The Definition of Points
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From: Virgil on 13 Apr 2007 14:19 In article <461fbab9(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Once they are ordered in whatever manner, they become sequences, trees, > or other structures, and it is only with such a recursive definition > that such an infinite structure can be created. That is a broad claim. Can you prove it? Oh! I forgot! You never prove anything. > Does Virgil forget what he cuts from the post? What do you think we were > discussing? I thought it was N specifically. TO's discusssions skip so wildly from one thing to another, it is easy to lose track. > > >>>> Order is defined by x<y ^ y<z -> x<z. > >>> Transitivity is one of the properties of most of the orderings we're > >>> talking about. But transitivity is not the only property that defines > >>> such things as 'partial order', 'linear order', 'well order'. > >>> > >> It defines order, in general. > > > > Only to TO. For everyone else, other properties are required. > > > > For example, in addition to transitivity, > > ((x>y) and (y>x)) -> x = y > > is a necessary property /every/ ordering. > > Um, that one is blatantly self-contradictory. x>y -> not y>x, always. I > suppose you meant: > ((x>=y) and (y>=x)) -> x = y > or: > (~(x>y) and ~(y>x)) -> x = y Nothing in TO's definition of "<" prohibits '(x>y) and (y>x)' from being true, so if he wishes to require such a prohibition, he must specifically add it to his transistivity requirement. > > > You're missing the point. MY point is that requiring only transistivity of a relation is not enough by itself to assure that one has an order relation. TO insists that transitivity is enough, which is wrong. > > The mechanics of "less than" depends on what standard of measurement one > > is using, so claiming that one measure measures all is a procrustean > > fallacy. > > You have a very negative attitude. Mathematics involves a lot of very careful nit picking. Those who regard such nit picking as "a very negative attitude" often have great problems with mathematics. > > >> There can always be a 1-1 correspondence defined between a set > >> with no end and its proper subset with no end, even if that > >> correspondence is so complicated so as to defy all attempts to define > >> it. > > > > Trivially false. > > > > Neither the set of reals nor the set of rationals has an end, and the > > rationals are a proper subset of the reals, but there is no bijection > > between them. > > Golly! Wasn't it you among others that was telling me how R was derived > from Q which was derived from N, but that they were all distinct sets, > and N and Q WEREN'T subsets of R? See, TO can pick a nit when it pleases him. In any model of the reals there is a unique minimal subfield which is field- isomorphic to the rationals. We might label that subfield as the rational reals, in which case: Neither the set of reals nor the set of rational reals has an end, and the rational reals are a proper subset of the reals, but there is no bijection between them. > Isn't R a set defined using Dedekind > cuts or Cauchy sequences, which neither naturals nor rationals are? But, > I disputed that, anyway, so you're right. There remains the difference > between countable and uncountable infinity, but that's just a > distinction between potential and actual infinity. The set of reals and the set of rational reals are equally potential and equally actual at being infinite. > > > > > And, given the axiom of choice, any well ordered uncountable set even > > has well ordered countable subsets with which it does not biject. > > Sure, uncountable vs. countable. I stand corrected. You were probably sitting, not standing, as you wrote that.
From: Virgil on 13 Apr 2007 14:21 In article <461fbc19(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > MoeBlee wrote: > > On Apr 12, 12:27 pm, Tony Orlow <t...(a)lightlink.com> wrote: > >> MoeBlee wrote: > > I really don't care what you work on. My point is that your commentary > > in these threads has virtually no formal mathematical import, as it > > comes down to a bunch of whining that your personal notions are not > > embodied in set theory even though you can't point to a formal system > > (either published or of your own, and the gibberish you've posted in > > threads and on your own site is not even a corhernt attempt toward a > > formal system) that does embody your personal notions and you can't > > even HINT at what such a system might be. > > > > MoeBlee > > > > What does any of your whining have to do with the definition of points? > This is tiresome. It is, among other things, noting the lack of any progress towards defining points by either you or Lester.
