The Definition of Points
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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
From: Tony Orlow on 13 Apr 2007 14:36 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." 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? Is that too much of an assumption to make, when exploring the meaning of truth? In ways yes, but for a start, no. 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. 01oo
From: Tony Orlow on 13 Apr 2007 14:40
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, 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". 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? 01oo |