Galileo's Paradox
in [Math]
|
From: stephen on 12 Dec 2006 00:21 Tony Orlow <tony(a)lightlink.com> wrote: > stephen(a)nomail.com wrote: >> Tony Orlow <tony(a)lightlink.com> wrote: >>> MoeBlee wrote: >>>> Tony Orlow wrote: >>>>> I claimed no such thing. I am saying his very reasonable approach >>>>> directly contradicts the very concept of the limit ordinals, which are >>>>> schlock, >>>> WHAT contradiction? Robinson uses classical mathematical and set theory >>>> all over the place. >>>> >> >>> Wonderful. Then there must be a smallest infinite number, omega, in his >>> theory. Oh, but there's not. For any infinite a, a=b+1, and b is >>> infinite. Can a smallest infinite exist, and not exist too? Nope. >> >> Can a smallest number exist and not exist? 1 is the smallest >> positive integer. There is no smallest positive real. That >> is exactly analogous to the supposed contradiction you are >> talking about. Ordinals are different types of numbers than >> Robinson's infinite numbers, just as integers are different >> types of numbers than real numbers. >> >> You seem to be purposefully trying to not understand >> these simple points. >> >> Stephen > I am purposefully trying to understand how it SHOULD be. There is no > smallest infinite. There is no smallest finite. There is no largest > finite. There is no largest infinite. There is also no largest or > smallest infinitesimal. That's the way it is. And what god told you that? > Omega is a phantom. There > are twilight zones between any two uncountably sepearted countable > neighborhoods. I have spoken. :) > Did that sound good and cranky. I am kind of cranky... It is not a thing to be strived for. But apparently who want to be just like Lester. It is rather sad actually. Stephen
From: Virgil on 12 Dec 2006 01:21 In article <457e1f4c(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > MoeBlee wrote: > You cannot have a smallest infinity, and also not have it. It is those who do not have it when working with ordinals who are in trouble. Just as those who have any infinite numbers within standard analysis are in trouble. > > 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. The discrepancy is between what mathematicians understand and what TO understands about mathematics.
From: Virgil on 12 Dec 2006 01:23 In article <457e203e(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > 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. Robinson needs a completed set of naturals, which gives him omega even if he does not call it that in order to construct his non-standard reals.
From: Virgil on 12 Dec 2006 01:28 In article <457e229a(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> MoeBlee wrote: > >>> Tony Orlow wrote: > >>>> I claimed no such thing. I am saying his very reasonable approach > >>>> directly contradicts the very concept of the limit ordinals, which are > >>>> schlock, > >>> WHAT contradiction? Robinson uses classical mathematical and set theory > >>> all over the place. > >>> > >> Wonderful. Then there must be a smallest infinite number, omega, in his > >> theory. Oh, but there's not. For any infinite a, a=b+1, and b is > >> infinite. Can a smallest infinite exist, and not exist too? Nope. > > > > There IS the smallest infinite ORDINAL omega in Robinson's work. > > Robinson is in classical mathematical logic and set theory. > > > > You are AGAIN conflating elements (that are called 'infinite elements' > > or 'infinite numbers') in a certain ordering with ordinals. Robinson > > doesn't eschew infinite ordinals. They're all over the place in > > Robinson's work. > > > > You really don't WANT to understand any of this, do you? Actually > > understanding would put you in the position of having to recognize that > > Robinson's work embraces classical mathematical logic and ZFC, and that > > just doesn't go with what you WANT Robinson's work to be. > > > > Can you cite where he uses omega in the development of NSA please? > > >>> What are you TALKING ABOUT? Read Robinson (which means reading the > >>> actual development, not just isolated passages), why don't you, instead > >>> of ignorantly spouting about what YOU THINK he does and does not need. > > > >> There is no need for omega in nonstandard analysis. Shows how little TO knows. > > > > Robinson works in classical mathematical logic and set theory, in which > > omega exists. IST includes Z set theory, in which omega exists. Or, if > > you want to point to so other treatment of non-standard analysis in > > which treatment does not also entail the existence of omega, then > > you're welcome to do it, but it ain't Robinson and it ain't IST. > > > > He ignores it. It would contradict his internally consistent theory, and > that would bother set theorists. Show any such contradiction. > > Read the VERY FIRST SENTENCE in Robinson's book, why don't you. > > Did you read "contemporary mathematical logic" to mean "transfinite set > theory"? Without the transfinite, there cannot be any non-standard analysis. The very ultrafilters on which it is based are transfinite sets.
From: Virgil on 12 Dec 2006 01:30
In article <457e2364$1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> The point is, omega cannot coexist with NSA. > > > > You just want to ignorantly say that over and over and over. > > > > Non-standard analysis is done in classical mathematical logic and with > > Z set theory (and, as far as I know, you can't do it without choice, > > whether with ultrafilters or otherwise). > > > > And Z set theory is SUBtheory of IST. Every theorem of set theory is a > > theorem of IST, incuding the existence of infinite ordinals. > > > > There is NO contradiction between the existence of a non-standard > > system of numbers and the existence of omega. Omega doesn't happen to > > be a member of the non-standard system, but that doesn't entail that > > there is a contradiction. Look, for that matter, omega isn't a member > > of the STANDARD reals. > > Of course it's not a standard real. Is it an infinite number? I guess > it's not an infinite "nonstandard" number, and there are no infinite > "standard" numbers. It's a limit ordinal number. It just doesn't mean > anything to me, or satisfy any intuitions, or seem the least bit > sensible, whereas Robinson makes sense and comes to all the right > conclusions. I kind of like that. :) Those conclusions are no more "right" than those of standard analysis, and none of them could exist without standard set theory as a basis. |