From: Virgil on 13 Apr 2007 14:24 In article <461fc017(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > MoeBlee wrote: > > On Apr 12, 2:36 pm, "MoeBlee" <jazzm...(a)hotmail.com> wrote: > > > >> Zermelo's motivation was to prove that every set is well ordered. > > > > Since that phrasing might be misunderstood, I should say that I mean: > > Zermelo's motivation was to prove that for every set, there exists a > > well ordering on it. > > > > MoeBlee > > > > I am not sure how the Axiom of Choice demonstrates that. > > Well Order the Reals! TO misses the point again. Existence proofs do not have to actually instantiate what they are proving exists. And the AOC allows an existence proof of a well ordering of any set without requiring that any such well orderings be actually created.
From: MoeBlee on 13 Apr 2007 14:24 On Apr 13, 10:56 am, Tony Orlow <t...(a)lightlink.com> wrote: > Well, of course, Moe's technically right, though I originally asked > Lester to define his meaning in relation to his grammar. Technically, > grammar just defines which statements are valid, to which specific > meanings are like parameters plugged in for the interpretation. That is completely wrong. You have it completely backwards. What you just mentioned is part of semantics not grammar. Grammar is syntax - the rules for formation of certain kinds of strings of symbols, formulas, sentences, and other matters related purely to the "manipulation" of sequences of symbols and sequences of formulas, and of such objects. On the other hand, semantics is about the interpretations, the denotations, the meanings of the symbols, strings of symbols, formulas, sentences, and sets of sentences. Mathematical logic includes the study of these two things - syntax and semantics - both separately and in relation to each other. > I asked > the question originally using truth tables to avoid all that, so that we > can directly equate Lester's grammar with the common grammar, on that > level, and derive whether "not a not b" and "not a or not b" were the > same thing. They seem to be. Truth tables are basically a semantical matter. Inspection of a truth table reveals the truth or falsehood of a sentential formula per each of the assigments of denotations of 'true' or 'false' to the sentence letters in the formula. MoeBlee
From: Tony Orlow on 13 Apr 2007 14:25
Mike Kelly wrote: > On 12 Apr, 19:54, Tony Orlow <t...(a)lightlink.com> wrote: >> Mike Kelly wrote: >>> On 2 Apr, 15:49, Tony Orlow <t...(a)lightlink.com> wrote: >>>> Mike Kelly wrote: >>>>> On 1 Apr, 04:44, Tony Orlow <t...(a)lightlink.com> wrote: >>>>>> cbr...(a)cbrownsystems.com wrote: >>>>>>> On Mar 31, 5:45 pm, Tony Orlow <t...(a)lightlink.com> wrote:> >>>>>>>> Yes, NeN, as Ross says. I understand what he means, but you don't. >>>>>>> What I don't understand is what name you would like to give to the set >>>>>>> {n : n e N and n <> N}. M? >>>>>>> Cheers - Chas >>>>>> N-1? Why do I need to define that uselessness? I don't want to give a >>>>>> size to the set of finite naturals because defining the size of that set >>>>>> is inherently self-contradictory, >>>>> So.. you accept that the set of naturals exists? But you don't accept >>>>> that it can have a "size". Is it acceptable for it to have a >>>>> "bijectibility class"? Or is that taboo in your mind, too? If nobody >>>>> ever refered to cardinality as "size" but always said "bijectibility >>>>> class" (or just "cardinality"..) would all your objections disappear? >>>> Yes, but my desire for a good way of measuring infinite sets wouldn't go >>>> away. >>> You seem to be implying that the existence and acceptance of >>> cardinality as one way of measuring infinite sets precludes the >>> invention of any other. This is patently false. There is an entire >>> branch of mathematics called "measure theory" which, roughly speaking, >>> examines various ways to measure and compare infinite sets. Measure >>> theory builds upon set theory. Set theory doesn't preclude mesure >>> theory. > > No response to this bit, of course. You're chronically incapable of > acknowledging this. > I've discussed that with you and others. It doesn't cover the cases I am talking about. The naturals have a "measure" of 0, no? So, measure theory doesn't address the relationship between, say, the naturals and the evens or primes. It's not as general as it should be. So, what do you want me to say? >>> Of course, if *your* ideas were to be formalised then first of all >>> you'd have to pull your head out of.. the sand, accept that you've >>> made numerous egregiously erroneous statements about standard >>> mathematics, learn how to communicate mathematically and learn how to >>> formalise mathematical ideas precisely. Look at NSA and the Surreal >>> numbers if you need evidence that non-standard ideas can be expressed >>> clearly and coherently within an existing framework of