Galileo's Paradox
in [Math]
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From: Tony Orlow on 11 Dec 2006 18:44 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. >> and can stay in their little paradise/cave, while the rest of >> the world starts to take a more reasonable and less mystical approach to >> the infinite. Ordinals and cardinals are unrelated to anything worth >> bothering with. > > You seem to want to bother with non-standard analysis and with IST. > Do you think? What a bother. >>>> Of course, he has no need for omega. It's illegitimate schlock, like I said. > > 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. > > MoeBlee > There is no need for omega in nonstandard analysis. There is no smallest infinite allowed at all. He makes reference to "countablility" but I haven't seen any alephs about yet.
From: Tony Orlow on 11 Dec 2006 18:45 Virgil wrote: > In article <457D6A64.7010008(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 12/6/2006 8:52 PM, Virgil wrote: >>> In article <4576EDC6.6090307(a)et.uni-magdeburg.de>, >>> Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: >>> >>>> On 12/5/2006 11:48 PM, Virgil wrote: >>>>> In article <457596BC.3040307(a)et.uni-magdeburg.de>, >>>>> Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: >>>>> >>>>>> On 12/4/2006 10:47 PM, David Marcus wrote: >>>>>>>> Standard mathematics may lack solid fundamentals. At least it is >>>>>>>> understandable to me. >>>>>>> If it is understandable to you, then convince us you understand it: >>>>>>> Please tell us the standard definitions of "countable" and >>>>>>> "uncountable". >>>>>> I do not like such unnecessary examination. >>>>> Then stop trying to examine others. >>>> I examined others here? >>> You try, but as you are proceeding from a false assumption: >>> that your own understanding of mathematics is superior to that of >>> thousands of others who have spent much more time and effort and talent >>> in gaining their understanding than you have. >> Concerning time you may be correct. However, do not underestimate the >> talents by Galilei, Spinoza, Gauss, Kronecker, Poincar�, Wittgenstein >> and many others. > > Just as Hilbert could improve on Euclid's geometry, those who follow can > improve on what those who have gone before have wrought, but not the > reverse. Hilbert's axioms need to be reviewed. I turned the first 8 into four more powerful ones, and I'm not even a mathematician. Is it better nutrition these days?
From: Tony Orlow on 11 Dec 2006 18:46 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.
From: Tony Orlow on 11 Dec 2006 18:48 Virgil wrote: > In article <457D6707.4010208(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 11/28/2006 9:30 PM, Tony Orlow wrote: >> >>>> The relations smaller, equally large, and larger are invalid for >>>> infinite quantities. >>>> >>> Galileo's conclusions notwithstanding, there are certainly relationships >>> among many countably and uncountably infinite sets which indicate >>> unequal relative measures. I certainly consider 1 inch to be twice as >>> infinitely many points >> Twice as infinitely many is Cantorian nonsense. > > Actually is is anti-Cantorian nonsense. TO is just of a different > faction of anti-Cantorianism nonsense purveyors than EB. > > I prefer to be called a "person of Bigulosity", or post-Cantorian. > Cantor was able to show >> himself that all natural and rational numbers do not have a different >> "size". Infinity is not a number. > > No mathmatician says "infinity" in that context. They will call some > sets "infinite", but that is not the same thing. > > >> It has been understood like something >> which cannot be enlarged and not exhausted either: > > Infinite sets can be enlarged in the sense that for each > of them there are proper supersets, at least in ZFC and NBG. > > They can be exhausted by taking away all their members. Huh! And I thought sets had static size....
From: Tony Orlow on 11 Dec 2006 19:02
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. |