Galileo's Paradox
in [Math]
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From: Tony Orlow on 12 Dec 2006 10:54 cbrown(a)cbrownsystems.com wrote: > Tony Orlow wrote: >> cbrown(a)cbrownsystems.com wrote: >>> Tony Orlow wrote: >>>> 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? >>> If you mean, does infinite(T) = 1, I don't know - that's why I'm >>> asking. It's your definition. >>> >> If T is infinite, it's greater than 1.72. > > And therefore if not T > 1.72, then not infinite(T). Thus, the urgency > of my question: is T > 1.72? If so, why? If not, why not? If you cannot > answer this question either yes or no, why can you not so answer? Your > definition provides no explicit resolution to these questions. > >> Is T supposed to be the area of the triangle? Then no. > > No, T is supposed to be an equilateral triangle with sides of length 1. > Is it your claim that the existence of such mathematical objects causes > a contradiction? > >>>> Does "1.72" refer to the number of points? >>> "1.72" refers to a real number; in other words, 1.72 in R. >>> >>> Cheers - Chas >>> >> Yeah, cheers. 1.72 what in R? Triangles don't exist in linear space. >> What are you talking about? > > I'm talking about your assertion:: > > (T1) infinite(x) <-> For all y in R, x>y > > ... and trying to understand why, when T is a unit equilateral > triangle, either infinite(T) = 0; or infinite(T) = 1; or infinite(T) = > 0.5, or infinite(T) = some specific "infinitesimal" e, or infinite(T) > is true or false, or /whatever you mean by it/ when T is an equilateral > triangle having side length 1? > > Eternally Hopeful - Chas > This still makes no sense. x and y are reals, or at least reside along the same metric from comparison. It's like asking whether a baseball is more or less than a washcloth. triangles are not quantities, but geometrical objects.
From: Lester Zick on 12 Dec 2006 12:33 On Tue, 12 Dec 2006 01:40:44 +0000 (UTC), 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. So striking common predicates we are left with Stephen's implied claim that the integer 1 is not real. Oh well. Onward and upward with modern math. ~v~~
From: Lester Zick on 12 Dec 2006 12:35 On Tue, 12 Dec 2006 05:21:45 +0000 (UTC), stephen(a)nomail.com wrote: >Tony Orlow <tony(a)lightlink.com> wrote: [. . .] >> Did that sound good and cranky. I am kind of cranky... > >It is not a thing to be strived for. But apparently who >want to be just like Lester. It is rather sad actually. Truth is often sad. It remains however true which is more than Lester can or modern mathematikers are willing to say for modern math and set "theory". ~v~~
From: MoeBlee on 12 Dec 2006 13:50 Tony Orlow wrote: > You cannot have a smallest infinity, and also not have it. You are actually TRYING NOT to listen. 'Smallest infinity' means two DIFFERENT things in two DIFFERENT contexts. I've explained too many times already now but you just keep going right past the point to make your same error over and over and over. MoeBlee
From: MoeBlee on 12 Dec 2006 13:58
Tony Orlow wrote: > 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. Page 52. Not only does he mention omega, but he explicitly uses it in a formula of ordinal arithmetic to state the order type of the nonstandard *N. Now will you PLEASE stop claiming that Robinson's nonstandard analysis is incompatible with having a least infinite ordinal? Maybe, just MAYBE, you'll take ONE MINUTE to think toward understanding what I've said to you so many times already about your confusing two different senses of 'infinite'. MoeBlee |