The Definition of Points
in [Math]
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From: Tony Orlow on 14 Apr 2007 14:23 Virgil wrote: > In article <461fd938(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: > >>> 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. >>> >> Yeah, actually, I misspoke, in a way. Your statement is still blatantly >> false, in any case. It's possible for x<y and y<x in a cyclical-type >> system, but those two facts together do not imply x=y. > > > But a "cyclical-type system" is not an "ordered system" in any standard > mathematical sense. Times of day have no order? Pulllease!!! > > For any in which "<" is to represent the mathematical notion of an order > relation one will always have > ((x<y) and (y<x)) implies (x = y) > Okay, I'm worried about you. You repeated the same erroneous statement. You didn't cut and paste without reading, did you? Don't you mean "<=" rather than "<". The statement "x<y and y<y" can only be true in two unrelated meanings of "<", or else "=" doesn't have usable meaning. > > >> It may be the >> case, for every x and y, even when x=y, that x<y and y<x, but that >> doesn't mean x=y. The statements I gave you are correct, assuming your >> premise is false above. > > Which "premise" of mine are you presuming is false? "It's possible for x<y and y<x in a cyclical-type system, but those two facts together do not imply x=y." In other words, "((x<y) and (y<x)) implies (x = y)" is not correct. Oy, lemme get back to the rest of this later. Must get offline and play with daughter, dontcha know? I'll try to remember where I left off. >>>> 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. >>> >>> >>> >> It is the start of order. > > But one can have transitivity in an order relation without its being an > order relation. > > For example, the equality relation is clearly transitive, but is clearly > NOT an order relation on any set of more than one member. > > > >>>>> 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. >>> >> How procrustean of you. > > It may be procrustean of mathematics to require such pickiness, but it > is not anything I impose on mathematics or on those attempting to learn > mathematics, it just is the way mathematics is. >>>>>> 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. >>> >> Is it nitpicking to point out the forks of your tongue? > > Nowadays I only use spoons. > >>>> 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. >> Yes, there is a strange discrepancy there. I've never been able to >> accept that, for every unit of quantity on R, there are aleph_0 >> rationals and 1 natural, and yet, N and Q are "equinumerous". It's >> hogwash. > > it is the inevitable consequence of defining "equinumerous for sets as > meaning capable of being bijected with each other. Just another of those > inevitable mathematical nits. > >> The rationals are defined by NxN, minus the redundancies in >> quantity within the matrix. > > That "matrix" is a geometric interpretation, which is quite irrelevant. > > A better definition for the rationals, based on I as the set of integers > and P as the set of strictly positive integers is the set IxP modulo the > "==" relation defined by (a,b) == (c,d) iff a*d = b*c. > > > >> Equinumerous to those redundancies, which >> are the vast majority of cells, are the irrationals. That's how it >> actually works. > > Is TO actually claiming that the irrationals form a subset of the > countable set NxN. > > That is NOT how it works in any standard mathematics.
From: Lester Zick on 14 Apr 2007 14:52 On Fri, 13 Apr 2007 14:40:40 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Thu, 12 Apr 2007 14:29:22 -0400, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >>> Lester Zick wrote: >>>> On Sat, 31 Mar 2007 21:14:27 -0500, Tony Orlow <tony(a)lightlink.com> >>>> wrote: >>>> >>>>> Yeah, "true" and "false" and "or" are kinda ambiguous, eh?" >>>> They are where your demonstrations of their truth are concerned >>>> because there don't seem to be any. You just trot them out as if they >>>> were obvious axiomatic assumptions of truth not requiring any >>>> mechanical basis whatsoever or demonstrations on your part. >>>> >>>> ~v~~ >>> So, you're not interested in classifying certain propositions as "true" >>> and others as "false", so each is either true "or" false? I coulda >>> swored you done said that....oh nebbe mine! >> >> It makes no difference how you classify proposition as true or false >> when you can't demonstrate how it is they're true or false to begin >> with. Just saying they're true or false is irrelevant unless you can >> show why and how. That's what I'm trying to point out to you. You seem >> stuck on merely assuming certain propositions are true or false. >> >> ~v~~ > >Look, Lester, if you're actually interested in the mechanics of logical >truth, then you are looking for general rules. These rules cover the >general case, all input combinations, all possibilities. When evaluating >a statement based on assumptions, Well there are "assumptions" Tony, and then there are "assumptions". I assume there are things which can be true or not whereas you assume you already understand which things are true or not and how to work with their truth in mechanical terms while I'm interesting in finding out how to demonstrate their truth in mechanical terms to begin with and how to work with that truth or lack of it in mechanical terms. > do please assess each of those >assumptions for certitude when assessing the statement's truth, but when >speaking of the "mechanics" of deduction or induction it doesn't help to >worry about "assuming". Of course, Tony. You just take assumptions for granted. I don't. That's the whole context of the discussion. Mathematikers and empirics can't be bothered to demonstrate their assumptions. > That's inherent to the process, if your process >has anything to do with science. You "assume" there is such a thing as >truth. Is there? What's the alternative? Well it's about time you started to ask questions, Tony, instead of making problematic proclamations of mathematical certitude. I don't assume there is any such thing as truth. I just posit a mechanism of universal tautological contradiction and show that one alternative is self contradictory so I assume the other alternative is universally characteristic of everything which is not self contradictory and that is what I call true just as I call self contradiction false. In other words it's the mechanics underlying the determination of true and false that I'm trying to get at and not just the proclamation of true and false and the binary mechanics of working with those results. You on the other hand seem hell bent on explaining the mechanics of working with the results of true and false without explaining the mechanics of determining those circumstances. Mathematikers can't bring themselves to call the results "true" and "false" with straight faces so they just call them "truth values" instead of true and false. 