Galileo's Paradox
in [Math]
|
From: Mike Kelly on 11 Dec 2006 19:09 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. -- mike
From: Tony Orlow on 11 Dec 2006 19:14 MoeBlee wrote: > Tony Orlow wrote: >>> 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, 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: > > You are REPEATING the mistake you've made from the beginning regarding > Robinson, because you never bothered to read it and understand it but > instead just jumped around to find things you only THINK say what you > think they should say. The fact that I'm repeating it might serve as a clue that it's not a mistake. > > In CONTEXT, when Robinson is talking about those infinite numbers, he > is NOT talking about ordinals such as omega. There are two DIFFERENT > senses of 'infinite' in play, one is infinite cardinality and the other > is that of a certain ordering of elements in certain sets, which is NOT > a cardinality ordering and does NOT apply to omega the way you have > answered. No kidding. Omega is the smallest infinite. It doesn't exist in NSA. My statement was that using omega as an infinite number in the context of NSA is contradictory, and so NSA contradicts transfinitology. Get it? > > And I can predict you're going to say something along the lines of that > you're not saying what I have said you said. But then your answer makes > NO SENSE as a response about OMEGA. > It's a comment on the uselessness of omega, and the fact that NSA is not built solely upon transfinitology, or there would be no contradiction in conclusions. It's an alternative view of infinite numbers, one which actually makes sense. >> "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. > > What does, "he has no for omega" mean? "no need" Sorry. What is illegitimate schlock? Um, I think in Jamaica, they call it "Mama blood clot." LOL :D > Robinson uses ZFC and classical mathematical logic all over the place. > > MoeBlee > Please cite where he employs or even references omega in the book. I'd be interested. I haven't finished it, so I can't swear he doesn't, but....I'm pretty sure he has absolutely no use for a concept that is impossible within his theory, at least while discussing his theory.
From: MoeBlee on 11 Dec 2006 19:15 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. > > 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. 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. > There is no smallest > infinite allowed at all. It's not a question of "allowed". You really understand NOTHING about this. In particular sets and systems that are proven to exist, ordinals are not members. So what? The ENTIRE theory in which this takes place DOES prove the existence of ordinals. Look, no ordinal is a complex number, but we construct the complex numbers in a theory in which ordinals do exist, even if ordinals are not complex numbers. No ordinal is a non-standard real. But the theory in which non-standard reals are proven to exist does also prove the existence of ordinals. > He makes reference to "countablility" but I > haven't seen any alephs about yet. The ordinals themselves are not members of the non-standard number system, but DERIVING the existence of a non-standard system takes place in a theory in which ordinals do exist. You can't just rip one part of a theory, like a shard, out of a whole theory. Perhaps there is a non-standard analysis that can be devised without classical mathematical logic and ZFC, but Robinson's work does NOT do that. He uses classcial mathematical logic and set theory all over the place in connection with results in non-standard analysis. And IST includes EVERY SINGLE theorem of Z set theory. Read the VERY FIRST SENTENCE in Robinson's book, why don't you. MoeBlee
From: MoeBlee on 11 Dec 2006 19:23 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. MoeBlee
From: cbrown on 11 Dec 2006 19:28
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. That assumes that in his theory, omega is a number. It's not; it's an ordinal. Cheers - Chas |