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From: Bastian Erdnuess on 17 Jun 2010 17:35 Rob Johnson wrote: >>>>>> Now suppose that w + x' = w' + x and y + z' = y' + z. >>>>> wy + xz + w'z' + x'y' = w'y' + x'z' + wz + xy (*) > As I mentioned in a follow-up message, I had viewed this problem as > a catalytic process; that is, add something that effects a change but > itself remains unchanged. Then we can remove the catalyst using the > cancellation property (iii). That is, you do (w y + x'y' + x z + w'z') + w'y = w y + x'y' + x z + (w'z' + w'y ) = w y + x'y' + (x z + w'z ) + w'y' = w y + (x'y' + x'z ) + w z + w'y' = (w y + x'y ) + x'z' + w z + w'y' = w'y + (x y + x'z' + w z + w'y') while the "cat" cycles w'y -> w'z -> x'z -> x'y -> w'y and thus cancels. Nice. Cheers, Bastian
From: Rob Johnson on 17 Jun 2010 18:30 In article <slrni1l58f.5da.earthnut(a)wh36-e604.wh36.uni-karlsruhe.de>, Bastian Erdnuess <earthnut(a)web.de> wrote: >Rob Johnson wrote: > >>>>>>> Now suppose that w + x' = w' + x and y + z' = y' + z. > >>>>>> wy + xz + w'z' + x'y' = w'y' + x'z' + wz + xy (*) > >> As I mentioned in a follow-up message, I had viewed this problem as >> a catalytic process; that is, add something that effects a change but >> itself remains unchanged. Then we can remove the catalyst using the >> cancellation property (iii). > >That is, you do > > (w y + x'y' + x z + w'z') + w'y > = w y + x'y' + x z + (w'z' + w'y ) > = w y + x'y' + (x z + w'z ) + w'y' > = w y + (x'y' + x'z ) + w z + w'y' > = (w y + x'y ) + x'z' + w z + w'y' > = w'y + (x y + x'z' + w z + w'y') > >while the "cat" cycles w'y -> w'z -> x'z -> x'y -> w'y and thus cancels. Indeed; that is the idea of the follow-up message I posted <http://groups.google.com/group/sci.math/msg/6a226feb3c89597e> >Nice. Thanks. Rob Johnson <rob(a)trash.whim.org> take out the trash before replying to view any ASCII art, display article in a monospaced font
From: David C. Ullrich on 18 Jun 2010 12:48 On 17 Jun 2010 15:04:47 -0400, Bill Dubuque <wgd(a)nestle.csail.mit.edu> wrote: >Gerry Myerson <gerry(a)maths.mq.edi.ai.i2u4email> wrote: >> David C. Ullrich <ullrich(a)math.okstate.edu> wrote: >>> >>> (Why? Because constructing the _positive_ >>> reals using Dedekind cuts (sets of positive >>> rationals) works out much more elegantly >>> than constructing the reals using Dedekind >>> cuts; negative numbers introduce unfortunate >>> complications in multiplication. No fair working >>> this out and publishing it, btw... you heard it >>> here first.) >> >> No, I read it (or something very much like it) first on pages 25-26 of >> Conway's On Numbers And Games. He says the best path from the positive >> integers to the reals is via the positive rationals and then the >> positive reals, and the reason he gives is precisely that if you go >> from the rationals to the reals by Dedekind cuts you have to make lots >> of special cases depending on signs when you get to the multiplication. > >It's an interesting (though tedious) exercise to prove that the two >different ways yield isomorphic structures. I recall seeing the >details worked out in a textbook, but the title escapes me right now. Of course they're both complete ordered fields (the wonderful thing about Dedekind cuts is that completeness is almost obvious; seems to me the most tedious thing is defining multiplication and showing it works right with the original construction). Showing that any two complete ordered fields are isomorphic doesn't seem so hard or tedious, not that I've ever seen or written down every detail. I've been intending to write down every detail sometime soon, so just for practice: [about ten minutes pass] Ok, never mind. Its straightforward but a little tedious. >--Bill Dubuque
