The Definition of Points
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From: cbrown on 13 Apr 2007 18:56 On Apr 13, 1:07 pm, Tony Orlow <t...(a)lightlink.com> wrote: > cbr...(a)cbrownsystems.com wrote: > > On Apr 13, 10:38 am, Tony Orlow <t...(a)lightlink.com> wrote: > >> MoeBlee wrote: > >>> On Apr 12, 2:36 pm, "MoeBlee" <jazzm...(a)hotmail.com> wrote: > >>>> Zermelo's motivation was to prove that every set is well ordered. > >>> Since that phrasing might be misunderstood, I should say that I mean: > >>> Zermelo's motivation was to prove that for every set, there exists a > >>> well ordering on it. > >>> MoeBlee > >> I am not sure how the Axiom of Choice demonstrates that. > > > AoC says (roughly) that we have a way to unambiguously choose an > > element from any set: i.e., for any set S, there exists a function f > > such that for any subset A of S, f(A) is a member of A (and of course > > therefore, a member of S). > > > "S is well ordered by <=" states that "<=" is a total order on S, and > > for any subset A of S, there is a specific least element of A. > > > So if "<=" well-orders S, then "the least element of A" is a function > > like the one that AoC tells us exists: it chooses a specific element > > from any subset A of S. So "every set can be well-ordered" implies > > "for every set S, there exists a choice function for S". > > > The converse ("for every set S, there exists a choice function for S" > > implies "every set can be well-ordered") is a bit more complicated. A > > proof online is at > > >http://planetmath.org/?op=getobj&from=objects&id=3359 > > > but you'll have to accept certain facts about ordinals (e.g., that > > they exist, that any set of them is well-ordered by inclusion, that > > transfinite induction over a well-ordered set is possible, etc.) which > > you have previously balked at. > > > The basic idea is to let f(S) be the smallest element of S, then f(S\ > > {f(S)}) be the next smallest element, and so on. Of course, > > transfinite induction is required for sets which are not countable. > > > Cheers - Chas > > Thanks for the reply, Chas. You should see mine to MoeBlee. I don't have > a lot of time now, because I have to tool on outta here, but I took a > quick look at your link, and found it rests on transfinite induction. I > don't recall exactly how that works right now, but I definitely recall > feeling it was very kludgy. So, I'm not sure I have to accept that > proof. I'll try to take a look at it in more dtail when I get a chance. Feel free to post questions; the proof at that link is very terse. > > Just to recap what I said to MoeBlee, I can't help feeling that, since a > countable union of countable sets is countable, either there will be an > uncountable number of partitions of an uncountable set, in which case > there will exist infinite descending chains before the first "limit > ordinal" in the well order (besides the first element of the first > partition), or there will exist at least one uncountable partition which > will produce infinite descending chains within itself, after all > countable partitions have been exhausted. DC or ACC seem acceptable, but > not AC, to me. > > Tony You're not alone in this "gut feeling". The mathematical joke goes: "The Axiom of Choice is obviously true, the Well-ordering principle is obviously false, and who knows about Zorn's lemma?" The axiom of choice, by itself, simply says : Let X be a set of non-empty sets. Then we can choose a single member from each set in X. That seems pretty clear-cut; who could doubt it? Except that it / logically/ implies all sorts of "horrible" (to some people's intuitions) things. Cheers - Chas
From: Lester Zick on 13 Apr 2007 19:00 On Fri, 13 Apr 2007 13:42:06 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Thu, 12 Apr 2007 14:23:04 -0400, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >>> Lester Zick wrote: >>>> On Sat, 31 Mar 2007 16:18:16 -0400, Bob Kolker <nowhere(a)nowhere.com> >>>> wrote: >>>> >>>>> Lester Zick wrote: >>>>> >>>>>> Mathematikers still can't say what an infinity is, Bob, and when they >>>>>> try to they're just guessing anyway. So I suppose if we were to take >>>>>> your claim literally we would just have to conclude that what made >>>>>> physics possible was guessing and not mathematics at all. >>>>> Not true. Transfite cardinality is well defined. >>>> I didn't say it wasn't, Bob. You can do all the transfinite zen you >>>> like. I said "infinity". >>>> >>>>> In projective geometry points at infinity are well defined (use >>>>> homogeneous coordinates). >>>> That's nice, Bob. >>>> >>>>> You are batting 0 for n, as usual. >>>> Considerably higher than second guessers. >>>> >>>> ~v~~ >>> That's okay. 0 for 0 is 100%!!! :) >> >> Not exactly, Tony. 0/0 would have to be evaluated under L'Hospital's >> rule. >> >> ~v~~ > >Well, you put something together that one can take a derivative of, and >let's see what happens with that. Or let's see you put something together that you can't take the deriviative of and let's see how you managed to do it. ~v~~
