The Definition of Points
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From: MoeBlee on 13 Apr 2007 17:32 On Apr 13, 12:51 pm, Tony Orlow <t...(a)lightlink.com> wrote: > MoeBlee wrote: > > 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. > > I didn't bring up "measure theory". Where do I begin: transitivity, ordering, recursion, axiom of infinity, non-standard analysis...on and on and on... > > 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 If you mean non-constructivity, then no one disputes that the axiom of choice is non-constructive. No one says that the axiom of choice proves the existence of a definable well ordering. But if you require constructivity then you can't without contradiction endorse Robinson's non-standard analysis. > >>> 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 > > What point did I miss? The MAJOR point - the hypothetical nature of mathematical reasoning (think about the word 'if' twice in the poster's paragraph) and the inessentiality of what words we use to name mathematical objects and their properties. I've been trying to get you to understand that for about two years now. > I don't take transfinite cardinality to mean > "size". You say I missed the point. You didn't intersect the line. You just did it AGAIN. We and the poster to whom you responded KNOW that you don't take cardinality as capturing your notion of size. The point is then just for your to recognize that IF by 'size' we mean cardinality, then certain sentences follow and certain sentences don't follow and that what is important is not whether we use 'size' or 'cardinality' or whatever word but rather the mathematical relations that are studied even if we were to use the words 'schmize' or 'shmardinal' or whatever. MoeBlee
From: Virgil on 13 Apr 2007 17:45 In article <461fde40(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > cbrown(a)cbrownsystems.com wrote: > > On Apr 13, 10:13 am, Tony Orlow <t...(a)lightlink.com> wrote: > >> Virgil wrote: > >>> 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? > > That structure doesn't look like a tree to you, which each node having > as children the elements of its power set? That's the picture that > appears to me. Then TO cannot have looked very carefully. In a tree, no node has more than one parent but P({a,b}) = {{}, {a}, {b} and {a,b}} requires that, (1) if {} is the root, then {a} and {b} are both parents of {a,b}, or (2) if {a,b} is the root then {a} and {b} are both parents of {}. The appropriate mathematical name for such a structure is a /lattice/. > > > > > 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? > > > > The Archimedean principle, as far as I can tell. Where does the Archimedean principle apply in that definition, and how is it recursive in that definition? > > >> In that sense, there is > >> no pure infinite set without some defining structure Previously claimed without proof and again claimed without proof. , .... > >> 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. > > > > Actually I corrected this response to Virgil. I misspoke a little, but > he's still wrong. :) Then according to TO mere transistivity make equality an order relation, as it satisfies TO's sole criterion. > > >> 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. > > > > Please do elaborate. if "<" and "=" are the same relation of equality, for instance. > > >> 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"? > > > > Where b<c, {a,b}<{a,c}. Not necessarily. An order relation on the members of a set does not necessitate any order relation of the subsets of that set. Nor vice versa. > > Could you state what the definition of "<= totally orders the set S" > > is again? There are three simply stated properties, IIRC. > > > > Lookitup. Transitivity does not define "total order", but is that start > of order. > >>> 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? > > > > That's a bunch of lines. Then it is not one 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 > > > > Again, there is order to the elements themselves. Which of a number of possible orders is "the" order on "the set of all finite subsets of the naturals having a prime number of elements"
From: MoeBlee on 13 Apr 2007 17:46 On Apr 13, 12:11 pm, Tony Orlow <t...(a)lightlink.com> wrote: > I had said that, hoping you might give some explanation, but you didn't > really. Since you posted that, I wrote a long post about the axiom of choice. Now it's not showing up in the list of posts. Darn! I went into a lot of detail and answered your questions; I don't want to write it all again; maybe it will show up delayed. MoeBlee
From: Lester Zick on 13 Apr 2007 18:33 On Fri, 13 Apr 2007 13:45:46 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Thu, 12 Apr 2007 14:31:52 -0400, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >>> Lester Zick wrote: >>>> On Sat, 31 Mar 2007 20:51:49 -0500, Tony Orlow <tony(a)lightlink.com> >>>> wrote: >>>> >>>>> A logical statement can be classified as true or false? True or false? >>>> You show me the demonstration of your answer, Tony, because it's your >>>> question and your claim not mine. >>>> >>>> ~v~~ >>> I am asking you whether that statement is true or false. If you have a >>> third answer, I'll be happy to entertain it. >> >> The point being, Tony, that you don't have a first answer much less a >> second or third. You can't tell me or anyone else what it means to be >> true in mechanically exhaustive terms. Mathematikers routinely demand >> students deal in the most exacting exhaustive mechanical terms with >> axioms, theorems, and doctrines of their own. Yet the moment they're >> required to deal with their own axioms, doctrines, and assumptions of >> truth in mechanically exhaustive terms they shy away with complaints >> no one can expect to prove the truth of what they assume to be true. >> >> You draw up all kinds of binary "truth" tables as if they meant or had >> to mean something in mechanically exhaustive terms and demand others >> deal with them in binary terms you set forth. Yet you can't explain >> what you mean by "truth" or "falsity" in mechanically exhaustive terms >> to begin with. So how do you expect anyone to deal with truth tables? >> >> ~v~~ > >Just answer the question above. What question? You seem to think there is a question apart from whether a statement is true or false. All your classifications rely on that presumption. But you can't tell me what it means to be true or false so I don't know how to answer the question in terms that will satisfy you. ~v~~
From: Lester Zick on 13 Apr 2007 18:37
On Fri, 13 Apr 2007 14:33:20 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >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. Only because you're not doing a ground up mechanical derivation of true or false. You're just telling me how you employ the terms true and false in particular contexts whereas what I'm interested in is how true and false are defined in mechanically reduced exhaustive terms. What you clearly laid out are the uses of true and false with respect to one another once established. But you haven't done anything to establish true and false themselves in mechanically exhaustive terms. ~v~~ |