Galileo's Paradox
in [Math]
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From: Virgil on 11 Dec 2006 21:47 In article <457dee2b(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> 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"? > >>> > >> What in math is not an assumption, or built upon assumption? What are > >> axioms but assumptions? He has postulated that he can form an extended > >> system by extending statements about N to *N, and works out the details > >> and conclusions of that assumption. Why do you ask? > >> > >>>> 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 need for omega. It's illegitimate schlock, like I > >>>> said. > >>> Do you really think Robinson is talking about ordinals? > >>> > >> Did you even read what I said? Of course he's not talking about omega > >> and the ordinals, he's talking about a sensible approach to the infinite > >> and infinitesimal for a change. Sheesh! > > > > Then your response about Robinson was a COMPLETE NON SEQUITUR. Sheesh! > > > > MoeBlee > > > > The point is, omega cannot coexist with NSA. Omega can exist in the foundations from which NSA is built. And has to.
From: Virgil on 11 Dec 2006 21:52 In article <457df483(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > The fact that I'm repeating it might serve as a clue that it's not a > mistake. To is in the habit of repeating his mistakes, ad nauseam, so that argument fails.
From: Tony Orlow on 11 Dec 2006 22:04 MoeBlee wrote: > Virgil wrote: >> But, as I said, is about as non-existent as a string can get. Any less >> existent and it wouldn't be a string at all. > > Okay, so there are degrees (degrees of more and of less) of existence? > > MoeBlee > Of course! I'm sure Virgil will readily agree that the null string is more existent than my uncountable-but-countably-infinite T-riffic digital strings, since a lot more people have heard of it and it's generally accepted, but less real than non-null strings, since he hasn't, and isn't, generally. :) Tony
From: Tony Orlow on 11 Dec 2006 22:16 MoeBlee wrote: > Tony Orlow wrote: >> Is omega considered the smallest infinite number? Omega then does not >> exist in nonstandard analysis. > > You'll have to define 'exist in non-standard analysis'. Sorry. Exists = Does not produce a contradiction. S is the set of all infinite values, xeS, yeS: ~(Ax Ey: ~(x=y) -> y<x) ^ (Ey Ax ~(x=y) -> y<x) You cannot have a smallest infinity, and also not have it. In the set > theory that is presupposed for the work of non-standard analysis, omega > exists. In classical mathematical logic that is the very hearth of > non-standard analysis, we suppose the existence of the set of natural > numbers. And in IST, omega exists. > > You seem to have the mistaken impression that non-standard analysis is > some kind of mathematics that is separable from classical mathematics > and set theory. > > MoeBlee > I certainly detect a discrepancy, yes.
From: Tony Orlow on 11 Dec 2006 22:20
Mike Kelly wrote: > Tony Orlow wrote: >> MoeBlee wrote: >>> Tony Orlow wrote: >>>> David Marcus wrote: >>>>> 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"? >>>>> >>>> What in math is not an assumption, or built upon assumption? What are >>>> axioms but assumptions? He has postulated that he can form an extended >>>> system by extending statements about N to *N, and works out the details >>>> and conclusions of that assumption. Why do you ask? >>>> >>>>>> 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 need for omega. It's illegitimate schlock, like I said. >>>>> Do you really think Robinson is talking about ordinals? >>>>> >>>> Did you even read what I said? Of course he's not talking about omega >>>> and the ordinals, he's talking about a sensible approach to the infinite >>>> and infinitesimal for a change. Sheesh! >>> Then your response about Robinson was a COMPLETE NON SEQUITUR. Sheesh! >>> >>> MoeBlee >>> >> The point is, omega cannot coexist with NSA. > > Wow. You know nothing about NSA. > Can you or anyone please cite where Robinson mentions omega in Nonstandard Analysis? I'm not saying it isn't there, but I haven't seen it. Granted, I'm not very far through it, but so far I see no need for it. |