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? And if it's just made in accordance with the manipulation of other "truth values" how are those "truth values" determined? Or is it all just a bunch of running "truth value" manipulations with no beginning or end? If that's all they are then you have no reason to call "truth values" "truth" values and you might just call them what they are 1's and 0's because that's all they really are. So what all this nonsense comes down to is that your "truth values" have no beginning in actual mechanical terms because they're given to you by assumption and not demonstrated in mechanical terms and all those conjunctions and conjunctive manipulations you describe are just so many arbitrary translation rules to work with otherwise meaningless 1's and 0's. ~v~~
From: Alan Smaill on 14 Apr 2007 14:56 Lester Zick <dontbother(a)nowhere.net> writes: > On Sat, 14 Apr 2007 13:56:37 +0100, Alan Smaill > <smaill(a)SPAMinf.ed.ac.uk> wrote: > >>Lester Zick <dontbother(a)nowhere.net> writes: >> >>> On Fri, 13 Apr 2007 16:10:39 +0100, Alan Smaill >>> <smaill(a)SPAMinf.ed.ac.uk> wrote: >>> >>>>Lester Zick <dontbother(a)nowhere.net> writes: >>>> >>>>> On Thu, 12 Apr 2007 14:23:04 -0400, Tony Orlow <tony(a)lightlink.com> >>>>> wrote: >>>>>> >>>>>>That's okay. 0 for 0 is 100%!!! :) >>>>> >>>>> Not exactly, Tony. 0/0 would have to be evaluated under L'Hospital's >>>>> rule. >>>> >>>>Dear me ... L'Hospital's rule is invalid. >>> >>> What ho? Surely you jest! >> >>Who, me? >> >>> Was it invalid when I used it in college? >> >>If you used it to work out a value for 0/0, then yes. > > Well the problem is that you didn't claim my application of > L'Hospital's rule was invalid. You claimed the rule itself was > invalid. So perhaps you'd like to show how the rule itself is invalid > or why my application of the rule is? Or both: The rule is invalid because that's what you find in Hospitals. Your use is invalid because the rule says nothing about the value of 0/0. > ~v~~ -- Alan Smaill
From: Lester Zick on 14 Apr 2007 14:57 On Fri, 13 Apr 2007 16:19:04 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On 13 Apr 2007 11:24:48 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >> >>> 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. >> >> If any and all these things are not demonstrably true and merely >> represent so many assumptions of truth why would anyone care what you >> think about what they are or aren't? I mean it really isn't as if >> truth is on your side to the exclusion of what others claim, Moe(x). >> >> ~v~~ > >Define "assumption". Any declarative judgment not demonstrated in mechanically exhaustive terms. > Do you "believe" that truth exists? Of course. > 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. > Is there such a thing as >truth, or falsity? Of course. > Does logic "exist". Yes. >There exists a set of assumptions, A, which are true. > >True? Yes. The difference is that to the extent they're undemonstrated assumptions we can have no idea which assumptions are true. ~v~~
From: Lester Zick on 14 Apr 2007 15:16
On Fri, 13 Apr 2007 14:36:12 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Thu, 12 Apr 2007 15:30:57 -0400, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >>> Lester Zick wrote: >>>> On Mon, 2 Apr 2007 16:12:46 +0000 (UTC), stephen(a)nomail.com wrote: >>>> >>>>>> 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. >>>> Is that true? >>>> >>>> ~v~~ >>> Yes, Lester, Stephen is exactly right. I am very happy to see this >>> response. It follows from the assumptions. Axioms have merit, but >>> deserve periodic review. >> >> What follows from the assumptions, Tony? Truth? >"that N does not have a size." I wasn't commenting on whether your assumptions are consistent with your axioms, Tony. I was asking whether your assumptions were true. >If the assumptions >> were true and could be demonstrated they wouldn't have to be assumed >> to begin with. > >Can we assume that a statement is either true, or it's false? Sure. Happens all the time. However if you're asking whether a statement must be one or the other the answer is no. There are problematic exceptions to the so called excluded middle. > Is that >too much of an assumption to make, when exploring the meaning of truth? >In ways yes, but for a start, no. Well your phrase "exploring the meaning of truth" is ambiguous, Tony, because what you're really doing is exploring consequences of truth or falsity given assumptions of truth or falsity to begin with, which is an almost completely trivial exercise in comparison with the actual determination of truth in mechanically exhaustive terms initially. >Mathematikers and empirics expect their students to use >> the most rigorous, exhaustive mechanics in extrapolating theorems and >> experimental methods from foundational assumptions. But the minute the >> same requirements of rigorous mechanics are laid on them and their own >> axioms and foundational assumptions they cry foul and claim no one can >> prove their assumptions and that even their definitions are completely >> arbitrary and can be considered neither true nor false. >> >> ~v~~ > >The question about axioms is whether each one is justifiable and >sufficiently general enough to be accepted as "true" in some universal >sense. No the actual question is whether each and every axiom is actually true and demonstrably so in mechanically exhaustive terms. Otherwise there's not much point to the exhaustively rigorous demonstration of theorems in terms of axioms demanded of students if axioms themselves are only assumed true. ~v~~ |