mathematical >>> expression. >>> You may be a lost cause though. You've spent, what, three years >>> blathering on Usenet and your mathematical understanding and maturity >>> hasn't improved a jot. It seems like you genuinely don't want to >>> learn. Is ranting incoherently just your way of blowing off steam? >> You're not entirely wrong, Mike. > > Natch. > >> I mean, you've been a jerk through all of this, but you have a point. > > Whereas you.. > >> In order to supplant what is currently >> the overly axiomatic bent of the field of mathematics and return to a >> balance between the deductive and the inductive sides of logic requires >> that one delve into the very foundations of logic itself, and form a new >> basis for determining what constitutes evidence in the field of >> mathematics, and how this evidence should be fed as deductively derived >> input into the inductive process of choosing axioms from which to build >> theorems. I am working on how to balance this, but it's not easy, and >> life's not easy, and I have other things to do. But, I'm doing this, >> too. I wish I had more time to research, but when I find the time to do >> this here, it's occasional. I promise to try harder in the future. > > There, there Tony. Aren't you just the cutest little mathematical > pioneer? > Yep. Jerk. >> My threads do get attention, because I raise some valid issues. > > Guess again. > Just a chance to be a jerk, and you can't pass it up? >> You want a solution? Help me out. :) > > Solution to what? All I'm trying to do here is to get you to realise > that you are wasting your time trying to develop new foundations for > mathematics or logic or whatever when you are utterly incapable of > understanding existing ideas, following proofs, writing proofs, > communicating mathematically... > > It'd be nice if some day you learned some math above high school > level. Seems a rather remote possibility though because you are > WILLFULLY ignorant. > And you are willfully obnoxious, but I won't take it seriously. I'm not the only one in the revolution against blind axiomatics. >>>>>> given the fact that its size must be equal to the largest element, >>>>> That isn't a fact. It's true that the size of a set of naturals of the >>>>> form {1,2,3,...,n} is n. But N isn't a set of that form. Is it? >>>> It's true that the set of consecutive naturals starting at 1 with size x has largest element x. >>> No. This is not true if the set is not finite (if it does not have a >>> largest element). >> Prove it, formally, please, from your axioms. > > I don't have a formal definition of "size". You understand this point, > yes? Then how do you presume to declare that my statement is "not true"? > > It's very easily provable that if "size" means "cardinality" that N > has "size" aleph_0 but no largest element. You aren't actually > questioning this, are you? No, have your system of cardinality, but don't pretend it can tell things it can't. Cardinality is size for finite sets. For infinite sets it's only some broad classification. > > It's rather disingenuous to ask me for a formal proof of something > that is couched in your informal terms. > Don't say "This is not true" if you can't disprove it. >>> It is true that the set of consecutive naturals starting at 1 with >>> largest element x has cardinality x. >> Forget cardinality. Can a set of naturals starting with 1 and with size >> X possibly have any other maximum value besides X? This is inductively >> impossible. > > (Just to be clear, we're talking very informally here. It's quite > obviously the only way to talk to you.) > > Tony, can you discern a difference between the following two > statements? > > a) A consecutive set of naturals starting with 1 with size X can not > have any maximum other than X. > b) A consecutive set of naturals starting with 1 with size X has > maximum X. Yes, the first allows that may be no maximum, but where there is a specific size for such a set, there is a specific maximum as well. I am not the one having the logical difficulty here. > > Seriously, do you comprehend that they are saying different things? > This is important. It would be if it had anything substantive to do with my point. Whatever the size of a set of consecutive naturals from 1 is, that is its maximal element. > > I'm not disputing a) (although you haven't defined "size" and thus > it's trivially incorrect). I'm disputing b). I don't think b) follows > from a). I don't think that all sets of naturals starting from 1 have > a maximum. So I don't think that "the maximum, if it exists, is X" > means "the maximum is X". Because for some sets of naturals, the > maximum doesn't exist. It is inductively provable that for all such sets the maximum is EQUAL to the size. Therefore, if one exists, then so does the other, in both directions. Do you know what EQUAL means? If a=b, can a exist and b