From: Lester Zick on 13 Apr 2007 19:34 On Fri, 13 Apr 2007 13:56:45 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On 12 Apr 2007 14:07:53 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >> >>> On Apr 12, 11:30 am, Tony Orlow <t...(a)lightlink.com> wrote: >>>> Lester Zick wrote: >>>>> What grammar did you have in mind exactly, Tony? >>>> Some commonly understood mapping between strings and meaning, >>>> basically. >>> Grammar is syntax, not meaning, which is semantics. What you just >>> described, an intrepative mapping from strings to meanings of the >>> strings is semantics, not grammar. >> >> Gee that's swell, Moe(x). Thanks for the lesson in semantics if not >> much of anything else. Next time we need a lesson in modern math don't >> call us we'll call you. >> >> ~v~~ > >Well, of course, Moe's technically right, though I originally asked >Lester to define his meaning in relation to his grammar. Technically, >grammar just defines which statements are valid, to which specific >meanings are like parameters plugged in for the interpretation. I asked >the question originally using truth tables to avoid all that, so that we >can directly equate Lester's grammar with the common grammar, on that >level, and derive whether "not a not b" and "not a or not b" were the >same thing. They seem to be. Well, Tony, are there any other assumptions of truth you'd care to make along the way that I can confirm or deny for you? I mean if you can't even get past the definition of truth in mechanically exhaustive terms how do you exactly expect to move on to assumptions of truth in conjunctions, grammar, syntax, semantics, semiotics, and symbolics? I mean you just seem to "assume" we have all these things the way you lay them out without so much as a nod to whether they're true or not. Has it ever occurred to you that maybe we might need a little truth in terms of what all these things are supposed to mean before we willy nilly begin to brandish them about as if we actually knew what we're talking about in demonstrably exhaustive terms? Does it even occur to you that truth itself represents the product and result of tautological mechanics and that without that there is no grammar, syntax, semantics, semiotics, and symbolics, or conjunctions, not to mention mathematics or truth tables to deal with or in? >One more time, is this correct, for the four combinations of a and b >being true or false? > >a b "not a not b" > >F F T >F T T >T F T >T T F > >01oo ~v~~
From: Brian Chandler on 14 Apr 2007 03:09 Tony Orlow wrote: > MoeBlee wrote: > > On Apr 13, 10:38 am, Tony Orlow <t...(a)lightlink.com> wrote: > >> MoeBlee wrote: > > > >>> Zermelo's motivation was to prove that for every set, there exists a > >>> well ordering on it. > > > >> I am not sure how the Axiom of Choice demonstrates that. > > > > You don't know how the axiom of choice is used to prove that for every > > set there exists a well ordering of the set? Virtually any set theory > > textbook will give a cycle of proofs showing equivalence of (not > > necessarily in order) the axiom of choice (in its various > > formulations), Zorn's lemma, the well ordering theorem, the numeration > > theorem, etc. > > > > Among those textbooks I recommend Stoll's 'Set Theory And Logic' as it > > accomplishes some of the proofs without using the axiom schema of > > replacement while other textbooks do use the axiom schema of > > replacement for certain of the proofs, though, I don't recommend > > Stoll's book for an overall systematic treatment since it jumps around > > topics too much and doesn't have the kind of "linear" format that > > Suppes does so well. > > > > Anyway, even if you don't know the details of the proofs, don't you at > > least have an intuition how a choice function would come in handy > > toward proving the well ordering theorem? > > > > MoeBlee > > > > Hi MoeBlee - > > I had said that, hoping you might give some explanation, but you didn't > really. However, before I sent that response I reminded myself on > Wikipedia exactly what AC says, and looked at the definitions of > Dependent Choice and Countable Choice as well, and descriptions of the > relationships between them. I didn't find anything objectionable in ACC > or DC. I think it is the broad statement of AC that any set is well > orderable that offends my sensibilities. What a good job then, that mathematics is not about "your