The Definition of Points
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From: Tony Orlow on 13 Apr 2007 13:56 Lester Zick wrote: > 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~~ 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. 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. One more time, is this correct, for the four combinations of a and b being true or false? a b "not a not b" F F T F T T T F T T T F 01oo
From: MoeBlee on 13 Apr 2007 14:01 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
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. |