Galileo's Paradox
in [Math]
|
From: Tony Orlow on 11 Dec 2006 12:23 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!
From: Tony Orlow on 11 Dec 2006 12:30 Virgil wrote: <snip> >>>> No. Reread the following: >>>> >>>>>> 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, 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." > > Does TO claim that the infinite numbers of Robinson's non-standard > analysis are in any way connected to the transfinite cardinals and > ordinals of Cantor's analyses? Pray tell what ultrafilter generates > standard cardinals and ordinals in the way that ultrafilters are needed > to construct Robinson's non-standard reals from standard reals. I claimed no such thing. I am saying his very reasonable approach directly contradicts the very concept of the limit ordinals, which are schlock, and can stay in their little paradise/cave, while the rest of the world starts to take a more reasonable and less mystical approach to the infinite. Ordinals and cardinals are unrelated to anything worth bothering with. >> Of course, he has no need for omega. It's illegitimate schlock, like I said. > <snip>
From: Bob Kolker on 11 Dec 2006 12:32 Eckard Blumschein wrote:> > No. Cantor again merely showed by contradiction that the power set is > not countable. The reason is: Already the entity of all natural numbers > is an uncountable fiction. By definition, the set of integers is countable. A countable infinite set is a set which can be put in one to one correspondence with the set of integers. Bob Kolker
From: Tony Orlow on 11 Dec 2006 12:32 Virgil wrote: > In article <457cc0ce(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> I suppose now you're going to tell me I'm using nonstandard language.... > > Almost always, as far as the meaning of standard mathematical terms goes. Did you at least learn what a formal language is now? I remember you taking issue with me about the "null string" as if no such thing existed. You may have some mathematical holes of your own to patch up. Don't we all?
From: Tony Orlow on 11 Dec 2006 12:35
Virgil wrote: > In article <457cc17f(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > > >> Axioms declaring existence of sets from nothing seem rather pretentious >> to me. > > Isn't that precisely what TO's various alternate number systems do, > declare something out of nothing? No more than Peano. The H-riffics start with a 0 or 1 declared, and then go on to generate all desired reals with two rules. The T-riffics are a simple extension of normal digital number systems, with a few additional rules added to the normal ones. My axioms of infinity don't say the sets exist. They say that if a set has this logical property, then it can be said to be internally or externally infinite. |