The Definition of Points
in [Math]
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From: Tony Orlow on 17 Apr 2007 13:02 Brian Chandler wrote: > Tony Orlow wrote: >> MoeBlee wrote: >>> On Apr 13, 10:38 am, Tony Orlow <t...(a)lightlink.com> wrote: >>>> MoeBlee wrote: >>>>> Zermelo's motivation was to prove that for every set, there exists a >>>>> well ordering on it. >>>> I am not sure how the Axiom of Choice demonstrates that. >>> You don't know how the axiom of choice is used to prove that for every >>> set there exists a well ordering of the set? Virtually any set theory >>> textbook will give a cycle of proofs showing equivalence of (not >>> necessarily in order) the axiom of choice (in its various >>> formulations), Zorn's lemma, the well ordering theorem, the numeration >>> theorem, etc. >>> >>> Among those textbooks I recommend Stoll's 'Set Theory And Logic' as it >>> accomplishes some of the proofs without using the axiom schema of >>> replacement while other textbooks do use the axiom schema of >>> replacement for certain of the proofs, though, I don't recommend >>> Stoll's book for an overall systematic treatment since it jumps around >>> topics too much and doesn't have the kind of "linear" format that >>> Suppes does so well. >>> >>> Anyway, even if you don't know the details of the proofs, don't you at >>> least have an intuition how a choice function would come in handy >>> toward proving the well ordering theorem? >>> >>> MoeBlee >>> >> Hi MoeBlee - >> >> I had said that, hoping you might give some explanation, but you didn't >> really. However, before I sent that response I reminded myself on >> Wikipedia exactly what AC says, and looked at the definitions of >> Dependent Choice and Countable Choice as well, and descriptions of the >> relationships between them. I didn't find anything objectionable in ACC >> or DC. I think it is the broad statement of AC that any set is well >> orderable that offends my sensibilities. > > What a good job then, that mathematics is not about "your > sensibilities". If you accept (use, adopt, whatever) the Axiom of > Choice then there is a proof that any set has a well-ordering. > Sensibilities don't come into it. > Sensibilities indeed play a role in what assumptions we do or don't accept to begin with. The relegation of mathematics to the purely deductive realm is a copout. Where a lot of intuition went into the formulation of ZFC, intuition may also reject it. >> Here's my intuition. If you have a set, and can partition it into >> mutually exclusive subsets, within which and between which exists an >> order, then you can linearly arrange the elements of the set by choosing >> the first element of the first partition, and repeatedly choosing the >> first unchosen element from the next partition, returning to the first >> partition when the last is used, and skipping any partitions from which >> all elements have been chosen. > > What if any of thes sets do not have either a first member (within the > ordering) or a last member? If the set is ordered to begin with, such that x<>y -> x<y v y<x, then any subset is also ordered accordingly. > Do your intuitions perhaps tell you that there always _is_ a "first" > member and a "last" member? Do you notice any similarity between this > particular "intuition" and the thing you were claiming to be proving > in the first place? > > > Does a well orderable set have a first member? What was I proving to begin with? Did you think this statement of intuition was a proof? > Where we have a countably infinite set, >> we may choose a finite number of partitions, at least one of which has a >> countably infinite number of elements, or we may choose all finite >> partitions, but a countably infinite number of them. In such a case, >> indeed, it's possible to define a well order. > > If by "countable" and "well order" you mean what mathematicians mean, > then this is a bizarre circumlocution, since a countable set has a > well-ordering more or less by definition. I suppose you mean something > subtly different by these terms - something you yourself understand > perfectly, yet are unable to convey to anyone with a grasp of actual > mathematics. > > <snip> > > Brian Chandler > http://imaginatorium.org > I am saying it's obvious that any countable set has a well ordering. It is not obvious for uncountable sets. Nice snip of the actual import of my post, BTW. Thanx. Tony Orlow http://realitorium.net ;)
From: Tony Orlow on 17 Apr 2007 13:09 Mike Kelly wrote: > On 13 Apr, 20:51, Tony Orlow <t...(a)lightlink.com> wrote: >> MoeBlee wrote: >>> On Apr 13, 11:25 am, Tony Orlow <t...(a)lightlink.com> wrote: >>>>> 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"? >> AC > > And what AC have to do with cardinality? What do any of the axioms of ZFC have to do with cardinality? Extensionality. But, cardinality is a Galilean, read "primitive", extension of Extensionality. > >>>>> 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. >> What point did I miss? I don't take transfinite cardinality to mean >> "size". You say I missed the point. You didn't intersect the line. > > 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". If I say there are countably infinitely many possible lemurs in the future, and countably infinitely many possible mammals in the future, are there equally infinitely many possible lemurs as mammals? It's unlikely that there will ever be a single mammalian species in the universe, much less that it be lemurs. Lemurs will always be a proper subset of mammals. There will always be more mammals than lemurs, until there are none of each. I guess 0 is countable, but not infinite... tony.