sensibilities". If you accept (use, adopt, whatever) the Axiom of Choice then there is a proof that any set has a well-ordering. Sensibilities don't come into it. > Here's my intuition. If you have a set, and can partition it into > mutually exclusive subsets, within which and between which exists an > order, then you can linearly arrange the elements of the set by choosing > the first element of the first partition, and repeatedly choosing the > first unchosen element from the next partition, returning to the first > partition when the last is used, and skipping any partitions from which > all elements have been chosen. What if any of thes sets do not have either a first member (within the ordering) or a last member? Do your intuitions perhaps tell you that there always _is_ a "first" member and a "last" member? Do you notice any similarity between this particular "intuition" and the thing you were claiming to be proving in the first place? Where we have a countably infinite set, > we may choose a finite number of partitions, at least one of which has a > countably infinite number of elements, or we may choose all finite > partitions, but a countably infinite number of them. In such a case, > indeed, it's possible to define a well order. If by "countable" and "well order" you mean what mathematicians mean, then this is a bizarre circumlocution, since a countable set has a well-ordering more or less by definition. I suppose you mean something subtly different by these terms - something you yourself understand perfectly, yet are unable to convey to anyone with a grasp of actual mathematics. <snip> Brian Chandler http://imaginatorium.org
From: Mike Kelly on 14 Apr 2007 07:20
On 13 Apr, 19:25, Tony Orlow <t...(a)lightlink.com> wrote: > Mike Kelly wrote: > > On 12 Apr, 19:54, Tony Orlow <t...(a)lightlink.com> wrote: > >> Mike Kelly wrote: > >>> On 2 Apr, 15:49, Tony Orlow <t...(a)lightlink.com> wrote: > >>>> Mike Kelly wrote: > >>>>> On 1 Apr, 04:44, Tony Orlow <t...(a)lightlink.com> wrote: > >>>>>> cbr...(a)cbrownsystems.com wrote: > >>>>>>> On Mar 31, 5:45 pm, Tony Orlow <t...(a)lightlink.com> wrote:> > >>>>>>>> Yes, NeN, as Ross says. I understand what he means, but you don't. > >>>>>>> What I don't understand is what name you would like to give to the set > >>>>>>> {n : n e N and n <> N}. M? > >>>>>>> Cheers - Chas > >>>>>> N-1? Why do I need to define that uselessness? I don't want to give a > >>>>>> size to the set of finite naturals because defining the size of that set > >>>>>> is inherently self-contradictory, > >>>>> So.. you accept that the set of naturals exists? But you don't accept > >>>>> that it can have a "size". Is it acceptable for it to have a > >>>>> "bijectibility class"? Or is that taboo in your mind, too? If nobody > >>>>> ever refered to cardinality as "size" but always said "bijectibility > >>>>> class" (or just "cardinality"..) would all your objections disappear? > >>>> Yes, but my desire for a good way of measuring infinite sets wouldn't go > >>>> away. > ><snip> > > > It'd be nice if some day you learned some math above high school > > level. Seems a rather remote possibility though because you are > > WILLFULLY ignorant. > > And you are willfully obnoxious, but I won't take it seriously. Look who's talking.. >I'm not the only one in the revolution against blind axiomatics. Blind axiomatics? So you think ZFC was developed by blindly? People picked the axioms randomly without any real consideration for what the consequences would be? Please. ZF(C) provides a foundation for virtually all modern mathematics. This didn't happen by accident. What's "blind" about ZF(C)? What great insight do you think is missed that you are going to provide, oh mighty revolutionary? What mathematics can be done with your non-existant foundation that can't be done in ZF(C)? > >>>>>> given the fact that its size must be equal to the largest element, > >>>>> That isn't a fact. It's true that the size of a set of naturals of the > >>>>> form {1,2,3,...,n} is n. But N isn't a set of that form. Is it? > >>>> It's true that the set of consecutive naturals starting at 1 with size x has largest element x. > >>> No. This is not true if the set is not finite (if it does not have a > >>> largest element). > >> Prove it, formally, please, from your axioms. > > > I don't have a formal definition of "size". You understand this point, > > yes? > > Then how do you presume to declare that my statement is "not true"? > > > 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. > > > It's rather disingenuous to ask me for a formal proof of something > > that is couched in your informal terms. > > Don't say "This is not true" if you can't disprove it. > > >>> It is true that the set of