Galileo's Paradox
in [Math]
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From: Tony Orlow on 10 Dec 2006 21:24 Virgil wrote: > In article <457c12f6(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> David Marcus wrote: >>> Tony Orlow wrote: > >>>> Well, I have been through much of that regarding such specific language >>>> approaches as the T-riffic digital numbers, but that's not necessary for >>>> this purpose. It suffices to say that, if a statement is proved true for >>>> all n greater than some finite k, that that also includes any postulated >>>> infinite values of n, since they are greater than any finite k. I don't >>>> need to construct these numbers. Consider them axiomatically declared. >>> Then list the axioms for them. >>> >> (sigh) >> >> infinite(x) <-> A yeR x>y > > This may serve in some sense as a definition, but does not imply that > any such x exists, so it is totally unsatisfactory as an axiom declaring > or requiring such x's existence. > > One can equally well say > > "x is Santa Claus if and only if ...", but no matter how one fills in > that blank, it doesn't mean any Santa exists. Axioms declaring existence of sets from nothing seem rather pretentious to me. It seems rather more appropriate for every axiom to be a rule involving logical implication, which may involve constructing sets from atoms and other sets. What do you mean by "existence" anyway?
From: Tony Orlow on 10 Dec 2006 21:40 cbrown(a)cbrownsystems.com wrote: > Tony Orlow wrote: >> David Marcus wrote: >>> Tony Orlow wrote: >>>> David Marcus wrote: >>>>> Tony Orlow wrote: >>>>>> David Marcus wrote: >>>>>>> Tony Orlow wrote: >>>>>>>> Why do you have a problem with the mere suggestion of an infinite value? >>>>>>> I don't have a problem with infinite values. However, you have to do >>>>>>> more than merely say "positive infinite n". Assuming your positive >>>>>>> infinite things aren't the same as something that we already know about >>>>>>> (in which case, you should just say so), you either need to give a >>>>>>> construction of these positive infinite things or you need to specify >>>>>>> their properties. You haven't done either. For example, how many of >>>>>>> these things are there? How do they relate to each other? How do they >>>>>>> interact with the natural numbers? Are the operations of addition and >>>>>>> multiplication defined for them? >>>>>> Well, I have been through much of that regarding such specific language >>>>>> approaches as the T-riffic digital numbers, but that's not necessary for >>>>>> this purpose. It suffices to say that, if a statement is proved true for >>>>>> all n greater than some finite k, that that also includes any postulated >>>>>> infinite values of n, since they are greater than any finite k. I don't >>>>>> need to construct these numbers. Consider them axiomatically declared. >>>>> Then list the axioms for them. >>>> (sigh) >>>> > > (T1) infinite(x) <-> A yeR x>y > >>>> infinite(x) <-> A yeR x>y >>> Is that your only axiom? If so, then state your first theorem about them >>> and give the proof. >>> >> That's the only one necessary for what defining a positive infinite n. A >> whole array of theorems pop forth... > > Before going there, you might want to start by adding the axiom: > > (T2) exists B such that infinite(B) > > Otherwise, who cares if you can prove a whole bunch of theorems about > something that doesn't exist? > > Cheers - Chas > What do you mean by "exist"? We can postulate the existence of such a set, and derive conclusions about it. It seems to me that it exists by virtue of being referenced.
From: Tony Orlow on 10 Dec 2006 21:43 Lester Zick wrote: > On Sun, 10 Dec 2006 12:50:43 -0500, Tony Orlow <tony(a)lightlink.com> > wrote: > >> Lester Zick wrote: >>> On Sat, 09 Dec 2006 19:28:44 -0500, Tony Orlow <tony(a)lightlink.com> >>> wrote: >>> >>>> David Marcus wrote: >>>>> Tony Orlow wrote: >>>>>> Lester Zick wrote: >>>>>>> On Fri, 08 Dec 2006 14:20:32 -0500, Tony Orlow <tony(a)lightlink.com> >>>>>>> wrote: >>>>>>> >>>>>>>>> Now we're all aware of phase transition rules for imaginary numbers. >>>>>>>>> c+ni is no greater or less than c alone unless n is even in which case >>>>>>>>> there is a transition from the i phase to the c phase and c+ni becomes >>>>>>>>> larger than c alone. >>>>>>>> I am not aware of that. Please give me a good link that explains it. It >>>>>>>> might prove quite important. Thanks. >>>>>>> Just complex algebra, Tony, where powers of i contribute to cardinal >>>>>>> magnitudes according to whether they're even or not. >>>>>>> >>>>>> Uh, why don't I see any powers of i in c+ni? Do you mean c+i^n? Yes, >>>>>> there's a funny interaction between the real and imaginary components of >>>>>> complex numbers which maps to angular and linear operations on the >>>>>> radius of a circle. What does this have to do with infinitesimals? They >>>>>> are not produced by calculus, but used to form the basis of it. A number >>>>>> infinitesimally close to x is considered equal to x in standard math. >>>>> I think Lester would like to avoid having a real number plus an >>>>> infinitesimal be a real number. So, he wants to keep them separate, like >>>>> we do when we add a real number and an