The Definition of Points
in [Math]
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From: Tony Orlow on 31 Mar 2007 21:23 Lester Zick wrote: > On Fri, 30 Mar 2007 12:31:08 -0600, Virgil <virgil(a)comcast.net> wrote: > >> In article <460d489b(a)news2.lightlink.com>, >> Tony Orlow <tony(a)lightlink.com> wrote: >> >>> Lester Zick wrote: >>>> On Thu, 29 Mar 2007 09:37:21 -0500, Tony Orlow <tony(a)lightlink.com> >>>> wrote: >>>> >>>>>> Just ask yourself, Tony, at what magic point do intervals become >>>>>> infinitesimal instead of finite? Your answer should be magnitudes >>>>>> become infintesimal when subdivision becomes infinite. >>>>> Yes. >>>> Yes but that doesn't happen until intervals actually become zero. >>>> >>>>> But the term >>>>>> "infinite" just means undefined and in point of fact doesn't become >>>>>> infinite until intervals become zero in magnitude. But that never >>>>>> happens. >>>>> But, but, but. No, "infinite" means "greater than any finite number" and >>>>> infinitesimal means "less than any finite number", where "less" means >>>>> "closer to 0" and "more" means "farther from 0". >>>> Problem is you can't say when that is in terms of infinite bisection. >>>> >>>> ~v~~ >>> Cantorians try with their lame "aleph_0". Better you get used to the >>> fact that there is no more a smallest infinity than a smallest finite, >>> largest finite, or smallest or largest infinitesimal. Those things >>> simply don't exist, except as phantoms. >> But all other mathematical objects are equally fantastic, having no >> physical reality, but existing only in the imagination. So any statement >> of mathematical existence is always relative to something like a system >> of axioms. > > Whew. Means you don't have to consider whether they're true. Quite > a relief I'd say. You can always take it up with someone who unlike > yourself isn't too lazy or stupid to think for a living. > > ~v~~ Oh, c'mon, Lester. If Virgie says it exists, why question it? He says 0 (zero) exists. I agree. 1 exists too. R exists, and N too, and |N|<|R|, and N "subset" R too. And 0eN and 1eN, so 0eR and 1eR. Also, 0<1. And if xeR and zeR and x<z, then E yeR such that x<y and y<z, kinda like 1/2 between 0 and 1. xeN and yeN -> x/yeQ, i.e. 1/2eQ. Seems that |N|<|Q|, since definitely N is a proper subset of Q. That's not what standard math tells us, given "cardinality", but we don't have to toe the line, but it is interesting to consider pi and e and other transcendentals and their place on the line. Can such a point be "pointed out"? And if we want to get all standardly stickly about it, x<y -> ~y<x. So, ain't no circles, and can't that |R|<|N|. Whew! That was close! :) But, questions remain. Really, when it comes to math, "in my book", it means methods of calculation, ways of figgerin'. When that addresses endlessnesses, one can't pretend with integrity to measure something endless, except as some formulaic expression of progress on the way "there". These are formulaic expressions.That's IFR and N=S^L. My friend, points do integrate into lines, according to the dimension over which they are integrated... Your intuition is not incorrect regarding the dependency of more limited dimensions on those more numerous, or even infinite, but that's a physical creation truth, not a mathematical one, based on waves. Some things need to be separated, from where I sit. But, this chair is getting comfortable...still I should go see what the sky is doing... 01oo
From: Tony Orlow on 31 Mar 2007 21:29 Mike Kelly wrote: > On 1 Apr, 00:58, Tony Orlow <t...(a)lightlink.com> wrote: >> step...(a)nomail.com wrote: >>> In sci.math Brian Chandler <imaginator...(a)despammed.com> wrote: >>>> step...(a)nomail.com wrote: >>>>> In sci.math Tony Orlow <t...(a)lightlink.com> wrote: >>>>>> If all other elements in the sequence are a finite number >>>>>> of steps from the start, and w occurs directly after those, then it is >>>>>> one step beyond some step which is finite, and so is at a finite step. >>>>> So you think there are only a finite number of elements between 1 and >>>>> w? What is that finite number? 100? 100000? 100000000000000000? >>>>> 98042934810235712394872394712349123749123471923479? Which one? >>>> None of the ones you've mentioned. Although it is, of course, a >>>> perfectly ordinary natural number, in that one can add 1 to it, or >>>> divide it by 2, its value is Elusive. Only Tony could actually write >>>> it down. >>> These Elusive numbers have amazing properties. According to >>> Tony, there are only a finite number of finite naturals. >>> There exists some finite natural Q such that the set >>> { 1,2,3,4,.... Q} >>> is the set of all finite natural numbers. But what of Q+1? >>> Well we have a couple of options: >>> a) Q+1 does not exist >>> b) Q+1 is not a finite natural number >>> c) { 1,2,3,4, ... Q} is not the set of all finite natural numbers >>> Tony rejects all these options, and apparently has some fourth >>> Elusive option. >>> Stephen >> Oy. The "elusive" option is that there is no acceptable "size" for N. >> That was really hard to figure out after all this time... > > Lucky, then, that set theory don't refer to the "size" of sets but > rather to their "cardinality". > > You still haven't figured that out after all this time. It's a very > strange mental block to have. > > -- > mike. > hi. mike. good thing, that. lucky, in fact. block. mental. to have, yes. I, nope don't. The question is to what extent set theoretical conclusions can be trusted, when the notion of order is introduced through such a suspect mechanism as the von Neumann ordinals. Just give '<' equal status with 'e' as a fundamental operator, and get real already! Sheesh! Heh! ;) ahem. tony.