From: Transfer Principle on 18 Jun 2010 22:59 On Jun 16, 5:32 pm, Gerry Myerson <ge...(a)maths.mq.edi.ai.i2u4email> wrote: > In article <g3nh16hp0mr5g3f0apdqnfq38b83750...(a)4ax.com>, > David C. Ullrich <ullr...(a)math.okstate.edu> wrote: > > (Why? Because constructing the _positive_ > > reals using Dedekind cuts (sets of positive > > rationals) works out much more elegantly > > than constructing the reals using Dedekind > > cuts; negative numbers introduce unfortunate > > complications in multiplication. No fair working > > this out and publishing it, btw... you heard it > > here first.) > No, I read it (or something very much like it) first on pages 25-26 of > Conway's On Numbers And Games. He says the best path from the positive > integers to the reals is via the positive rationals and then the > positive reals, and the reason he gives is precisely that if you go > from the rationals to the reals by Dedekind cuts you have to make lots > of special cases depending on signs when you get to the multiplication. Believe it or not, even _I_ have heard of constructing the reals via D-cuts of _positive_ rationals rather than all the rationals. And where have I heard it from? Metamath (a computer theorem prover created by Norm Megill). Megill writes: "To construct the complex numbers, we start with the finite ordinals (natural numbers) of set theory and successively build temporary positive integers, temporary positive rationals, temporary positive reals (based on Dedekind cuts), temporary signed reals, and finally the actual complex numbers." Not only is multiplication easier to define on the positive D-cuts rather than signed D-cuts, but it's easier to define positive _rationals_ as equivalence classes of ordered pairs of positive _naturals_, then have to include special cases to ensure that the second elements (denominator) of the rationals aren't zero. So Megill avoids the problems with negatives and the problem with zero until after the positive D-cuts. (Of course, if we were to use Cauchy sequences instead of D-cuts, we can at least avoid the problem of defining multiplication for negative reals.)
From: Bill Dubuque on 19 Jun 2010 12:40
rob(a)trash.whim.org (Rob Johnson) wrote: > Bill Dubuque <wgd(a)nestle.csail.mit.edu> wrote: >>rob(a)trash.whim.org (Rob Johnson) wrote: >>>Bill Dubuque <wgd(a)nestle.csail.mit.edu> wrote: >>>>rob(a)trash.whim.org (Rob Johnson) wrote: >>>>>David C. Ullrich <ullrich(a)math.okstate.edu> wrote: >>>>>> >>>>>>Say P is a structure with an addition and >>>>>>multiplication satisfying >>>>>> >>>>>> (i) x+y = y+x >>>>>> (ii) x+(y+z) = (x+y)+z >>>>>> (iii) If x+z = y+z then x = y. >>>>>> (iv) (P, multiplication) is an abelian group >>>>>> (v) x(y+z) = xy+xz. >>>>>> >>>>>>I gather that although the definitions are not quite standard, >>>>>>this might be called a semi-ring plus cancellation. >>>>>> >>>>>>Now suppose that w + x' = w' + x and y + z' = y' + z. >>>>>>Does it follow that >>>>>> >>>>>>(*) wz + xy + x'y' + w'z' = w'z' + x'y' + xy + wz ? >>>>>> >>>>>>If we had subtraction then this would be clear, >>>>>>since (*) just says that (w-x)(y-z) = (w'-x')(y'-z') >>>>>>and we're given that w-x = w'-x' and y-z = y'-z' >>>>>>(btw if (*) is _not_ the same as (w-x)(y-z) = (w'-x')(y'-z') >>>>>>then it's a typo in (*)). Seems like the cancellation >>>>>>property (iii) should be a substitute for subtraction, >>>>>>but I've gone around in circles with no luck. >>>>>> >>>>>>(If anyone's curious, this has to do with showing >>>>>>that the obvious definition of multiplication is >>>>>>well-defined if we apply the Grothendiek construction >>>>>>to try to get a ring from the semi-ring. The question >>>>>>doesn't really matter, since in the context I have >>>>>>in mind there are