From: ags@seaman.cc.purdue.edu (Dave Seaman) Subject: Re: Zeno gets naked :-) Date: 14 Jul 2000 10:13:35 -0500 Newsgroups: sci.math Summary: [missing] In article <396ece7e_3@data.wt.net>, Pax wrote: >"Dave Seaman" wrote in message >news:8kmcsm$cnu@seaman.cc.purdue.edu... >> Now compare that with our other field, the field obtained from the ring S >> of Cauchy sequences of rationals by considering the equivalence relation >> ~ that I described previously. >Must query: Cauchy uses rings? If so, I do understand rings relatively well. The "Cauchy" in this discussion comes from the fact that our sequences (the members of S) all satisfy the Cauchy property: For each epsilon > 0 there exists N > 0 such that | a_m - a_n | < epsilon for all m, n > N. It so happens that the set of Cauchy sequences of rationals, with elementwise addition and multiplication, forms a commutative ring with unity. I have no idea who it was who first made that particular observation -- probably not Cauchy. If you know about rings, then perhaps you also know about ideals. Let I be the subset of S consisting of all the sequences that tend to zero. Then I is an ideal in S, and it makes sense to talk about the quotient ring S/I = R. The quotient ring is a field, because I is a maximal ideal in S. >> We have >> Integers mod 7 Reals >> -------------- ----- >> A commutative ring: Z S >> An RST relation: == (mod 7) ~ >> An RST class: [1] [1] >> An identity: [2] = [-5] [1] = [0.999...] An ideal: 7Z = all multiples of 7 I The quotient ring: Z/7Z S/I = R >Thank you for your patience, Mr. Seaman. You've been very informative and >extremely helpful. :) :) >Will gladly listen to whatever other information with which you might care to >enlighten me. (For instance: On the Cauchy Sequence and/or the Dedikind Cut? >MathWorld is NOT terribly helpful.) A Dedekind cut is a pair of sets (L,R), a left-set and a right-set, having the properties: 1) L and R are nonempty, 2) L and R are disjoint. 3) The union of L and R is Q, the set of rationals, 4) For each a in L and each b in R, we have a < b, 5) L has no largest member. Some examples of Dedekind cuts are given by: a) L = { x in Q : x < 1 } and R = Q\L. b) R = { x in Q : x^2 >= 2 } and L = Q\R. c) L = { x in Q : x < sum_(i=0 to n) 1/i! for some n >= 0 }, and R = Q\L. The real numbers, according to this model, are exactly the Dedekind cuts. The particular real numbers represented by these three examples are 1, sqrt(2) and e, respectively. Notice what the model is implicitly saying: if x = (L,R) and y = (L',R') are distinct real numbers (cuts), then we can find a rational that separates them. That is, L differs from L' (and R from R') by at least one member of Q. We can represent 0.999... by saying it is the cut with L = { x in Q : x < 1 - 1/10^n for some n >= 0 }. It's not hard to see that this cut turns out to be identical to the one mentioned in example (a), and therefore 0.999... = 1. In other words, since we can't find a rational number that separates the two, they must be the same cut. What's going on here is that we want a field R that contains Q, and looks as much like Q as possible, but with one additional property: every Cauchy sequence in R converges to some member of R. That is the completeness property (logically equivalent to the least upper bound principle that I stated earlier). We don't have more than one real number for the same cut in the rationals, because that would be introducing more complexity than we need in order to force every Cauchy sequence to converge. Remember Einstein's statement about simplicity. R is called the "completion" of Q. >And... if you could also point me to a link or a book that has a listing of >notations? >Kind regards - Pax Many math textbooks have listings of notations that are used in that particular text. If you read different texts on the same subject, you will sometimes come across different notations for the same concept. -- Dave Seaman dseaman@purdue.edu Amnesty International calls for new trial for Mumia Abu-Jamal