consecutive naturals starting at 1 with > >>> largest element x has cardinality x. > >> Forget cardinality. Can a set of naturals starting with 1 and with size > >> X possibly have any other maximum value besides X? This is inductively > >> impossible. > > > (Just to be clear, we're talking very informally here. It's quite > > obviously the only way to talk to you.) > > > Tony, can you discern a difference between the following two > > statements? > > > a) A consecutive set of naturals starting with 1 with size X can not > > have any maximum other than X. > > b) A consecutive set of naturals starting with 1 with size X has > > maximum X. > > Yes, the first allows that may be no maximum, but where there is a > specific size for such a set, there is a specific maximum as well. I am > not the one having the logical difficulty here. > > > Seriously, do you comprehend that they are saying different things? > > This is important. > > It would be if it had anything substantive to do with my point. Whatever > the size of a set of consecutive naturals from 1 is, that is its maximal > element. > > > I'm not disputing a) (although you haven't defined "size" and thus > > it's trivially incorrect). I'm disputing b). I don't think b) follows > > from a). I don't think that all sets of naturals starting from 1 have > > a maximum. So I don't think that "the maximum, if it exists, is X" > > means "the maximum is X". Because for some sets of naturals, the > > maximum doesn't exist. > > It is inductively provable that for all such sets the maximum is EQUAL > to the size. Therefore, if one exists, then so does the other, in both > directions. Do you know what EQUAL means? If a=b, can a exist and b not? > > > Do you agree that some sets of consecutive naturals starting with 1 > > don't have a maximum element (N, for example)? Do you then agree that > > a) does not imply b)? > > No, b) implies that such sets also do not have a size. Get it? OK so all of the above comes down to you demanding that we don't call cardinality "size". If we don't call cardinality "size" then all your objections to cardinality disappear. I'm bored of trying to get you to realise that logic doesn't care what label we give to concepts. We have this definition called "cardinality" which is to do with which sets are bijectible. Some people think it seems like a fair notion of "size", but it's immaterial whether you agree with them or not. Cardinality is still perfectly well defined. You seem incapable of grasping this point. Moving on... At this point you're probably going to say "cardinality works but it's not sufficient. I want a richer way of measuring infinite sets, so I can say the evens are half the naturals... blah blah blah". Of course, there is nothing stopping you doing this. Certainly, cardinality doesn't stop development of other ways of measuring sets (see measure theory for example [note: I'm not saying measure theory does what you want. It was an *example* of another way of measuring sets that isn't precluded by set theory, rather it builds upon it.]). Of course, it's not clear WHY you want to develop "Bigulosity". It tells us NOTHING interesting about sets. It doesn't lead to any new mathematics. It (pupportedly) matches one persons intuitions better than cardinality. Woo hoo. > >>> 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. > >> Given the definition of the naturals, given any starting point 0, a set > >> of consecutive naturals of size y has maximum element x+y. > > > x+y? Typo I guess. > > Yes, I was going to say "starting point x", then changed that part and > not the other (which would have needed a "-1", anyway). > > >> Does the set of naturals have size aleph_0? If so, then aleph_0 is the maximal natural. > > > 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. Sigh. But some people do. And some people don't. Some people don't care, because "size" is inherently vague. It's no problem whatsoever to use different definitions as we like, so long as we are always clear which definitions we are using. Unless we have squirrels living in our head and can't distinguish between how we label a thing from the thing itself. Then we run into problems. > > Under some definitions of "size" your statement is true. Under others > > (such as cardinality) it isn't. So you can't use your statement about > > SOME definitions of size to draw conclusions about ALL definitions of > > size. Not all sets of naturals starting at 1 have a maximum element > > (right?). Your statement is thus obviously wrong about any definition > > of "size" that gives a size to non-finite sets. > > It's wrong in any theory that gives a size to any countably infinite > set Yes. Well done. > , except as a formulaic relation