An Ultrafinite Set Theory
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From: Marshall on 12 Mar 2010 12:14 On Mar 12, 6:09 am, Patricia Shanahan <p...(a)acm.org> wrote: > > There has been a long term trend in computer science to assume that the > current limit on hardware integer arithmetic is so large that there is > no need to allow for integer overflow. So far, it has not worked. I love the understatedness of "has not worked." I wonder if it wouldn't be useful for a programming language to have both arbitrary-precision integers and naturals, as well as machine-sized signed and unsigned ints. In a sufficiently expressive language, it would be possible to formalize the properties of these as rings. Marshall
From: Patricia Shanahan on 12 Mar 2010 12:27 Marshall wrote: > On Mar 12, 6:09 am, Patricia Shanahan <p...(a)acm.org> wrote: >> There has been a long term trend in computer science to assume that the >> current limit on hardware integer arithmetic is so large that there is >> no need to allow for integer overflow. So far, it has not worked. > > I love the understatedness of "has not worked." > > I wonder if it wouldn't be useful for a programming language to > have both arbitrary-precision integers and naturals, as well as > machine-sized signed and unsigned ints. In a sufficiently > expressive language, it would be possible to formalize > the properties of these as rings. Many languages do have arbitrary range integers. For example, Java has java.math.BigInteger. Haskell has Integer. Those types cannot overflow, but your program may run out of memory. Unfortunately, they tend to be much worse performance, and bigger memory footprint, than the bounded range machine integers. The machine integers are usually used for most housekeeping activities, such as indexing data structures. Patricia
From: tchow on 12 Mar 2010 13:10 In article <0f-dne3xYo6a1gfWnZ2dnUVZ_jidnZ2d(a)earthlink.com>, Patricia Shanahan <pats(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. What exactly do you mean by "avoids the idea"? Do you just mean that the formalism makes no mention of infinite sequences? Even if the formalism avoids it, the "idea" behind the formalism might secretly be motivated by infinite sequences. Indeed, it's hard to imagine any treatment of calculus that cannot be thought of in terms of infinite sequences, at least informally. Depending on how strict your criteria are, formalizing calculus without explicit mention of infinite sequences is not difficult. It can certainly be done in first-order Peano arithmetic, and I think much of it can even be done in primitive recursive arithmetic. -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences
From: Transfer Principle on 12 Mar 2010 14:36 On Mar 11, 10:58 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > So it may be helpful to > find a theory T such that V_7 is a model of T. Correction: It may be helpful to find a theory T such that V_7 is the largest possible model of T. In other words, T has no model of strictly larger cardinality than that of V_7, and so in particular, it has no infinite models.
From: Transfer Principle on 12 Mar 2010 14:40
On Mar 12, 10:10 am, tc...(a)lsa.umich.edu wrote: > In article <0f-dne3xYo6a1gfWnZ2dnUVZ_jidn...(a)earthlink.com>, > 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. > Depending on how strict your criteria are, formalizing calculus without > explicit mention of infinite sequences is not difficult. It can certainly > be done in first-order Peano arithmetic, and I think much of it can even > be done in primitive recursive arithmetic. But the formalization of calculus must be relatively _simple_ -- perhaps nearly as simply as it can be done in ZFC. Otherwise, the standard theorists will point out that the theory is less "powerful" and more "cumbersome" to use than just doing calculus with ZFC and a complete ordered field. Indeed, we recall how Marshall Spight has already mentioned how the so-called "cranks" are trying to some up with theories that are "less powerful" and "more work to use" just to avoid some counterintuitive notion (in this case infinitude). So we need an ultrafinitist theory that _doesn't_ fit Spight's description of "less powerful" and "more work to use." |