Galileo's Paradox
in [Math]
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From: Tony Orlow on 11 Dec 2006 22:30 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. > > 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. >> 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. > Ahem. He "proves" it cannot exist, just like the monkeys prove there's no largest banana. There's no tiniest giant. He's talking about something on the same continuum as the reals and naturals and rationals. He's not talking about any extra dimensions of specification like with complex numbers. Ordinals and cardinals are naturals in their finite state. Do the infinite forms of them jut off in other directions? Apparently so. Robinson's do not. Isn't it time Nonstandard Analysis became the Standard? >> 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. Yeah, over there, and they can't play this game. They don't belong to this continuum. 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. Ae you saying there is no contradiction between standard and nonstandard analysis, no cnclusions that are different? > > Read the VERY FIRST SENTENCE in Robinson's book, why don't you. Did you read "contemporary mathematical logic" to mean "transfinite set theory"? You can do better than that. > > MoeBlee >
From: Tony Orlow on 11 Dec 2006 22:33 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. :) > > MoeBlee > ToeKnee
From: Tony Orlow on 11 Dec 2006 22:34 cbrown(a)cbrownsystems.com 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. > > That assumes that in his theory, omega is a number. It's not; it's an > ordinal. > > Cheers - Chas > Oh, sorry, Chas. I must have got confused. Somewhere along the line I thought I heard ordinals referred to as numbers. That must be the confusion.... ;)
From: Tony Orlow on 11 Dec 2006 22:37 MoeBlee wrote: > Tony Orlow wrote: >> MoeBlee wrote: >>> Tony Orlow wrote: >>>> A formal language is a set of strings: >>>> http://en.wikipedia.org/wiki/Formal_language >>>> I suppose now you're going to tell me I'm using nonstandard language.... >>> Wikipedia. What a lousy basis for the subject of formal languages. >>> >>> MoeBlee >>> >> It was the first thing appropriate I saw in Google, and it contained the >> necessary information to explain the basics to Virgil. He's free to >> google "formal language" or take out a book, as he chooses. > > Yeah, right, like you have any idea. > > MoeBlee > Huh! Pfff!!! Tsss!!
From: cbrown on 11 Dec 2006 22:46
Tony Orlow wrote: > 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. Interesting. Most mathematicians assume it means "there is a proof of its existence". You seem to mean "there is no proof of its non-existence". Kind of a glass half full / glass half empty sort of thing. Does this solve those troublesome fulminations of Godel? Or does it just mean there are false things that cannot be proven to be false? Cheers - Chas |