The Definition of Points
in [Math]
|
Prev: Guide to presenting Lemma, Theorems and Definitions
Next: Density of the set of all zeroes of a function with givenproperties
From: Tony Orlow on 31 Mar 2007 22:11 stephen(a)nomail.com wrote: > In sci.math Tony Orlow <tony(a)lightlink.com> wrote: >> stephen(a)nomail.com wrote: >>> In sci.math Brian Chandler <imaginatorium(a)despammed.com> wrote: >>>> stephen(a)nomail.com wrote: >>>>> In sci.math Tony Orlow <tony(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. > > None of the options mention "size" Tony. What does "size" have > to do with a, b or c? > Ugh. Me already tell you, nth one is n, then there are n of them. So easy, even a caveman can do it. Size is difference between. >> That was really hard to figure out after all this time... > > Have you finally figured it out then? I somehow doubt it. > As you have been repeatedly told, for years now, "size" is > not a term used in set theory. Ah I see. Size not matter. Ummm....okay. You tell yourself that. That good for me. You have sister? > > So Tony, yes or no, does there does exist a finite natural Q such > that {1, 2, 3, ... Q } is the set of all finite naturals? > > Stephen > No, there not be place of ending the count. There no biggest, and there no place to call "this big". There always more. Then, after that, even more. Even more bigger than all the things that end. The things don't end. Us make fire and dance around for that. Ugh. Tony
From: Tony Orlow on 31 Mar 2007 22:14 Lester Zick wrote: > On Fri, 30 Mar 2007 12:49:54 -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: >>> >>>>> This is why science is so useful because you stop arguing isolated >>>>> problems to argue demonstrations instead which subsume those isolated >>>>> problems. There's simply no point to arguing such problems >>>>> individually as to whether "not" is universally true of everything or >>>>> whether there are such things as conjunctions not reducible to "not" >>>>> in mechanically exhaustive terms unless the demonstration itself is >>>>> defective and not true. And just claiming so per say won't cut it. >>>>> >>>> Your "not a not b" has an assumed OR in it. >>> The problem is not whether it has or doesn't, Tony, but how do you >>> know and how can you demonstrate the truth of that claim. I mean there >>> is no visible indication what the relation between A and B is. You >>> might consider the relation between them is "or" but we have no >>> evidence that this conjecture is right and not just rank speculation. >>> I mean there are plenty of people out there who insist that relations >>> between any two items like A and B are theistic, deistic, or even the >>> product of aliens and UFO's. >> >> Please choose true or false, if you didn't do it last time: >> >> a b not a not b >> >> true true true or false? >> true false true or false? >> false true true or false? >> false false true or false? > > Kinda hard to tell what these terms mean, Tony? Are there clues or do > we just wing it? > Yeah, "true" and "false" and "or" are kinda ambiguous, eh?" >>> Consequently it's not my assumption of any relation between A and B >>> but my demonstrations of relations between them that matters. Sure I >>> can assume anything I want. And on previous occasions I certainly have >>> assumed the relation between them was a functional if not explicit or >>> because it seems to me the most plausible mechanical relation likely. >>> But that doesn't mean it's necessarily true. >> You need to define what relation your grammar denotes, or there is no >> understanding when you write things like "not a not b". > > Of course not. I didn't intend for my grammar to denote anything in > particular much as Brian and mathematikers don't intend to do much > more than speak in tongues while they're awaiting the second coming. > Then, what, you're not actually saying anything? >>> However the fact is that given two different things A and B we can >>> combine them with compoundings of "not" and when we do certain >>> conjunctive relations between them fall out the first of which is >>> "and" and the next of which is "or". That's how we can tell what the >>> originary implications between two distinct items is and has to be. >> Not if you assumed OR to begin with. In that case, you're as circular as >> anyone else, and more. Better to build up from true() and false() as >> 0-place predicates. > > Of Christ. Don't you and the zero place predicates wait up for us. > I'll leave the light on. haha >>> But that doesn't mean there is any assumption of "or" between them >>> only that given two distinct things like A and B we can determine any >>> conjunctive relations between them without the implicit assumption of >>> or explicit use of conjuctions. And that means conjunctions and so on >>> are "in here" and not "out there" among distinct things themselves. >> Choose true or false above, and I guarantee you'll see it's the relation OR. > > Of course it is, Tony. I just tried to slip one over on you. OR Brian > OR Virgil OR Stephen OR PD OR David Or Mikey Or someone else > without saying so. > > ~v~~ I think you mean AND. ;) 01oo
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 |