Galileo's Paradox
in [Math]
|
From: Tony Orlow on 11 Dec 2006 20:19 MoeBlee wrote: > Tony Orlow wrote: >>> Fortunately, TO's opinion is of no weight. >> But it has a great value range. > > Tony makes an inside joke about his own Tonyology. No narcissist, he. > Not much. > > MoeBlee > Nople Dopeys.
From: Tony Orlow on 11 Dec 2006 20:22 MoeBlee wrote: > Tony Orlow wrote: >> MoeBlee wrote: >>> Tony Orlow wrote: >>>> You might want to look into Internal Set Theory, a partial >>>> axiomatization of Nonstandard Analysis. >>> Why do you say 'parital'? >> I said "partial", and said that because that's what I've read. I am not >> sure what parts of nonstandard analysis are not included in the axioms >> of IST. >> >>>> Both infinitesimal and infinite >>>> values are "nonstandard", and no reference to "standard" values is >>>> allowed in the definition of any set. >>> Not ANY set. IST includes standard sets too. You do realize that IST is >>> an EXTENSION of ZF, right? >>> >>> MoeBlee >>> >> Sorry, "standard" is not allowed in the definition of any *internal* set. > > You didn't answer the question. > > 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. If there is no smallest infinity in IST or NSA, but there is in ZFC, how do you explain that? Is there an infinite less than omega in NSA?
From: MoeBlee on 11 Dec 2006 20:35 Tony Orlow wrote: > > 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. Read the damn axioms of IST then! > If there is no smallest infinity in IST or > NSA, but there is in ZFC, how do you explain that? Is there an infinite > less than omega in NSA? I explained to you how you are confusing two DIFFERENT senses of 'infinte' about A HUNDRED TIMES already! MoeBlee
From: stephen on 11 Dec 2006 20:40 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
From: Bob Kolker on 11 Dec 2006 20:57
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 |