more axioms that help). >>>>> >>>>> Now that I have seen the reply from Bastian Erdnuess, I understand >>>>> that the question is really to show that >>>>> >>>>> wy + xz + w'z' + x'y' = w'y' + x'z' + wz + xy (*) >>>>> >>>>> After having worked this through, I realize that it is the samw as >>>>> Bill Dubuque's answer, but I will post it anyway since a second way >>>>> of looking at something is sometimes helpful. >>>>> >>>>> (wy + xz + w'z' + x'y') + (x'y + w'z + w'y + x'z) >>>>> >>>>> = (wy + x'y) + (xz + w'z) + (w'z' + w'y) + (x'y' + x'z) >>>>> >>>>> = (w'y + xy) + (x'z + wz) + (w'z + w'y') + (x'y + x'z') >>>>> >>>>> = (xy + wz + w'y' + x'z') + (w'y + x'z + w'z + x'y) >>>>> >>>>> Now we just cancel the right summand from the far-left and far-right >>>>> sides to get (*): >>>>> >>>>> wy + xz + w'z' + x'y' = xy + wz + w'y' + x'z' >>>> >>>>By removing the motivational remarks from my proof, and by inlining >>>>the lemmas, you've completely obfuscated the essence of the matter. >>>>In case it was not clear, my proof is just a special case of the >>>>well-known proof that congruences can be multiplied, e.g. >>>> >>>>LEMMA a = A, b = B (mod n) => ab = AB (mod n) >>>> >>>>PROOF n|a-A,b-B => n | (a-A)b + A(b-B) = ab - AB >>> >>> I removed nothing because I wrote my article based on the typo noted >>> by Bastian Erdnuess [...] in fact, I had not even read your post >>> until just before I posted >> >>Ok, then I'm quite interested in the motivation behind your proof. >>Why did you choose those particular expressions? Why add them, etc? >>As you said, a second way of looking at it may be helpful, so I'm >>interested in understanding the key idea(s) of your "second way". > > As I mentioned in a follow-up message, I had viewed this problem as > a catalytic process; that is, add something that effects a change but > itself remains unchanged. Then we can remove the catalyst using the > cancellation property (iii). In the equations above, I tried to > exhibit this process using parentheses. The four partial processes > (annotated in the follow-up) are given by > > (wy + x'y) = (w + x')y = (w' + x)y = (w'y + xy) [1] > > Equation [1] converts wy to xy while rotating catalyst x'y to w'y > > (xz + w'z) = (x + w')z = (x' + w)z = (x'z + wz) [2] > > Equation [2] converts xz to wz while rotating catalyst w'z to x'z > > (w'z' + w'y) = w'(z' + y) = w'(z + y') = (w'z + w'y') [3] > > Equation [3] converts w'z' to w'y' while rotating catalyst w'y to w'z > > (x'y' + x'z) = x'(y' + z) = x'(y + z') = (x'y + x'z') [4] > > Equation [4] converts x'y' to x'z' while rotating catalyst x'z to x'y > [...] But of course there are obvious symmetries that can be exploited to streamline the proof. But that does not answer my questions. Why did you choose those particular expression to start with? Why did you add them? As-is your proof is pulled out of a hat with no motivation at all for its genesis. That is why I had assumed that you had started with one of the earlier posted proofs when you mentioned that you saw them before posting. When you said you had a "second way" I thought you meant that you had a second conceptual way of viewing the proof, not merely a syntactic rearrangement gotten by stripping the motivational remarks and inlining the lemmas. Why such obfuscations appeal to you continues to puzzle me, just as it has in our many earlier exchanges on such. > take out the trash before replying I've tried everything, even teaching you "green" proof methods, with energy-saving reusable lemmas, ROHS-compliant construction materials, and insightful abstractions that promote optimal reuse. However, you continue to generate more trash than I can take out. Where did you learn those transfinite trash generation techniques? |