The Definition of Points
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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?
From: MoeBlee on 17 Apr 2007 15:29
On Apr 17, 11:22 am, Tony Orlow <t...(a)lightlink.com> wrote: > 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? "Abuse". Come on. > If I am a smartass, well, > I've got more smarts than you have in your little finger. (Did that come > out right?) ;) I've never questioned your intelligence. I have no idea as to our comparative intelligence, and not much interest in it. I guess that most people who have an interest in mathematics are more intelligent, or at least more adept at mathematical problem solving, than I am. I surmise that your problem is not intelligence but something else altogether. I don't necessarily subscribe to such rubrics as 'narcissistic personality disorder', but I'd bet that you'd be get a high evaluation for it. It strikes me that whatever it is, there is something in your inflated sense of your self that blocks you from appreciation that to understand the subject of mathematics and logic requires learning it the way everyone else learns it - textbook by textbook, chapter by chapter, defintion by defintion, theorem by theorem, proof by proof. MoeBlee |