Galileo's Paradox
in [Math]
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From: Tony Orlow on 11 Dec 2006 22:49 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 is also no largest or smallest infinitesimal. That's the way it is. Omega is a phantom. There are twilight zones between any two uncountably sepearted countable neighborhoods. I have spoken. :) Did that sound good and cranky. I am kind of cranky... Tony
From: Tony Orlow on 11 Dec 2006 22:53 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) So, it's not infinite, then.....
From: Tony Orlow on 11 Dec 2006 22:58 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. Is T supposed to be the area of the triangle? Then no. >> 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?
From: cbrown on 11 Dec 2006 23:08 Tony Orlow wrote: > 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.... ;) Right; and that confusion is easily remedied. Usually when I talk about "a number", I mean a real number. But sometimes, due to considerations of brevity, I mean a natural number. And sometimes I mean an ordinal number. And sometimes I mean a complex number. And sometimes I mean what Robinson is calling numbers. If you're not sure what I mean, it's the usual thing to ask "yo Chas, old chap; by 'x is a number' do you mean to say that x is a real number?!?". Or some such "interrogative". Then the "original meaning" can be "made clear" to all. Best Regards - Chas
From: cbrown on 12 Dec 2006 00:05
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 |