>> equivalence relation (reflexive, symmetric, transitive). Every >>> equiavlance relation on set induces a partition of the set; that >>> partition being a set of equivalence classes. Then two objects are >>> equivalent iff they are both members of the same equivalence class, >>> which is to say that they bear the equivalence relation (the >>> reflexive, symmetrc, transitive relation) to each other. Notice that >>> identity is a an equivalence relation but not all equivalence >>> relations are the identity relation. >> Okay, distinction noted. It would seem to apply well to cardinality, >> where we have classes of sets that match, even if they are not equal as >> sets, eh? > > I wouldn't put it that way, but I think you're onto the basic idea. > Yes, bijectability (equinumeroisity) is an equivalance relation on any > set. > Bijectibility creates a set of equivalence classes. It doesn't determine equality, though. That's why words like "equinumerosity" irk me so. >>>> Is it not, in the tradition of Leibniz, a >>>> matter of detecting a distinction between two objects, or not? That is, >>>> if we can detect no difference between two objects, if we can find no >>>> attribute which distinguishes them, then are they not "equal" or >>>> "equivalent", or "identical"? Ultimately, if we can not say, in one >>>> sense or another, that x<y or y<x, then do we not consider that x=y? >>> One direction of Leibniz's principle (the indiscerniblity of >>> identicals) is acheived by an axiom that if x = y then whatever holds >>> for x holds for y. But, in first order, we cannot generalize that the >>> other direction (the identity of indiscernbiles) can be stated even as >>> an axiom schema. Thus a stipulated semantics is given for the identity >>> symbol. >> And this is because first order logic does not allow the universal >> quantifier applied to sets or properties, but only elements, is that >> right? > > Again, I wouldn't put it that way, but you're on to the right idea. In > first order, we don't quantify over predicates (we don't follow the > universal or existential quantifier by a variable that ranges over > sets of n-tuples (n>0) of members of the domain of discourse, since we > have no such variables). > Is there an advantage to excluding such statements? >> Is it not possible to concoct a logic where the first order of >> analysis is on the property, versus object, level? I'll have to dream >> about that a bit...what do you think? > > I don't know what the point would be. If you are going to quantify > over predicates, then why not just quantify over both predicates and > individuals, which is what second order logic is. > What is the point of just quantifying over objects, rather than starting with both? >>>> Defining "=" depends, as far as I can tell, on defining "<". Is this >>>> wrong, in your "opinion"? >>> In what context? In set theory, given an appropriate axiomatization, >>> we can define '=' from just 'e' (the membership symbol). By the way, >>> that does not contradict my remarks about first order not being able >>> to "capture" identity, since what that says is that it is not the case >>> that for ANY langauge or theory we can "capture" identity. For CERTAIN >>> languages and theories (viz. those with only finitely many predicate >>> symbols) we can "capture" so as to make a definition for the identity >>> symbol that conforms to the basic semantics that 'x=y' is true iff >>> denotation of x and the denotation of y are identical. >>> MoeBlee >> In a very real sense, for all xeX, x<X, so defining '=' from 'e' is >> pretty similar to defining it from '<'. > > That's the definition of '<' for ordinals. It's not for sets in > general. And it wouldn't work for equality of sets in general as long > as we have regularity, since it is never the case that xey & yex. The > actual definition can be made by the subset relation: x=y <-> (x > subset_of y & y subset_of x). > or (X=Y) = (Ax xeX = xeY) >> I understand that this is not >> pure first order logic, but first order with an added operator with its >> own properties. First order by itself can't distinguish among anything >> but truth values, > > I'm not sure what you mean by that. It only processes logical truth values as input. Every variable is true or false. > >> so if those pertain to some objects or properties, the >> additional relationships between those need to be handled with operators >> that take as parameters whatever type of objects or properties one has, >> rather than just truth values. In the context of whatever additional >> operators are added to first order logic, indeed, if every statement >> true of each is true of the other, then those two objects or properties >> are considered identical. > > In set theory, since there are only finitely many predicate symbols, > we CAN and we DO implement both directions of Leibniz's principle. >> That's not to say that an additional operator >> might not produce a difference between two previous equated elements, is it? > > I don't follow what you meant about operators. But why don't you get > 'Logic: Techniques Of Formal Reasoning' by Kalish, Montague, and Mar? > I'm not saying that challengingly (this time). I am sincerely > suggesting that such books would give you start toward a solid basis > upon which to develop and communicate your own ideas and possible > innovations with things such as operators. > > MoeBlee > Thanks. I noted the title. We'll see when I can get my hands on it. TOEknee
