The Definition of Points
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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.
From: cbrown on 13 Apr 2007 14:30 On Apr 13, 10:13 am, Tony Orlow <t...(a)lightlink.com> wrote: > Virgil wrote: > > In article <461e8...(a)news2.lightlink.com>, > > Tony Orlow <t...(a)lightlink.com> wrote: > > >> MoeBlee wrote: > >>> On Mar 31, 5:18 am, Tony Orlow <t...(a)lightlink.com> wrote: > >>>> Virgil wrote: > >>>>> In article <1175275431.897052.225...(a)y80g2000hsf.googlegroups.com>, > >>>>> "MoeBlee" <jazzm...(a)hotmail.com> wrote: > >>>>>> On Mar 30, 9:39 am, Tony Orlow <t...(a)lightlink.com> wrote: > >>>>>>> They > >>>>>>> introduce the von Neumann ordinals defined solely by set inclusion, > >>>>>> By membership, not inclusion. > >>>>> By both. Every vN natural is simultaneously a member of and subset of > >>>>> all succeeding naturals. > >>>> Yes, you're both right. Each of the vN ordinals includes as a subset > >>>> each previous ordinal, and is a member of the set of all ordinals. > >>> In the more usual theories, there is no set of all ordinals. > > >> Right. Ordinals are...ordered. Sets aren't. > > > Ordinals have a unique ordering by reason of their being ordinals. > > Sets in general have all sorts of orderings, but none which is as > > inherent in their being sets as the ordinal order is in sets being > > ordinals. > > 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. Why is a recursive definition required? Given any set S, the set of all subsets of S can be (partially) ordered as follows: for subsets A, B of S, define A <= B iff every member of A is a member of B. What is recursive about that definition? Alternatively, S be the set of all subsets of the naturals (note that S is not countable). If A, B are in S, define A <= B if there is a natural number m in B such that m is not in A, and for all n < m, n in A and n in B. What is recursive about that definition? > In that sense, there is > no pure infinite set without some defining structure, so whatever > conclusions one thinks they have come to regarding infinite sets without > structure have no basis for comparison. Powerset(S) is 2^|S| sets, no > matter the size of S. That is a specific case of N=S^L, which applies to > symbolic strings and alphabets, as well as power sets where elements can > have S different levels of truth, not just 2. There are 3^log2(n) as > many ternary strings of length n as there are binary strings of length > n, be n finite or infinite. But, that involves a discussion of structure. > > >>>> Anyway, my point is that the recursive nature of the definition of the > >>>> "set" > >>> What recursive definition of what set? > > >> Oh c'mon! N. ala Peano? (sigh) What kind of question is that? > > > Does TO seem to thing that N is the only set defineable recursively or > > that "successor" is the only recursively defineable operations on sets? > > Does Virgil forget what he cuts from the post? What do you think we were > discussing? I thought it was N specifically. > I thought it had something to do with the real line, and orderings. > >>>> 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 don't see how this follows only from your assertion "x < y and y < z -> x < z". You stated: > >>>> Order is defined by x<y ^ y<z -> x<z. Or do you mean that there is /more/ to the definition of an order "<" than "x < y and y < z -> x < z"? If so, that was exactly Virgil's point. > suppose you meant: > ((x>=y) and (y>=x)) -> x = y > or: > (~(x>y) and ~(y>x)) -> x = y > These two statements are not equivalent. In some situations, the first can hold, while the second does not. > Yes, if neither x<y or y<x is true, that is, if no order can be > determined, then x=y for the purposes of that order. Let's order the subsets of {a,b,c} by inclusion. Then: {a,b} < {a,b,c}. {a,c} < {a,b,c}. {a} < {a,b}. {a} < {a,c}. But not ({a,b} < {a,c}); and not ({a,c} < {a,b}). Does that mean that {a,b} = {a,c} "for the purposes of that order"? > That defines '=' in > terms of '<'. It defines a point on the line, more or less, to get back > to the original question. > Could you state what the definition of "<= totally orders the set S" is again? There are three simply stated properties, IIRC. > > > > Also there are lots of transitive relations which are not orderings, at > > least as usually understood. E.g., universal relations, which hold true > > for all x and y in the relevant set. > > > So that TO's notion of an ordering does not necessarily order anything. > > You're missing the point. All I said was that one starts with inequality > defining the line itself. Is every set a line? Is the set of all triangles in the Euclidean plane a line? > Then one defines equality. Defining equality > where there is no relative order doesn't make sense. So, it makes no sense to say that the set of all finite subsets of the naturals having a prime number of elements is equal to itself? Cheers - Chas
From: Tony Orlow on 13 Apr 2007 14:33 Lester Zick wrote: > On Thu, 12 Apr 2007 14:35:36 -0400, Tony Orlow <tony(a)lightlink.com> > wrote: > >> Lester Zick wrote: >>> On Sat, 31 Mar 2007 20:58:31 -0500, Tony Orlow <tony(a)lightlink.com> >>> wrote: >>> >>>> How many arguments do true() and false() take? Zero? (sigh) >>>> Well, there they are. Zero-place operators for your dining pleasure. >>> Or negative place operators, or imaginary place operators, or maybe >>> even infinite and infinitesimal operators. I'd say the field's pretty >>> wide open when all you're doing is guessing and making assumptions of >>> truth. Pretty much whatever you'd want I expect.Don't let me stop you. >>> >>> ~v~~ >> Okay, so if there are no parameters to the function, you would like to >> say there's an imaginary, or real, or natural, or whatever kind of >> parameter, that doesn't matter? Oy! It doesn't matter. true() and >> false() take no parameters at all, and return a logical truth value. >> They are logical functions, like not(x), or or(x,y) and and(x,y). Not >> like not(). That requires a logical parameter to the function. > > Tony, you might just as well be making all this up as you go along > according to what seems reasonable to you. My point was that you have > no demonstration any of these characteristics in terms of one another > which proves or disproves any of these properties in mechanical terms > starting right at the beginning with the ideas of true and false. > > ~v~~ Sorry, Lester, but that's an outright lie. I clearly laid it out for you, starting with only true and false, demonstrating how not(x) is the only 1-place operator besides x, true and false, and how the 2-place operators follow. For someone who claims to want mechanical ground-up derivations of truth, you certainly seem unappreciative. 01oo
From: MoeBlee on 13 Apr 2007 14:36
On Apr 13, 11:25 am, Tony Orlow <t...(a)lightlink.com> wrote: > 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? Nothing, really, until you learn the mathematics you're pretending to know about. > > 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. Nothing to which you responded "pretends" that cardinality "can tell things it can't". What SPECIFIC theorem of set theory do you feel is a pretense of "telling things that it can't"? > > 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. Wow, that is about as BLATANTLY missing the point of what you are in immediate response to as I can imagine even you pulling off. MoeBlee |