The Definition of Points
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From: Tony Orlow on 31 Mar 2007 22:23 Virgil wrote: > In article <460ee90d(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <460e56a5(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>>> Virgil wrote: >>> >>>>> 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. >>>> Sure, but the question is whether any such assumption of existence >>>> introduces nonsense into your system. >>> It has in each of TO's suggested systems so far. >>> >> If thou so sayest, Sire. >> >>>> With the very basic assumption >>>> that subtracting a positive amount from anything makes it less >>> That presumes at least a definition of "positive" and a definition of >>> "amount" and a definition of "subtraction" and a definition of "less" >>> before it makes any sense at all. >> Yes, it does. > > Too late. Such definition have to precede, not follow, the claims. Is that how it works in your chronological theory of mathematics?
From: Tony Orlow on 31 Mar 2007 22:27 stephen(a)nomail.com wrote: > In sci.math Tony Orlow <tony(a)lightlink.com> wrote: >> stephen(a)nomail.com wrote: >>> In sci.math Tony Orlow <tony(a)lightlink.com> wrote: >>>> stephen(a)nomail.com wrote: >>>>> In sci.math Tony Orlow <tony(a)lightlink.com> wrote: >>>>>> stephen(a)nomail.com wrote: >>>>> So in other words >>>>>>>>>>> An actually infinite sequence is one where there exist two elements, one >>>>>>>>>>> of which is an infinite number of elements beyond the other. >>>>> is not your "correct" definition of an "actually infinite sequence", >>>>> which was my point. You are so sloppy in your word usage that you >>>>> constantly contradict yourself. >>>>> >>>>> If all you mean by "actually infinite" is "uncountable", then >>>>> just say "uncountable". Of course an "uncountable sequence" >>>>> is a contradiction, so you still have to define what you mean >>>>> by a "sequence". >>>>> >>>>> >>>> Please do expliculate what the contradiction is in an uncountable >>>> sequence. What is true and false as a result of that concept? >>> A infinite sequence containing elements from some set S is a function >>> f: N->S. There are only countably infinite many elements in N, >>> so there can be only countably infinite many elements in a sequence. >>> If you want to have an uncountable sequence, you need to provide >>> a definition of sequence that allows for such a thing, and until >>> you do, your use of the word "sequence" is meaningless, as nobody >>> will know what you are talking about. >>> > >> Oh. What word shall I use? Supersequence? Is that related to a >> subsequence or consequence? > > As long as you define your terms it does not matter to much what you > call it. You could just call it an uncountably infinite sequence, but you > need to define what that is if you want anyone to know what you are > talking about. Why are you so reluctant to define your terms? > I did that, and was told no such thing exists. Gee, then, don't talk about unicorns or alephs. >>>>>>>> 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? >>>>>>> >>>>>> Aleph_0, which is provably a member of the set, if it's the size of the >>>>>> set. Of course, then, adding w to the set's a little redundant, eh? >>>>> Aleph_0 is not a finite number. Care to try again? >>>>> >>>> It's also not the size of the set. Wake up. >>> It is the cardinality of a set. > >> Is that a number? > > What is your definition of "number"? aleph_0 is called a transfinite > number, but definitions, not names, are the important thing. > A number is a symbolic representation of quantity which can be manipulated to produce quantitative results in the form of symbols. I might be wrong, but I'm sure you can apprise me of the official meaning of "number", mathematically. ;) >> There is no standard definition >>> of "size", as you have been told countless times for a couple >>> of years now. Size is an ambiguous word in any situation, and >>> there is no argument in set theory that depends on the word "size". >>> >>> > >> Oh ambiguous... > > Yes, ambiguous. What is the size of a person? What is the > size of a dozen eggs? > > <snip> > >>>>> But the question is not about the number of elements up and including >>>>> any finite element of N. I asked how many elements are between 1 and w >>>>> in the set {1, 2, 3, ..., w }. >>>> w-2 are between w and 1. x-2 are between 1 and x. >>> What is w-2? Remember, I am talking about the standard definition >>> of w. The set I am talking about does not contain a w-2. It >>> contains all the finite elements of N, and the element w. >>> > >> How convenient. You can't move left from w. Well, that simplifies your >> dance, now, doesn't it? > > What does "moving" have to do with anything. We are talking about > sets, not locations. > >>>> w is not an element of N, nor is it finite. >>>> Oh, then why mention it? >>> Is there some rule saying that we can only mention finite elements, >>> or elements of N? I can describe all sorts of sets such as >>> N U { 1/2 }, or N U { w } or N U { {1, 2}, {2, 3}, {3, 4} ... }. >>> > >> Describe away - just don't expect it to prove anything if it's not >> pertinent. > > >>> The reason I mentioned it is because the set {1, 2, 3, ... w } >>> has the property that there exist two elements between which >>> there is an infinite number of elements, namely 1 and w. I know >>> that you do not consider {1, 2, 3, ... , w} an actually >>> infinite set, so I brought this up as an example of the fact >>> that even you do not agree with your own statement, which was: >>> >>>>> An actually infinite sequence is one where there exist two elements, one >>>>> of which is an infinite number of elements beyond the other. > >> Prove to me, logically, that there exist more than any finite number of >> elements between 1 and w. > > According to whose definition of finite? Anyway, let Q be a finite > natural greater than 1. The naturals {2,3, .. 2*Q } are all between > 1 and w. The number of elements in {2,3, ... 2*Q} is 2*Q-1. 2*Q-1 > Q > for all Q greater than 1. So for any finite natural Q greater than 1, > the number of elements between 1 and w is greater than Q. > > Here is another way to look at it. Suppose there are only Q > elements between 1 and w, where Q is a finite natural. These > elements are { 2, 3, 4, 5, ... Q, Q+1 }. Now Q is a finite natural, > so Q+1 is also a finite natural, and Q+2 is a finite natural, > and 1 < Q+2 < w. So there are more than Q elements between 1 and w. > >>> And of course that was my whole point. Despite the fact that >>> you posted that as a definition of an actually infinite sequence, >>> even you do not think it is the definition of an actually infinite >>> sequence. >>> > >> I do not think your example qualifies, logically. Sorry. > > So you are back to claiming there are only a finite number of elements > between 1 and w? So what is that finite number? I asked before, > and you did not name a finite number. > >>>>> I know you are incapable of actually thinking about all the elements of N, >>>>> but that is your problem. In any case, N is not an element of N. >>>>> Citing Ross as support is practically an admission that you are wrong. >>>>> >>>>> Stephen >>>>> >>>> Sure, of course, agreeing with someone who disagrees with you makes me >>>> wrong. I'll keep that in mind. Thanks.. >>>> Tony >>> No, agreeing with someone who makes absolutely no sense, such as >>> Ross, is tantamount to admitting you are wrong. > >> Whether Ross makes any sense or not is a personal judgment, based on >> whether what he says jibes with anything one may or may not think. Some >> of what he says jibes for me. So Ross doesn't make no sense, from where >> I sit, even if he doesn't have a system that I completely grok. His is >> not incompatible with mine. > > <snip> > >>> If you think Ross makes sense, explain his null axiom theory. >>> >>> Stephen >>> > >> I don't understand a theory without axioms, but I do understand the >> sentiment, and it's not dissimilar to Lester's. It's all about getting >> to the roots of the Tree of Knowledge, without undue assumptions. It's a >> worthy endeavor, even if fraught with entanglement and personal woe. The >> problem is, there's always two roots to every sprout...so let's all get >> used to it. > >> Tony > > But Ross claims to have a theory without axioms, whatever that means. > You apparently do not know what it means either, yet agree with > its consequences. > > Stephen >
From: Tony Orlow on 31 Mar 2007 22:29 Lester Zick wrote: > 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~~ Well, no, I don't wonder. 01oo
From: Tony Orlow on 31 Mar 2007 22:33 Virgil wrote: > In article <460ef650(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Bob Kolker wrote: >>> Tony Orlow wrote:>> >>>> Measure makes physics possible. >>> On compact sets which must have infinite cardinality. >>> >>> The measure of a dense countable set is zero. >>> >>> Bob Kolker >> Yes, some finite multiple of an infinitesimal. > > In any consistent system in which there are infinitesimals, none of > those infinitesimals are zero. On the finite scale, any countable number of infinitesimals has zero measure.
From: Tony Orlow on 31 Mar 2007 22:34
Virgil wrote: > 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. Is that wrong? haha. Anyway... You have seen two apples, and three? Nice 2cu again |