The Definition of Points
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From: Alan Smaill on 13 Apr 2007 11:10 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. > ~v~~ -- Alan Smaill
From: stephen on 13 Apr 2007 12:52 In sci.math 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. > 01oo Everything follows from the assumptions and definitions. People have been telling you this for well over a year now. If you change the axioms, or change the definitions, you will get different results. However the old axioms, definitions and results remain just the same as before. N has a cardinality. If "size" is defined as cardinality, N has a "size". If "size" is defined differently, N still has a cardinality. Stephen
From: Lester Zick on 13 Apr 2007 13:05 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~~
From: Lester Zick on 13 Apr 2007 13:16 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? If the assumptions were true and could be demonstrated they wouldn't have to be assumed to begin with. 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~~
From: Tony Orlow on 13 Apr 2007 13:13
Virgil wrote: > In article <461e879d(a)news2.lightlink.com>, > Tony Orlow <tony(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. 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. >>>> 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 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. That defines '=' in terms of '<'. It defines a point on the line, more or less, to get back to the original question. > > 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. Then one defines equality. Defining equality where there is no relative order doesn't make sense. It's like a point, by itself. Points are either defined as n-tuples assuming a space with measure, or maybe as intersections of lines, in each case requiring lines for their definition. Otherwise they're nothing. So, x<y has to come before x=y, for that reason, philosophically. >>>> I suppose >>>> this is one reason why I think a proper subset should ALWAYS be >>>> considered a lesser set than its proper superset. It's less than the >>>> superset by the very mechanics of what "less than" means. > > > 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. :) >> 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? 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. > > 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. |