The Definition of Points
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From: Lester Zick on 12 Apr 2007 18:36 On 12 Apr 2007 14:07:53 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >On Apr 12, 11:30 am, Tony Orlow <t...(a)lightlink.com> wrote: >> Lester Zick wrote: > >> > What grammar did you have in mind exactly, Tony? >> >> Some commonly understood mapping between strings and meaning, >> basically. > >Grammar is syntax, not meaning, which is semantics. What you just >described, an intrepative mapping from strings to meanings of the >strings is semantics, not grammar. Gee that's swell, Moe(x). Thanks for the lesson in semantics if not much of anything else. Next time we need a lesson in modern math don't call us we'll call you. ~v~~
From: Lester Zick on 12 Apr 2007 18:51 On Thu, 12 Apr 2007 14:31:52 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Sat, 31 Mar 2007 20:51:49 -0500, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >>> A logical statement can be classified as true or false? True or false? >> >> You show me the demonstration of your answer, Tony, because it's your >> question and your claim not mine. >> >> ~v~~ > >I am asking you whether that statement is true or false. If you have a >third answer, I'll be happy to entertain it. The point being, Tony, that you don't have a first answer much less a second or third. You can't tell me or anyone else what it means to be true in mechanically exhaustive terms. Mathematikers routinely demand students deal in the most exacting exhaustive mechanical terms with axioms, theorems, and doctrines of their own. Yet the moment they're required to deal with their own axioms, doctrines, and assumptions of truth in mechanically exhaustive terms they shy away with complaints no one can expect to prove the truth of what they assume to be true. You draw up all kinds of binary "truth" tables as if they meant or had to mean something in mechanically exhaustive terms and demand others deal with them in binary terms you set forth. Yet you can't explain what you mean by "truth" or "falsity" in mechanically exhaustive terms to begin with. So how do you expect anyone to deal with truth tables? ~v~~
From: Mike Kelly on 13 Apr 2007 07:06 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. > > 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? > My threads do get attention, because I raise some valid issues. Guess again. >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. > >>>> 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? 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? It's rather disingenuous to ask me for a formal proof of something that is couched in your informal terms. > > 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. Seriously, do you comprehend that they are saying different things? This is important. 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. 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)? > > 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. >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. 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. 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. b) Some notions of "size" give a "size" to sets of naturals without maximum elements. c) Some notions of "size" give a "size" to sets of consecutive naturals starting from 1 without a maximum element. d) The "size" that these notions give cannot be the maximum element, because those sets don't *have* a maximum element. 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. > > 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, > >> 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"..) > 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.
From: Mike Kelly on 13 Apr 2007 07:32 On 12 Apr, 20:17, Tony Orlow <t...(a)lightlink.com> wrote: > Mike Kelly wrote: > > On 2 Apr, 17:12, step...(a)nomail.com wrote: > >> In sci.math Mike Kelly <mikekell...(a)googlemail.com> wrote: > > >>> On 2 Apr, 15:49, Tony Orlow <t...(a)lightlink.com> wrote: > >>>> Mike Kelly wrote: > >>>>> 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). > >>> It is true that the set of consecutive naturals starting at 1 with > >>> largest element x has cardinality x. > >>> 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. > >> To be fair to Tony, he said "size", not "cardinality". If Tony wishes to define > >> "size" such that set of consecutive naturals starting at 1 with size x has a > >> largest element x, he can, but an immediate consequence of that definition > >> is that N does not have a size. > > >> Stephen > > > Well, yes. But Tony wants to use this line of reasoning to then say > > "and, therefore, if N has size aleph_0 then aleph_0 is the largest > > element, which is clearly bunk". This is where his claims that > > "aleph_0 is a phantom" come from. But, obviously, this line of > > reasoning doesn't apply to all notions of "size". > > mike, what I am saying is that, assuming inductive proof works for all > equalities, this equality holds: |N|=max(N). I guess inductive proof doesn't work for all equalities then, because it's obvious to my inerrant intuition that |N| != max(N) because max(N) doesn't exist. Thankyou for a clear demonstration of why anyone with even vaguely sensible intuitions would reject the idea that "inductive proof works for all equalities". > If this is the case, then > aleph_0 cannot be infinite, while also being a member of N, and > therefore is self-contradictory, and cannot exist any more than a > largest finite. If |N|=max(N) is the case then "|N|" does not denote cardinality, so what you are saying has nothing to do with aleph_0. -- mike.
From: Bob Kolker on 13 Apr 2007 10:06
Mike Kelly wrote: > > 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. Whoa! If a -consectutive-(!!!) set of naturals with 1 as its least element has cardinality X (an integer), then its last element must be X. A simple induction argument will show this to be the case. Can you show a counter example? Bob Kolker |