An Ultrafinite Set Theory
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From: Patricia Shanahan on 13 Mar 2010 09:42 Frederick Williams wrote: > Patricia Shanahan wrote: > >> Many, many years ago, I was given a basic definition for the limit of a >> sequence: A sequence x_1, x_2, x_3, ... tends to the limit L if, and >> only if, for every epsilon > 0 there exists n such that, for all m >= n, >> abs(x_m-L) < epsilon. I've been puzzling over the effects on that idea >> of a largest integer and a smallest positive real. > > If you were to conclude that delta-epsilonics requires that there is no > largest integer or smallest positive real would that be so bad? > Not necessarily. It would mean that the ultrafinite based mathematics would have to use another approach to deriving calculus, and that approach would have to be shown to work with a largest integer. Patricia
From: RussellE on 13 Mar 2010 16:49 On Mar 13, 6:42 am, Patricia Shanahan <p...(a)acm.org> wrote: > Frederick Williams wrote: > > Patricia Shanahan wrote: > > >> Many, many years ago, I was given a basic definition for the limit of a > >> sequence: A sequence x_1, x_2, x_3, ... tends to the limit L if, and > >> only if, for every epsilon > 0 there exists n such that, for all m >= n, > >> abs(x_m-L) < epsilon. I've been puzzling over the effects on that idea > >> of a largest integer and a smallest positive real. > > > If you were to conclude that delta-epsilonics requires that there is no > > largest integer or smallest positive real would that be so bad? > > Not necessarily. It would mean that the ultrafinite based mathematics > would have to use another approach to deriving calculus, and that > approach would have to be shown to work with a largest integer. I think we can make some minor modifications to delta-epsilon. Epsilon becomes the smallest real number. Let M be our finite set of all natural numbers. By limiting the number of terms in the sequence to |M|, the set of all n >= |M| is the empty set. Given a sequence for Sum(1/2^n) and M = {1,2,3} we can say the limit is 2 and show abs(2 - (1/2+1/4+1/8)) < epsilon (1/3). Russell - 2 many 2 count
From: FredJeffries on 14 Mar 2010 12:16 On Mar 13, 2:49 pm, RussellE <reaste...(a)gmail.com> wrote: > > I think we can make some minor modifications to delta-epsilon. > Epsilon becomes the smallest real number. > Let M be our finite set of all natural numbers. > By limiting the number of terms in the sequence to |M|, > the set of all n >= |M| is the empty set. > > Given a sequence for Sum(1/2^n) and M = {1,2,3} we can say the > limit is 2 and show abs(2 - (1/2+1/4+1/8)) < epsilon (1/3). > If {1,2,3} is your set of natural numbers and 1/3 is your smallest real number then there is no such thing as 2^2, 2^3, 1/4, 1/8, and you've got four terms in calculation when you've said that only three are allowed, ... Shouldn't it be, according your rules: Sum(1/2^n) = 1/2 (Actually, you haven't yet shown us how to get the existence of 1/2 when {1,2,3} is the set of natural numbers...) If you are seriously interested in this subject, I refer you to Jan Mycielski, "Analysis without actual infinity", Journal of Symbolic Logic 46:3, Sept 1981, pp. 625-633. And Shaughan Lavine, Understanding the Infinite, Harvard University Press, 1994 http://books.google.com/books?id=GvGqRYifGpMC
From: FredJeffries on 14 Mar 2010 17:37 On Mar 12, 7:09 am, Patricia Shanahan <p...(a)acm.org> wrote: > > I'm really looking forward to seeing a good theory of limits and > calculus that completely avoids the idea of an infinite sequence. I > think that may be even harder than calculating the largest possible > intermediate result. > Donald Knuth envisions teaching calculus using big O notation: http://www.ams.org/notices/199806/commentary.pdf http://www-cs-staff.stanford.edu/~uno/ocalc.tex See discussion at http://micromath.wordpress.com/2008/04/14/donald-knuth-calculus-via-o-notation/
From: Reinier Post on 19 Mar 2010 18:21
RussellE wrote: >As a programmer, I have to admit I am sometimes horrified >by the way set theorists encode things. A programmer who >made regular use of Godel numbering would probably be fired. Not only that. Sets are relatively rare, expensive objects to programmers. We use finite collections and infinite enumerations all the time, but we rarely deduplicate them - doing so is often unwanted or too expensive to just ignore. A foundational based on enumeration, rather than sets, is much more natural to us - e.g. lambda calculus. -- Reinier |