The Definition of Points
in [Math]
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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
From: Tony Orlow on 31 Mar 2007 22:38 Virgil wrote: > In article <460ef839(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <460ee056(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>> Please do expliculate what the contradiction is in an uncountable >>>> sequence. What is true and false as a result of that concept? >>> A mathematical sequence is a function with the naturals as domain. >>> If TO wishes to refer to something which is not such a function, he >>> should not refer to it as a sequence if he wishes to be understood in >>> sci.math. >>> >>> >> Pray tell, what term shall I use???? > > TO is so inventive in so many useless ways that I cannot believe that > his imagination will fail him in such a trivially useful way. >>>>> 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.. >>> >>> It is not so much that Ross disagrees with one person, it is that he >>> disagrees with everyone, frequently including himself. >> Ross has a vision, even if not axiomatically expressed. In fact, he's >> entirely honest about that, expounding an axiom free system. I like >> Ross. So do you. Admit it. :) >> > > > Like Russell? > > What is there about him to like? You don't like Russell?
From: cbrown on 31 Mar 2007 22:39 On Mar 31, 3:30 pm, Tony Orlow <t...(a)lightlink.com> wrote: > cbr...(a)cbrownsystems.com wrote: > > On Mar 31, 5:30 am, Tony Orlow <t...(a)lightlink.com> wrote: > >> Virgil wrote: > > >>> In standard mathematics, an infinite sequence is o more than a function > >>> whose domain is the set of naturals, no two of which are more that > >>> finitely different. The codmain of such a function need not have any > >>> particular structure at all. > >> That's a countably infinite sequence. Standard mathematics doesn't allow > >> for uncountable sequences like the adics or T-riffics, because it's been > >> politically agreed upon that we skirt that issue and leave it to the > >> clerics. > > > That's false; > > Please elucidate on the untruth of the statement. It should be easy to > disprove an untrue statement. > I did in the continuation of that sentence; but I'll repeat myself. You claimed that mathematics doesn't "allow for uncountable sequences" (for which you agree you've given no real useful definition). But on the contrary, "people" (aka, mathematicicians) have studied all sorts of ordered sets, finite, countable, and uncountable; and functions from them (whether they use the term "uncountable sequence" or not). So your claim that ordered sets which are not countable have not been studied is false; and therefore your comments that the reasons /why/ they have not been studied (political or religious) are non-sequiturs. The obvious question is why haven't /you/ studied them; instead of making vague and uninformed statements about them (regardless of what you choose to call these ordered sets). > people have examined all sorts of orderings, partial, > > > total, and other. The fact that you prefer to remain ignorant of this > > does not mean the issue has been skirted by anyone other than > > yourself. > > There have always been religious and political pressures on this area of > exploration. > How would you know what has "always" been the case in this area? Is there an example please of a /religious/ or /political/ pressure that you can cite? Besides the quite reasonable recommendation to educate yourself regarding the subject matter, using the freely available material relating to the subject, e.g., http://en.wikipedia.org/wiki/Order_theory ? > >> However, where every element of a set has a well defined > >> successor and predecessor, it's a sequence of some sort. > > > Let S = {0, a, 1, b, 2, c}. > > > Let succ() be defined on S as: > > > succ(0) = 1 > > succ(1) = 2 > > succ(2) = 0 > > succ(a) = b > > succ(b) = c > > succ(c) = a > > Okay you have two sequences. > Why two? Why not one, or three, or six? Your definition fails to say. Is S = {0,1}, succ(0)=1, succ(1)=0 a "sort of sequence"? It has a well- defined successor and predeccessor for each element. How about S = {0}, and succ(0) = 0? > > > > Every element of S has a well-defined successor and predecessor. What > > "sort of sequence" have I defined? Or have you left out some parts of > > the /explicit/ definition of whatever you were trying to say? > > > Cheers - Chas > > Yes, I left out some details. Given that you are claiming that your definition is somehow being surpressed by religious or political forces, why not take the opportunity to provide these details (in which we all know the devil resides)? As it stands, I have no real idea what you're talking about; and quite frankly, I doubt you yourself have a clear idea of what you are talking about. Other people have thought through similar ideas and presented a comprehensive structure for understanding. See, e.g., http://en.wikipedia.org/wiki/Order_theory in case you'd actually like to try to /learn/ something about the subject. Cheers - Chas
From: Virgil on 31 Mar 2007 22:42
In article <460f1799(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > > > > Too late. Such definition have to precede, not follow, the claims. > > Is that how it works in your chronological theory of mathematics? Using a definition before it exists is, at best, logically inept. |