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 20:59 Lester Zick wrote: > On Fri, 30 Mar 2007 12:27:40 -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: >>> >>>>> 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. > > Does anyone really care? > > ~v~~ You can only answer that for yourself. What was the topic again? 01oo
From: Virgil on 31 Mar 2007 20:59 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.
From: stephen on 31 Mar 2007 20:56 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? >>>>>>> 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. > 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: Virgil on 31 Mar 2007 21:04 In article <460eeb06(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <460e571f(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Mike Kelly wrote: > >>> On 30 Mar, 18:25, Tony Orlow <t...(a)lightlink.com> wrote: > >> ~v~~ > >>>> An actually infinite sequence is one where there exist two elements, one > >>>> of which is an infinite number of elements beyond the other. > >>>> > >>>> 01oo > >>> Under what definition of sequence? > >>> > >>> -- > >>> mike. > >>> > >> A set where each element has a well defined unique successor within the > >> set. Good enough? > > > > No! if we define the successor of x as x + 1, as we do for the ntaurals, > > then the set of rationals and the set of reals, with their usual > > arithmetic, both satisfy your definition of sequence. > > > > A sequence should at least be well ordered and have only one member, its > > first, without a predecessor. > > I agree that what are considered normal sequences have first elements, > but I don't see that the integers, or the adics, aren't a sequence in a > broader sense, if we choose any arbitrary starting point. We can say a > sequence has some single element without predecessor, or some element > without successor, or both, so as to limit the line to a ray or segment. > But the most general rule is that it may go one forever in both > directions, as y -> Ex Ez : x<y<z, ala Archimedes. Yesno? :D > > Tony In standard mathematics one uses standard definitions. When you intend those standard terms to convey non-standard meanings, you will fail.
From: Virgil on 31 Mar 2007 21:06
In article <460eecfe(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <460e5899$1(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> stephen(a)nomail.com wrote: > > > >>> It is not even true in Tony's mathematics, at least it was not true > >>> the last time he brought it up. According to this > >>> definition {1, 2, 3, ... } is not actually infinite, but > >>> {1, 2, 3, ..., w} is actually infinite. However, the last time this > >>> was pointed out, Tony claimed that {1, 2, 3, ..., w} was not > >>> actually infinite. > >>> > >>> Stephen > >> No, adding one extra element to a countable set doesn't make it > >> uncountable. > > > > Countability is a straw man. > > > > yurnngghhh? ;o > > Virgule no sense making. The sense is there for those with the wit to find it. > > > The issue is whether adding one element converts a > > "not actually infinite" set into an "actually infinite" set. > > Of course it can't. It only adds one extra position to a sequence where > all positions are finite, and a finite plus one more is still a finite, > no? Which element can w come after in that set, in any order, which is > in a finite position, and yet, has an infinite position immediately > following it? Can't happen, can it? Nope, can't. I not to buy that. Then stop trying to sell it. |