not? > > Do you agree that some sets of consecutive naturals starting with 1 > don't have a maximum element (N, for example)? Do you then agree that > a) does not imply b)? > No, b) implies that such sets also do not have a size. Get it? >>> It is not true that the set of consecutive naturals starting at 1 with >>> cardinality x has largest element x. A set of consecutive naturals >>> starting at 1 need not have a largest element at all. >> Given the definition of the naturals, given any starting point 0, a set >> of consecutive naturals of size y has maximum element x+y. > > x+y? Typo I guess. Yes, I was going to say "starting point x", then changed that part and not the other (which would have needed a "-1", anyway). > >> Does the set of naturals have size aleph_0? If so, then aleph_0 is the maximal natural. > > It has CARDINALITY aleph_0. If you take "size" to mean cardinality > then aleph_0 is the "size" of the set of naturals. But it simply isn't > true that "a set of naturals with 'size' y has maximum element y" if > "size" means cardinality. I don't believe cardinality equates to "size" in the infinite case. > > Under some definitions of "size" your statement is true. Under others > (such as cardinality) it isn't. So you can't use your statement about > SOME definitions of size to draw conclusions about ALL definitions of > size. Not all sets of naturals starting at 1 have a maximum element > (right?). Your statement is thus obviously wrong about any definition > of "size" that gives a size to non-finite sets. It's wrong in any theory that gives a size to any countably infinite set, except as a formulaic relation with N. > > I find it hard to beleive you don't understand this. Indicate the > point(s) where you disagree. > > a) Not all consecutive sets of naturals starting from 1 have maximum > elements. agree > b) Some notions of "size" give a "size" to sets of naturals without > maximum elements. disagree, personally. I can't accept transfinite cardinality as a notion of "size". > c) Some notions of "size" give a "size" to sets of consecutive > naturals starting from 1 without a maximum element. same > d) The "size" that these notions give cannot be the maximum element, > because those sets don't *have* a maximum element. agree - that would appear to be the rub > e) Your statement about "size" does not apply to all reasonable > definitions of "size". In particular, it does not apply to notions of > "size" that give a "size" to sets without a largest element. > It does not apply to transfinite cardinality. The question is whether I consider it a "reasonable" definition of size. I don't. >>> Do you see that changing the order of words in a statement can change >>> the meaning or that statement? Do you see that one statement can be >>> true, and another statement with the same words in a different order >>> can be false? >> This is not quantifier dyslexia, and I am not interested in entertaining >> that nonsense, thanx. > > It is doublethink though. You are simultaneously able to hold the > contradictory statements "Not all sets of naturals have a largest > element" with "All sets of naturals must have a largest element" to be > true, > No, "All sets of naturals WITH A SIZE must have a largest element", or more specifically, "All sets of consecutive naturals starting from 1 have size and maximal element equal." Equal things either both exist, or both don't. >>>> Is N of that form? >>> N is a set of consecutive naturals starting at 1. It doesn't have a >>> largest element. It has cardinality aleph_0. >> If aleph_0 is the size, then aleph_0 is the maximal element. aleph_0 e N. > > Wrong. If aleph_0 is the "size", AND the set HAS a maximal element > then aleph_0 is the maximal element. But N DOESN'T have a maximal > element so aleph_0 can be the size without being the maximal element. > > (speaking very informally again as Tony is incapable of recognising > the need to define "size"..) > I defined formulaic Bigulosity long ago. I've also made it clear that I don't consider transfinitology to be a valid analog for size in the infinite case. I've offered IF and N=S^L in the context of infinite-case induction, which contradicts your little religion, and may seem offensive, but is really far less absurd and paradox-free. :) >> Or, as Ross likes to say, NeN. > > Here's a hint, Tony : Ross and Lester are trolls. They don't beleive a > damn word they say. They are jerks getting pleasure from intentionally > talking rubbish to solicit negative responses. Responding to them at > all is pointless. Responding to them as though their "ideas" are > serious and worthy of attention makes you look very, very silly. > > -- > mike. > Yes, it's very silly to entertain fools, except when they are telling you the Earth is round. One needn't be all like that, Mike. When you argue with a fool, chances are he's doing the same. tony. |