The Definition of Points
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From: Tony Orlow on 17 Apr 2007 14:22 MoeBlee wrote: > On Apr 14, 7:08 pm, Tony Orlow <t...(a)lightlink.com> wrote: > >> Maybe I don't understand Robinson as well as I should, but it seems to >> me the basis of his analysis was semantic, regarding statements that >> would be considered true of *N if true of N. But, do go on... > > "But do go on..." Hey, smartass, YOU'RE the one who waves non-standard > analysis like banner when you don't know ANYTIHNG about it. > > If you want to know anything about non-standard analysis, then you'd > start by learning basic predicate calculus, set theory, and > mathematical logic (personally, I'd recommend that order). > >>> 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 guess, by "inessentiality", you mean any attribute that one could >> assign to any object... IF that's what you mean, or not.... > > No, smartass, months ago, several times, I explained in detail to you, > and with respect to the technicalities, what I mean. > >>> I've been trying to get you to understand that for about two years >>> now. >> Perhaps what you mean is exactly what I am trying to get across to >> Lester. If the truth table is the same, then it's the same logical >> function. It doesn't matter what the parameters are, it always works the >> same way. The pattern defines the relationship. > > That's getting closer. I won't quibble with it. But I'll just say that > in mathematical logic we have even more precise ways of saying it. > >> On the subject of ifs, "if this then that" means logical implication, or >> causality, and sometimes it's hard to tell which is meant, or if they're >> being confused. Is that what you "mean"? > > No. > >>>> 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. > > THAT, to which you did not respond, is, at least at this informal > level of discussion with you, good enough for a start as to what I > mean. > > MoeBlee > Why would I respond to that kind of abuse? If I am a smartass, well, I've got more smarts than you have in your little finger. (Did that come out right?) ;) TOEKnee
From: Alan Smaill on 17 Apr 2007 15:03 Tony Orlow <tony(a)lightlink.com> writes: > Alan Smaill wrote: >> Lester Zick <dontbother(a)nowhere.net> writes: >> >>> On Sun, 15 Apr 2007 21:34:09 +0100, Alan Smaill >>> <smaill(a)SPAMinf.ed.ac.uk> wrote: >>> >>>>>>>>>>>> Dear me ... L'Hospital's rule is invalid. >>>>> So returning to the original point, would you care to explain your >>>>> claim that L'Hospital's rule is invalid? >>>> haha! >>> Why am I not surprized? Remarkable how many mathematiker opinions on >>> the subject of mathematics don't quite hold up to critical scrutiny. >> knew you wouldn't get it, >> irony is not a strong point with Ziko. >> nor indeed do you bother defending your own view that you can use >> Hospital to work out the value for 0/0. >> well, there you go. >> >>> ~v~~ >> > > In all fairness to Lester, I am the one who said 0 for 0 is > 100%. T'was a joke, of course! > ala L'Hospital's theft from the Bernoullis, and > the division by 0 proscription. > > :) and Zick was the one who claimed that he would use l'Hospital to work out the right answer for 0/0. such a japester, eh? > > Tony -- Alan Smaill
From: MoeBlee on 17 Apr 2007 15:16 On Apr 17, 10:58 am, Tony Orlow <t...(a)lightlink.com> wrote: > Any false statement implies any statement, true or false, > as long as you're not an intuitionist. For intuitionists too. f |- P holds for intuitionistic logic. MoeBlee
From: Virgil on 17 Apr 2007 15:19 In article <4624fa54(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > cbrown(a)cbrownsystems.com wrote: > > To me, that "looks" more like a fish net or a giant diamond than a > > tree. > > > > At any rate, my question was: what is recursive about that definition? > > > > May I restate the definition, for the purposes of this elucidation? > > 1. E S e P(S) > 2. A X e P(S) A x e X E Y e P(S) | A y e X (x<>y -> y e Y) > > > Does this not recursively define a tree of subsets of P(S)? > > You may consider S to be the maximal element of P(S). What definition of "tree are you using, TO? One definition of a tree would be a set of objects, say S, such that (1) there is a unique object, r, in S called the root (2) there is a function P from S\{r} to S (called the parent function) such that, under iteration, every s in S has r as an ancestor. i.e., if we denote P^1(x) = P(x) and P^(n+1)(x) = P(P^n(x)), inductively, then for each s in S\{r} there is a unique n in N such that P^n(x) = r > > In what way does the assertion: > > > > For all positive real numbers r, there exists a natural number n such > > that n*r > 1? > > > > apply to definition above, which is about sets of naturals, and never > > mentions real numbers at all? > > > > "Given any Real number c, there exists a natural number n such that n > > c". If all naturals are reals, then this may be restated as "A neN E meN > | m>n". Sound a little Peanoesque to you? Not at all. It is clearly Archimedean, preceding Peano by millennia. > > >> > >>>> 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. > >> Points and lines, anyway. And points on R in N. > > > > On R, and yet in N? I don't understand what you mean. > > > > On the real line R, in the subset of real points called N. > > >> Or, whatever. > > > > Indeed. > > > >> > >> > >>>>>>>> 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. > >> Actually I corrected this response to Virgil. I misspoke a little, but > >> he's still wrong. :) > >> > >>>> 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. > >> > > > > See the example I posted below regarding subsets of {a,b,c}. > > > >>>> 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}. > > > > Since "<" indicates subset inclusion, it is the case that not b < c. > > > > If a, b and c are members of a set such that xeS ^ yeS -> x<y v y<x v > x=y, then either b<c v b<c v b=c. As stated, it is possible that b<c ^ b<c ^ b=c, unless TO intends, contrary to standard notation, to have "v" to represent "xor". > > > > Consider that all elements of N, x and y, are such that not(equal(x,y)) > -> x<y v y<x. Does TO intend "v" to mean "xor" ?
From: Virgil on 17 Apr 2007 15:24
In article <4624ff57(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Mike Kelly wrote: > > And what does AC have to do with cardinality? > > What do any of the axioms of ZFC have to do with cardinality? > Extensionality. Which of those axioms, or combination of axioms, is extensionally equivalent to cardinality? |