From: Tony Orlow on 17 Apr 2007 13:12 Lester Zick wrote: > On Fri, 13 Apr 2007 16:11:22 -0400, Tony Orlow <tony(a)lightlink.com> > wrote: > >>>>>>>>> Constant linear velocity in combination with transverse acceleration. >>>>>>>>> >>>>>>>>> ~v~~ >>>>>>>> Constant transverse acceleration? > >>>>>>> What did I say, Tony? Constant linear velocity in combination with >>>>>>> transverse acceleration? Or constant transverse acceleration? I mean >>>>>>> my reply is right there above yours. >>>>>>> >>>>>>> ~v~~ >>>>>> If the transverse acceleration varies, then you do not have a circle. >>>>> Of course not. You do however have a curve. >>>>> >>>>> ~v~~ >>>> I thought you considered the transverse acceleration to vary >>>> infinitesimally, but that was a while back... >>> Still do, Tony. How does that affect whether you have a curve or not? >>> Transverse a produces finite transverse v which produces infinitesimal >>> dr which "curves" the constant linear v infinitesimally. >>> >>> ~v~~ >> Varying is the opposite of being constant. Checkiddout! > > I don't doubt "varying" is not "constant". So what? The result of > "constant" velocity and "varying" transverse acceleration is still a > curve. > > ~v~~ I asked about CONSTANT transverse acceleration. Oy! More exactly, linearly proportional velocity and transverse acceleration produce the circle. It can speed up and slow down, as long as it changes direction at a rate in proportion with its change in velocity. Close your eyes, and watch.... 01oo
From: Tony Orlow on 17 Apr 2007 13:29 Lester Zick wrote: > 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. > Which class do you fall into? Lester e Mathematikers, or Lester e Empirics, or Lester e Bullshitters? What have you demonstrated, besides exhaustive somethingorother? >> 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. You don't answer questions, Lester. Did you this time? ... 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. Oh. You don't assume. You posit. Huh! Is that like depositing, as opposed to withdrawing? You're certainly not withdrawing, that I can see. What does "posit" mean? http://dictionary.reference.com/browse/posit Huh!!!! It means "assume"!!! Wow, that's strange.... > > 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. That's because the premises are...posited and not proven. > 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. And if it's just made in > accordance with the manipulation of other "truth values" how are those > "truth values" determined? That's an inductive matter, based on evidence, and in the case of math, the acceptability of the conclusions derived from the posited assumptions. 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. You cannot deduce conclusions without inducing assumptions, for the sake of logical consideration. > > 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~~ 0 and 1 are meaningful. They are nothing and all. 01oo
From: Tony Orlow on 17 Apr 2007 13:33
Lester Zick wrote: > 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. > Prove it in "mechanically exhaustive terms". >> 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. >> Is there such a thing as >> truth, or falsity? > > Of course. > "Prove" logic exists, in terms that precede logic. >> Does logic "exist". > > Yes. > Prove it from first principles. Unless, of course, you're just "positing". >> 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~~ The difference between which duck? Quack, Lester. Quack. 01oo |