with N. Que? > > I find it hard to beleive you don't understand this. Indicate the > > point(s) where you disagree. > > > a) Not all consecutive sets of naturals starting from 1 have maximum > > elements. > agree > > b) Some notions of "size" give a "size" to sets of naturals without > > maximum elements. > > disagree, personally. I can't accept transfinite cardinality as a notion > of "size". Blah blah blah. Labels aren't important. > c) Some notions of "size" give a "size" to sets of consecutive > > naturals starting from 1 without a maximum element. > same > > d) The "size" that these notions give cannot be the maximum element, > > because those sets don't *have* a maximum element. > > agree - that would appear to be the rub > > > > e) Your statement about "size" does not apply to all reasonable > > definitions of "size". In particular, it does not apply to notions of > > "size" that give a "size" to sets without a largest element. > > It does not apply to transfinite cardinality. The question is whether I > consider it a "reasonable" definition of size. I don't. Who cares? Set theory doesn't claim "cardinality is a reasonable definition of size". It uses cardinality to denote which sets are bijectible. That's all. You don't have to "consider cardinality a reasonable definition of size" to use it in set theory. Is your only objection to cardinality is that some people call it "size"? > >>> Do you see that changing the order of words in a statement can change > >>> the meaning or that statement? Do you see that one statement can be > >>> true, and another statement with the same words in a different order > >>> can be false? > >> This is not quantifier dyslexia, and I am not interested in entertaining > >> that nonsense, thanx. > > > It is doublethink though. You are simultaneously able to hold the > > contradictory statements "Not all sets of naturals have a largest > > element" with "All sets of naturals must have a largest element" to be > > true, > > No, "All sets of naturals WITH A SIZE must have a largest element", or > more specifically, "All sets of consecutive naturals starting from 1 > have size and maximal element equal." Equal things either both exist, or > both don't. OK, so let's call cardinality "bijection class" or something. Now, you have no objections? > >>>> Is N of that form? > >>> N is a set of consecutive naturals starting at 1. It doesn't have a > >>> largest element. It has cardinality aleph_0. > >> If aleph_0 is the size, then aleph_0 is the maximal element. aleph_0 e N. > > > Wrong. If aleph_0 is the "size", AND the set HAS a maximal element > > then aleph_0 is the maximal element. But N DOESN'T have a maximal > > element so aleph_0 can be the size without being the maximal element. > > > (speaking very informally again as Tony is incapable of recognising > > the need to define "size"..) > > I defined formulaic Bigulosity long ago. Bullshit. Vague mumblings are not definitions. > I've also made it clear that Idon't consider transfinitology to be a valid analog for size in the > infinite case. Who cares? Why should anyone care what you "consider" if it's just an aesthetic preference? > I've offered IF and N=S^L in the context of infinite-case > induction, which contradicts your little religion, and may seem > offensive, Religion? No, I am quite comfortable with the idea of adopting whatever axiom system seems interesting or useful for intellectual exploration or practical application. I have nothing invested in ZFC other than a recognition that it is coherent and useful (maybe even consistent!). You are the one who is chronically hung-up over the fact that your intuitions sometimes get violated. > but is really far less absurd Your ideas violate my intuition! Bigulosity seems very absurd to me. Now what? > and paradox-free. :) You think countable sets can have different sizes. That's a paradox to me. Some never-ending sequences end before other never-ending sequences? Ahaha, most amusing.. > >> Or, as Ross likes to say, NeN. > > > Here's a hint, Tony : Ross and Lester are trolls. They don't beleive a > > damn word they say. They are jerks getting pleasure from intentionally > > talking rubbish to solicit negative responses. Responding to them at > > all is pointless. Responding to them as though their "ideas" are > > serious and worthy of attention makes you look very, very silly. > > Yes, it's very silly to entertain fools, except when they are telling > you the Earth is round. One needn't be all like that, Mike. When you > argue with a fool, chances are he's doing the same. Nope. Ross and Lester are trolls. They are laughing at you when you agree with their fake online personas. Continue wasting your time on them if you like. -- mike. |