Galileo's Paradox
in [Math]
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From: Virgil on 12 Dec 2006 01:31 In article <457e2427(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > 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!! TO being eloquent!
From: Virgil on 12 Dec 2006 01:38 In article <457e272f(a)news2.lightlink.com>, 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 may not be in Lower Slobovia where you reside, but there is anywhere that ZFC or NBG is allowed. > There is also no largest or > smallest infinitesimal. That is an entirely different issue. Where there is an infinite set of smallest cardinality or a minimal ordinal, there are no infinitesimals at all, and vice versa. That's the way it is. Omega is a phantom. It is no more of a phantom than are the finite ordinals. > There > are twilight zones between any two uncountably sepearted countable > neighborhoods. I have spoken. :) And, as usual, spoken falsely. > > > Tony
From: Virgil on 12 Dec 2006 01:39 In article <457e27fd$1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Bob Kolker wrote: > > Tony Orlow wrote: > >> > >> > >> Is omega considered the smallest infinite number? Omega then does not > >> exist in nonstandard analysis. > > > > Omega is the smallest infinite ordinal. It is the limit ordinal of the > > set of finite ordinals. > > > > Bob Kolker > > > > You have GOT to be kidding. That is so, like, news to me!!! (giggle) TO is regressing again. Probably off his meds!
From: Mike Kelly on 12 Dec 2006 05:10 Tony Orlow 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. Your proclomation that "omega cannot coexist with NSA" is making me doubt something I'd always assumed when repsonding to you - that you are not actually stupid. But I'm left with little choice; you're either stupid or *intentionally* not understanding what people are telling you. Neither option is particularly attractive. You're either an idiot or you're dishonest with yourself. This really is very simple. When you say "there is no smallest 'infinte number' in NSA" you are refering to a *very specific* *type* of "infinite number" that Robinson has constructed and described. It is true that there is no smallest <<infinite number>. of the type that Robinson has constructed in NSA. To then make the leap that "anything that could reasonably be described as an "infinite number" in natural language must obey the same rule or contradict NSA is simply nonsense. Infinite ordinals are not the same type of thing as the <<infinite numbers>> that Robinson describes, even though they could both be reasonably be called "infinite numbers" in natural language. Both of these are true in NSA : There is a least infinite ordinal. There is no least <<infinite number>>. It is also the case that one could reasonably refer to both infinite ordinals and <<infinite numbers>> as "infinite numbers" in natural language. Now, where is the contradiction? Do you *really* not understand that our ability to refer to both infinite ordinals and <<infinite numbers>> as "infinite numbers" simply DOES NOT imply that they have to have all the same properties in NSA to be non-contradictory? This really is analogous to saying "there is a least natural number but there is no least real number therefore there both is and is not a least number --> contradiction". It's just wrong. And I think, somehow, you're just not allowing yourself to understand the truth. It's quite pitiful, and kind of sad. Just look at this exchange : >> IST is an extension of ZF. IST includes every theorem of ZF (plus more >> theorems). IST is not a theory that contradicts or even excludes ZF. >> IST is a theory of which ZF is INCLUDED. I'm just curious whether you >> know that, since you reject ZF but then I read you recommending that >> people look into IST. >> MoeBlee >I don't really believe that. You don't *believe* that? Well, sorry, but it's a FACT. If you don't believe it then you are deluding yourself to protect some fantasy you have that IST and NSA agree with your silly viewpoints. >If there is no smallest infinity in IST or NSA, but there is in ZFC, how do you explain that? There is no smallest <<infinite number>> in NSA or IST (there are no <<infinite numbers>> in ZFC.) There is a smallest infinte ordinal in IST, NSA and in ZFC. How do I explain that? Uh, explain what? I can't explain "that" because there's nothing to explain! These simple statements of fact about these theories are self-explanatory. I can explain your confusion, however : you are conflating different meanings of 'infinite number' in a vain attempt to rationalise your views. >Is there an infinite less than omega in NSA? Is there an equilateral triangle less than the Cauchy distribution? Is there a Banach space greater than the Riemann-Zeta function? Apples and oranges, Tony. There is no infinite ordinal less than Omega in NSA. Is there an <<infinite number>> less than Omega? As far as I can tell, there just isn't an obvious ordering including both the <<infinite numbers>> and the ordinals, so your question as it stands is non-sensical. .... Wake up from your self-delusional haze Tony and get this through your head : NSA and Omega are *in no way* contradictory. -- mike.
From: Eckard Blumschein on 12 Dec 2006 08:02
On 12/11/2006 9:21 PM, Virgil wrote: > In article <457D504B.2040402(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 12/6/2006 9:08 PM, Virgil wrote: >> > In article <4576F7E0.3090102(a)et.uni-magdeburg.de>, >> > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: >> >> >> > And oo is NaN, 2^oo has no meaning. >> >> >> >> You are a knowing-all. >> > >> > oo is, at best, ambiguous. >> >> In what respect? > > Which of the several separate notions of infinite is it supposed to > represent? Cardinal? Ordinal? Real compactification? Geometric? Etc. If Galilei was correct, then there is no reason to further believe in the bizarre system by Cantor. Then there is only one unambiguous (fictitious) notion of infinity: Something that cannot be enlarged and also not exhausted. Simply forget the cardinal nonsense. E>>>>>>> In case of an infinite set, there are not all elements available. V>>>>>>> "Available"? E>>>>>> Yes. You cannot apply the algorithm until you have all numbers. V>>>>> What "algorithm? E>>>> For calculating binominal coefficients. V>>> What does one need that for in accepting the axiom of the power set? Axioms are arbitrary. Accepting axioms is bound to reasonable basics. This basis is lacking in case of ZFC. Therefore your question has a distracting aspect. The axiom of power set was constructed in order to make sure that there are subsets of the real numbers. What characterizes real numbers? Do they have subsets at all? Therefore my following question was quite necessary: E>> Is there evidence for the reals as characterised by DA2 to actually fit >> ZFC? The reals according to DA2 are categorically different from the reals according to mandatory definitions. V> How is that relevant to your "calculating binomial coefficients" claim? The whole ZFC issue tries to maintain what I consider the D&C illusion that real numbers are quite normal numbers (in the language of C. numbers with full civil rights within the kingdom ...). It is obvious that one can only calculate all variations if one has actually all elements. > Avoiding a question by asking an unrelated one is a copout. I did not avoid questions on condition they are meant honestly and do not just waste our time. > >> >> This is Bolzano's religious thinking. > > It is far preferrable to EB's religious thinking. Bolzano was a priest, and his mathematics was closely related to his belief in god. |