Galileo's Paradox
in [Math]
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From: MoeBlee on 11 Dec 2006 19:28 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. Aside from that being FLAT OUT INCORRECT, your response about non-standard analysis was a non sequitur and does not give defense for your misrepresenting it. MoeBlee
From: MoeBlee on 11 Dec 2006 19:30 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
From: David R Tribble on 11 Dec 2006 19:47 Eckard Blumschein wrote: >> Twice as infinitely many is Cantorian nonsense. Cantor was able to show >> himself that all natural and rational numbers do not have a different >> "size". Infinity is not a number. It has been understood like something >> which cannot be enlarged and not exhausted either: >> >> oo * 2 = oo. > Tony Orlow wrote: > THAT'S the Cantorian nonsense. What I said is not Cantorian, nor in line > with transfinitology, as is your statement. My statement is a statement > of the result I and others intuit, that there are twice as many reals in > twice as large an interval, like there are twice as many naturals as > even naturals. How? Please demonstrate how for every real in, say [0,1] there are two reals in [0,2], and for every real in [0,2] there is only one real in [0,1]. Otherwise your "twice as many" has no meaning. > That can be accomodated mathematically, but not by set > theory as it stands. Maybe that should tell you something.
From: MoeBlee on 11 Dec 2006 20:05 Tony Orlow wrote: > 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. You have an odd notion of what gives evidence. Your propounding an error over and over and over is NOT evidence that you are correct. > > 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? It isn't an element of certain sets and number systems. But it does exist in the same theory in which non-standard analysis is a part. For example, as an analogy, omega isn't a member of the set of complex numbers, but omega does exist in a theory in which we construct the complex numbers. For that matter, omega doesn't exist as a member of the set of natural numbers, but omega does exist in the theory in which we "construct" (scare quotes important) the natural numbers. > > 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. Robinson's work is built SOLELY on classical mathematical logic and set theory (and even though he himself doesn't use a first order theory for the construction, it is still classical mathematical logic, which includes not just first order logic). READ THE VERY FIRST SENTENCE OF THE BOOK. > >> "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. Of course he does. Read his writings. > 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. It's NOT impossible in "his theory", because "his theory" is all in classical mathematical logic (even as he sometimes uses logic that is not just first order) and set theory. READ THE VERY FIRST SENTENCE OF HIS BOOK. As to specific passages, I would have to go to the library to get the book. And, in addition, to the book, read from his collected papers as they are published in bound volumes. In one essay he explictitly endorses classical mathematics and his many papers are all steeped in it. And as to other approaches inspired by his work, look at the actual mechanics and axioms of IST, or at model theoretic constructions such as given a nice synopsis in Enderton's text, or look at the ultrafilter approaches - all in classical mathematics. Even BEFORE Robinson, non-standard models were discovered through model theoretic notions that observe Lowenheim-Skolem-Tarski and compactness and all kinds of things about infinite sets. ONE MORE TIME: Don't confuse two DIFFERENT senses of the word 'infinite'. One applies to cardinality, the other applies to certain kinds of ordering. If you DON'T confuse these two different senses, then you will see that there is NOT a conflict between non-standard analysis and infinite ordinals. MoeBlee
From: Tony Orlow on 11 Dec 2006 20:18
cbrown(a)cbrownsystems.com wrote: > Tony Orlow wrote: >> cbrown(a)cbrownsystems.com wrote: > >>> (T1) infinite(x) <-> A yeR x>y > >>> Tony Orlow wrote: > >>>>>> infinite(x) <-> A yeR x>y >>>>> Is that your only axiom? If so, then state your first theorem about them >>>>> and give the proof. >>>>> >>>> That's the only one necessary for what defining a positive infinite n. A >>>> whole array of theorems pop forth... >>> Before going there, you might want to start by adding the axiom: >>> >>> (T2) exists B such that infinite(B) >>> >>> Otherwise, who cares if you can prove a whole bunch of theorems about >>> something that doesn't exist? >>> >>> Cheers - Chas >>> >> What do you mean by "exist"? > > That's what I get for letting sloppy notation confuse me :). > > I'll put it another way: When you assert "infinite(x) <-> Ay in R, x > > y", what are we supposed to think you mean by "x > y"? > > For example, let T be an equilateral triangle with unit length sides. > Is T > 1.72? > > Cheers - Chas > Is T infinite? Does "1.72" refer to the number of points? |