From: stephen on 19 Apr 2007 11:48 In sci.math Tony Orlow <tony(a)lightlink.com> wrote: > stephen(a)nomail.com wrote: >> In sci.math Tony Orlow <tony(a)lightlink.com> wrote: >>> stephen(a)nomail.com wrote: >>>> In sci.math Tony Orlow <tony(a)lightlink.com> wrote: >>>>> Mike Kelly wrote: >>>>>> The point that it DOESN'T MATTER whther you take cardinality to mean >>>>>> "size". It's ludicrous to respond to that point with "but I don't take >>>>>> cardinality to mean 'size'"! >>>>>> >>>>>> -- >>>>>> mike. >>>>>> >>>>> You may laugh as you like, but numbers represent measure, and measure is >>>>> built on "size" or "count". >>>> What "measure", "size" or "count" does the imaginary number i represent? Is i a number? >>>> The word "number" is used to describe things that do not represent any sort of "size". >>>> >>>> Stephen >> >>> Start with zero: E 0 >>> Define the naturals: Ex -> Ex+1 >>> Define the integers: Ex -> Ex-1 >>> Define imaginary integers: Ex -> sqrt(x) >> >>> i=sqrt(0-(0+1)), so it's built from 0 and 1, using three operators. It's >>> compounded from the naturals. >> >> That does not answer the question of what "measure", "size" or "count" i represents. >> And it is wrong on other levels as well. You just pulled "sqrt" out of the >> air. You did not define it. Claiming that it is a primitive operator seems >> a bit like cheating. And if I understand your odd notation, the sqrt(2) >> is an imaginary integer according to you? And sqrt(4) is also an imaginary integer? > No, but sqrt on the negatives produces imaginary numbers. Besides, sqrt > can be defined, like + or -, geometrically, through construction. You cannot define the sqrt(-1) geometrically. You are never going to draw a line with a length of i. >> >> You also have to be careful about about claiming that i=sqrt(-1). It is much safer >> to say that i*i=-1. If you do not see the difference, maybe you should explore the >> implications of i=sqrt(-1). >> >> So what is wrong with >> Start with zero: E 0 >> Define the naturals: Ex -> Ex+1 >> Define omega: Ax >> I did that using only one operator. >> >> > Ax? You mean, Ax x<w? That's fine, but it doesn't mean that w-1<w is > incorrect. No, I meant Ax. All of the natural numbers. I know you are incapable of actually imagining "all", but others do not have that limitation. Of course, who knows what your notation is really supposed to mean. What is (Ax)-1 supposed to be? How do you subtract one from all the naturals? >>> A nice picture of i is the length of the leg of a triangle with a >>> hypotenuse of 1 and a leg of sqrt(2), if that makes any sense. It's kind >>> of like the difference between a duck. :) >> >> That does not make any sense. There is no point in giving a nonsensical >> answer, unless you are aiming to emulate Lester. >> >> Stephen >> >> > It's not nonsensical, and may even apply to uses of imaginary numbers in > practice, but you can ignore it as I knew you would. That's okay. > Tony It is nonsense. Such a triangle does not exist. A # # # # # C############B Are you claiming this is a triangle? Are you claiming the distance from A to C is i? How exactly is this supposed to be a picture of i? What is i measuring in this picture? If this is not what you meant, please draw your picture of i. And you still have not answered what "size", "count" or "measure" i represents. Is i a number, or not? Stephen
From: Lester Zick on 19 Apr 2007 12:00 On Wed, 18 Apr 2007 14:57:50 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Tue, 17 Apr 2007 13:29:44 -0400, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >>>> Let me ask you something, Tony. When you send off for some truth value >>>> according to "true(x)" and it returns a 1 or 0 or whatever, how is the >>>> determination of that "truth value" made? >>> From the truth values of the posited assumptions, of course, just like >>> yours. >> >> So you just posit truth values and wing it whereas I'm more inclined >> to demonstrate the truth of what I posit instead? >> >> ~v~~ > >What truth have you demonstrated without positing first? And what truth have you demonstrated at all? ~v~~
From: Lester Zick on 19 Apr 2007 12:01
On Wed, 18 Apr 2007 15:25:30 -0600, Virgil <virgil(a)comcast.net> wrote: >In article <462669b3(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Lester Zick wrote: >> > On Tue, 17 Apr 2007 13:33:39 -0400, Tony Orlow <tony(a)lightlink.com> >> > wrote: >> > >> >>>> >> >>>> Is there a set >> >>>> of statements S such that forall seS s=true? >> >>> No idea, Tony. There looks to be a typo above so I'm not sure exactly >> >>> what you're asking. >> > >> >> I am asking, in English, whether there is a set of all true statements. >> > >> > No. There are predicates to which all true statements and all false >> > statements are subject respectively but no otherwise exhaustively >> > definable set of all true or false statements because the difference >> > between predicates and predicate combinations in true or false >> > statements is subject to indefinite subdivision. >> > >> > ~v~~ >> >> Uh, what? > >Just Zickism at its most opaque. As opposed to a Virgilism at its most transparent? ~v~~ |