From: Lester Zick on 31 Mar 2007 21:38 On Fri, 30 Mar 2007 13:02:44 -0500, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Thu, 29 Mar 2007 09:37:21 -0500, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >>> Your "not a not b" has an assumed OR in it. >> >> Tony, let me ask you something: without an AND or any other >> conjunction how would you mechanize the OR conjunction you >> claim I assume? And if it's just there as a basic circumstance of >> nature how do you get from there to other conjunctions and logic >> especially when you consider there's no necessity for conjoined >> components to be present together at the same time? >> >> ~v~~ > >Logical Mechanics 101: > >0-place predicates: > >false() 0 >true() 1 > >1-place predicates: > >x 0 1 > >false(x) 0 0 >x 0 1 >not(x) 1 0 >true(x) 1 1 > >2-place predicates: > >xy 00 01 10 11 > >false(x,y) 0 0 0 0 >and(x,y) 0 0 0 1 >not(x->y) 0 0 1 0 >x 0 0 1 1 >not(y->x) 0 1 0 0 >y 0 1 0 1 >xor(x,y) 0 1 1 0 >or(x,y) 0 1 1 1 >not(or(x,y)) 1 0 0 0 >not(xor(x,y)) 1 0 0 1 >not(y) 1 0 1 0 >y->x 1 0 1 1 >not(x) 1 1 0 0 >x->y 1 1 0 1 >not(and(x,y)) 1 1 1 0 >true(x,y) 1 1 1 1 Okay, Tony. I think I see what the problem is here. I say "not not" is false because it is self contradictory and "not" is true of everything because "not" is the tautological alternative to "not not" and those tautological alternatives are exhaustive of truth. You on the other hand say "true is true" and "false is false" because "true" is in truth tables and "false" is not true. Kinda reminisces me of the conversation I had recently with Dirk van der Putz who's quite happy to discuss the significance of time as long as time means whatever he wants it to mean and as long as he could grunt and point to measures of time which turns out he had quite some difficulty keeping running and measuring time. And you wonder why I don't exactly take your arguments on mechanics seriously? ~v~~
From: Virgil on 31 Mar 2007 21:38 In article <460ef25b(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > Every "sequence" must be a totally ordered set which is order isomorphic > > either to the ordered set of naturals, if it has a first element, or to > > the ordered set of integers, if it does not have a first element. > > Okay. How would you boil down that statement, and in which cases would > you say it applies? To all cases in which "sequence" is to be taken in its mathematical sense.
From: Virgil on 31 Mar 2007 21:41
In article <460ef372$1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <460e82b1(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > > > >> As I said to Brian, it's provably the size of the set of finite natural > >> numbers greater than or equal to 1. No, there is no last finite natural, > >> and no, there is no "size" for N. Aleph_0 is a phantom. > > > > All numbers are equally phantasmal in the physical world and equally > > real in the mental world. > > Virgule, you don't really believe that, do you? You're way too smart for > that... :) While I have seen numerals in the physical world, I have never seen any of the numbers of which they are only representatives. And I suspect that any who claim to have done so have chemically augmented their vision. |