imaginary number. >>>>> >>>> Yes, but I'm not sure why he wants that. Like I said in another post, I >>>> think it has to do with his infinitesimal change in radius as dr/dt or >>>> something, uh, nonstandard like that. >>> Not really, Tony. That was just a collateral observation. The point is >>> that finite numbers of infinitesimals don't alter finite magnitudes >>> any more than odd powers of i alter finite magnitudes. >> Finite numbers of infinitesimals don't change finite magnitudes in any >> finite way, but infinite numbers of infinitesimals can sum to finite or >> infinite values. > > But that's not what I remember you saying previously, Tony. I seem to > recall you indicating r+dr>r. > But, only infinitesimally, for infinitesimal "dr". >> Odd powers of i equal i, and all powers of 1 equal 1. >> That's not particularly related, as I see it. > > It's related if you maintain r+dr>r. > How? >> This effort was >>> simply an attempt to analyze the problem in terms analogous to the >>> treatment of complex numbers. If you have a better way to analyze the >>> problem by all means have at it. But you need to have something better >>> than constituent points and infinitesimals where there don't appear to >>> be any. >>> >>> ~v~~ >> There appear to be infinitesimals on the real line. Where you have some >> infinite number of reals in a unit interval, and assume a constant >> density in the continuum, you can handily declare such a constant unit >> infinity as a count per unit, and tie measure to infinite count of >> infinitesimal intervals. I see no reason to place them somewhere else. >> It's just that, in calculus, the derivative is the measure of change, >> perpendicular to the x direction along which we are taking the >> derivative, or the slope. The change in that change is the curvature of >> the function at a given point, or the change perpendicular to the graph >> at that point. But, just because those changes are "perpendicular" >> doesn't indicate to me that infinitesimals values are "perpendicular" to >> real numbers. Infinitesimals were a tool in deriving the calculus. I >> don't see that this application of them makes other interpretations wrong. > > So flat lines don't have a derivative 0dr? Sure, or some infinitesimal times "dr", whatever dr really is. > > ~v~~
From: David Marcus on 10 Dec 2006 23:26 Tony Orlow wrote: > Virgil wrote: > > In article <457c1fa0(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > >>> In article <457b8ccf(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>> > >>>> If the expressions used can themselves be ordered using > >>>> infinite-case induction, then we can say that one is greater than the > >>>> other, even if we may not be able to add or multiply them. Of course, > >>>> most such arithmetic expressions can be very easily added or multiplied > >>>> with most others. Can you think of two expressions on n which cannot be > >>>> added or multiplied? > >>> I can think of legitimate operations for integer operations that cannot > >>> be performed for infinites, such as omega - 1. > >> Omega is illegitimate schlock. Read Robinson and see what happens when > >> omega-1<omega. > > > > I have read Robinson. On what page of what book does he refer to omega - > > 1 in comparison to omega? I do not find any such reference. > > He uses the assumption that any infinite number can have a finite number > subtracted, "Assumption"? Why do you say "assumption"? > and become smaller, like any number except 0, so there is no > smallest infinite, just like you do with the endless finites. > Non-Standard Analysis, Section 3.1.1: > > "There is no smallest infinite number. For if a is infinite then a<>0, > hence a=b+1 (the corresponding fact being true in N). But b cannot be > finite, for then a would be finite. Hence, there exists an infinite > numbers [sic] which is smaller than a." > > Of course, he has no for omega. It's illegitimate schlock, like I said. Do you really think Robinson is talking about ordinals? -- David Marcus
From: Virgil on 11 Dec 2006 00:04
In article <457cbd17(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <457c658c$1(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> David Marcus wrote: > >>> Tony Orlow wrote: > >>>> > >>>> infinite(x) <-> A yeR x>y > >>> Is that your only axiom? If so, then state your first theorem about > >>> them and give the proof. > >>> > >> That's the only one necessary for what defining a positive infinite > >> n. > > > > It is a definition, not an axiom. And as it does not imply existence of > > anything, one has no more that other axioms will supply. > > > > > > > > Since it is a statement of implication, it's a rule, and an axiom. Many definitions are stated in the form of an equivalence. For example the Dedekind definition of infiniteness of a set: The set S is infinite if and only if there exists a function from S to a proper subset of S. Thus such definitions are not axioms, but merely rules of when the abbreviations described may be used. > That > it serves as the definition of infinite doesn't make it not a rule for > determining what values are infinite, once finites are defined. A definition which does not provide such a rule is hardly an adequate definition. |