The Definition of Points
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From: Lester Zick on 13 Apr 2007 18:48 On 12 Apr 2007 15:05:53 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> 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. A well ordering on it? Is this a joke, Moe(x)? ~v~~
From: Lester Zick on 13 Apr 2007 18:50 On 12 Apr 2007 15:20:24 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >On Apr 12, 3:17 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: >> On 12 Apr 2007 14:43:15 -0700, "MoeBlee" <jazzm...(a)hotmail.com> wrote: >> >> >I really don't care what you work on. My point is that your commentary >> >in these threads has virtually no formal mathematical import, >> >> Well no formal modern mathematical support perhaps, Moe(x), but I >> don't think you can say no formal mathematical import. > >I said 'virtually none'. As opposed to what, Moe(x), virtually some formal mathematical import? ~v~~
From: Lester Zick on 13 Apr 2007 18:51 On Fri, 13 Apr 2007 12:21:41 -0600, Virgil <virgil(a)comcast.net> wrote: >In article <461fbc19(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> MoeBlee wrote: >> > On Apr 12, 12:27 pm, Tony Orlow <t...(a)lightlink.com> wrote: >> >> MoeBlee wrote: > >> > I really don't care what you work on. My point is that your commentary >> > in these threads has virtually no formal mathematical import, as it >> > comes down to a bunch of whining that your personal notions are not >> > embodied in set theory even though you can't point to a formal system >> > (either published or of your own, and the gibberish you've posted in >> > threads and on your own site is not even a corhernt attempt toward a >> > formal system) that does embody your personal notions and you can't >> > even HINT at what such a system might be. >> > >> > MoeBlee >> > >> >> What does any of your whining have to do with the definition of points? >> This is tiresome. > > >It is, among other things, noting the lack of any progress towards >defining points by either you or Lester. My purpose was not to define points. ~v~~
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~~ |