An Ultrafinite Set Theory
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From: tchow on 12 Mar 2010 16:15 In article <7e9a6f87-1bd4-4f95-bd49-29ef9401e053(a)e7g2000yqf.googlegroups.com>, Transfer Principle <lwalke3(a)lausd.net> wrote: >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. I was specifically asking Patricia Shanahan; perhaps she doesn't care about these criteria. Anyway, if I take your use of the term "ZFC" literally, ZFC is actually very cumbersome. Encoding functions, ordered pairs, sequences, etc., using pure set theory is a pain in the neck. Not any less painful than encoding things using integers. -- 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: Patricia Shanahan on 12 Mar 2010 17:05 tchow(a)lsa.umich.edu wrote: > 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. I realize looking at your question that I did not say exactly what I meant. I am more concerned about the fixed upper bound on integers than about finiteness. With no infinity but with no specific largest integer, one could view a sequence whose limit is to be calculated rather like a Turing machine tape, which can be modeled as being always finite but always long enough. 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. Patricia
From: RussellE on 12 Mar 2010 17:18 On Mar 11, 10:58 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > On Mar 10, 12:37 pm, RussellE <reaste...(a)gmail.com> wrote: > > Let N={0,1,2,3} be a set of small natural numbers. > > Define "finite" addition as no more than one addition per member of a > > set. > > How large does a set have to be to "complete" finite addition > > for a set of small natural numbers? > > The largest finite addition for N is 3+3+3+3. > > So, we can always complete finite addition for set N > > with a set of size |N| * (|N|-1). > Both AP and RE realize that if M is the cardinality of their model, > then sqrt(M) is an upper bound on the numbers which can be multiplied > in their models. Sqrt(M) is the size of "small" numbers that can be added together in my theory. The set of small natural numbers would be more like Log(Log(M)) for multiplication. > AP would say something like, multiplication stops > working past 10, while RE writes that three is the largest "small" > natural number. An ultrafinite theory must do more than limit how big a number can be. An UST must also limit how often we can perform an operation. The simplest example of the need for this requirement is the Lamp paradox. http://en.wikipedia.org/wiki/Thomson's_lamp If we toggle a lamp on and off an infinite number of times, is the lamp on or off? What happens in AP's system if we allow 10+10+10+10+... ? My idea is to limit the number of times an operator can be applied. I assume we can apply an operator like addition once for each member of some set. Set theory often assumes we can perform an operation at least once for every member of a set. For example, a bijection assumes we can find a unique member of set B for every member of set A. I also use the idea of finite addition being "complete" for small natural numbers. By complete, I mean we can safely add any numbers in set S as long as we are limited to |S| additions. For example, we can add all of the elements of S together. The idea is for finite addition to be just like normal addition for small natural numbers, but undefined for larger natural numbers. Russell - 2 many 2 count
From: RussellE on 12 Mar 2010 17:34 On Mar 12, 1:15 pm, tc...(a)lsa.umich.edu wrote: > In article <7e9a6f87-1bd4-4f95-bd49-29ef9401e...(a)e7g2000yqf.googlegroups.com>, > Transfer Principle <lwal...(a)lausd.net> wrote: > > >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. > > I was specifically asking Patricia Shanahan; perhaps she doesn't care about > these criteria. > > Anyway, if I take your use of the term "ZFC" literally, ZFC is actually > very cumbersome. Encoding functions, ordered pairs, sequences, etc., > using pure set theory is a pain in the neck. Not any less painful than > encoding things using integers. 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. Russell - The universe is one dimensional
From: Frederick Williams on 13 Mar 2010 04:28
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? -- I can't go on, I'll go on. |