mm-192 Or, big bully brother simply grabs the whole cake and nobody dares to> challenge him.>> But I didn't intend to bring the president into this.>> But nice mathematically modeled children would not behave like that,> would they?> >> Indeed uprighteous Republicans, transcending beyond democracy> don't behave like gross plebeian Democrats who don't even recount.>> That's a bit over my head since I am from the other side of the> Atlantic. I guess that it refers to the rather fun election you guys> had last year.>Fun??? Oh yea, the joke is on US.> But I suppose if politicians can get in such a mess over something as> simple (conceptually) as an election then we should not expect perfect> behaviour from children sharing cakes. Maybe we let them sort it out> themselves as useful practice for later life. They can learn how to> deal with bullies while the only thing at stake is a cake.>What's the EU gonna do about Bully Bush?Perhaps your Brit in which case,drifting toward being on topic,can you prove or disprove the proportion Blair is to Bush as Thacher is to RayGun ? == Or, big bully brother simply grabs the whole cake and nobody dares >to> challenge him.> >> But I didn't intend to bring the president into this.> > >> But nice mathematically modeled children would not behave like >that,> would they?>> Indeed uprighteous Republicans, transcending beyond democracy> don't behave like gross plebeian Democrats who don't even recount.>> That's a bit over my head since I am from the other side of the> Atlantic. I guess that it refers to the rather fun election you guys> had last year.>> Fun??? Oh yea, the joke is on US.> But I suppose if politicians can get in such a mess over something as> simple (conceptually) as an election then we should not expect perfect> behaviour from children sharing cakes. Maybe we let them sort it out> themselves as useful practice for later life. They can learn how to> deal with bullies while the only thing at stake is a cake.>> What's the EU gonna do about Bully Bush?> Perhaps your Brit in which case,> drifting toward being on topic,> can you prove or disprove the proportion> Blair is to Bush as Thacher is to RayGun ?Yes, I am a Brit. I am a bit wary of getting into a politics discussion. I like the oddbit of fun slipping into some threads but if we start discussingpolitics then it may move beyond fun.J ==> I'm trying to 'nd integer coef'cients for a polynomial which has>> sqrt{3} + sqrt{2} as a root. Any ideas?>> x^4 - 10*x^2 +1 has roots the 4 roots +/-sqrt(2) +/-sqrt(3)> x = sqr 3 + sqr 2> x^2 = 5 + 2 sqr 6> (x^2 - 5)^2 = 24>> Is there an integer polynomial in Z[x] with roots sqr 2, sqr 3?That depends... if you require the polynomial to have sqrt 2 and sqrt3 as _the_ roots, then the answer is no. Clearly one polynomial withthese roots is f(x) = (x-sqrt(2))(x-sqrt(3)) = x^2 -(sqrt(2)+sqrt(3))x + sqrt(6). Any other polynomial with exactlysqrt(2) and sqrt(3) as roots is a constant times this polynomial. NowIf you require the polynomial to belong to Z[x], then you can onlymultiply f by integers; but since sqrt(6) is irrational, no polynomialin Z[x] is an integer multiple of f.If you only require that sqrt(2) and sqrt(3) are roots of thepolynomial, then the answer is clearly yes. Try g(x) =(x-sqrt(2))(x+sqrt(2))(x-sqrt(3))(x+sqrt(3))/Rasmus-- == Is there an integer polynomial in Z[x] with roots sqr 2, sqr 3?>> That depends... if you require the polynomial to have sqrt 2 and sqrt> 3 as _the_ roots, then the answer is no. Clearly one polynomial with> these roots is f(x) = (x-sqrt(2))(x-sqrt(3)) = x^2 -> (sqrt(2)+sqrt(3))x + sqrt(6). Any other polynomial with exactly> sqrt(2) and sqrt(3) as roots is a constant times this polynomial. Now> If you require the polynomial to belong to Z[x], then you can only> multiply f by integers; but since sqrt(6) is irrational, no polynomial> in Z[x] is an integer multiple of f.>> If you only require that sqrt(2) and sqrt(3) are roots of the> polynomial, then the answer is clearly yes. Try g(x) == (x-sqrt(2))(x+sqrt(2))(x-sqrt(3))(x+sqrt(3))>Indeed, however (x^2 - 2)(x^2 - 3) is reducible.Is there a irreducible poly including sqr 2, sqr 3 as roots?I think not. Now Q(sqr 6) subset Q(sqr 2, sqr 3) but equality seems most unlikely.Thus for 'eld F it appears, F(u,v) can't be reduced to F(w) for some w. == I'm trying to 'nd integer coef'cients for a polynomial which has> sqrt{3} + sqrt{2} as a root. Any ideas?> See pages 5 and 6 of my notes on algebraic number theory for a method> of solving all such problems (and an example slightly harder than this).(For the OP):I ended up there while googling for a proof that the algebraicintegers form a ring. Most such proofs (as Prof. Chapman does)establish a connection between an algebraic integer and a matrix for which the algebraic number is an eigenvalue, and they show how to construct that matrix explicitly. Given algebraic integersa and b, the matrixes A and B are constructed so that Av = av and Bv = bv, with a common eigenvector v.Then (a+b) is an eigenvalue of (A+B) with eigenvector v(and ab is an eigenvalue of AB). The construction showsyou how to relate the matrix to the minimal polynomial. - Randy == I'm trying to 'nd integer coef'cients for a polynomial which has> sqrt{3} + sqrt{2} as a root. Any ideas?> See pages 5 and 6 of my notes on algebraic number theory for a method> of solving all such problems (and an example slightly harder than this).Very helpful indeed. I have a related question: It is quite simple to show that 1/x is an algebraic number if x is an algebraic number other than zero. It is also quite easy to show that the square root of a non-integer rational number is an algebraic number, but not an algebraic integer. Now the question: Given an arbitrary algebraic number x, for example by specifying a polynomial P with integer coef'cients with P(x) = 0, is there a way to 'nd whether or not x is an algebraic integer? I would guess that x is not an algebraic integer if P(x) = 0 where P is a non-monic polynomial with integer coef'cents that cannot be divided by any integer constant or integer polynomial other than +/- 1. But I wouldn't have any idea how to prove this. ==Given an arbitrary algebraic number x, for example by specifying a >polynomial P with integer coef'cients with P(x) = 0, is there a way to >'nd whether or not x is an algebraic integer? Factor P over the integers. Exactly one of the factors will have P as a root. The coef'cient of the highest power of the variable in this factor is 1 or -1 iff x is an algebraic integer.Robert Israel israel@math.ubc.caDepartment of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 ==Given an arbitrary algebraic number x, for example by specifying a >polynomial P with integer coef'cients with P(x) = 0, is there a way to > >'nd whether or not x is an algebraic integer? > Factor P over the integers. Exactly one of the factors will have P as a > root. The coef'cient of the highest power of the variable in this factor >> is 1 or -1 iff x is an algebraic integer. http://mathworld.wolfram.com/AlgebraicNumber.htmlit says (not in these words): If x is the root of an (irreducible) polynomial with integer coef'cents then the other roots are called the >conjugates> of x. If the same x is the root of _any_ other polynomial with integer coef'cients then its conjugates are roots as well (proved by Conway and Guy in 1996. That is quite recent so I guess it must be quite hard to prove? ) svr.pol.co.uk>,> http://mathworld.wolfram.com/AlgebraicNumber.html> it says (not in these words): If x is the root of an (irreducible) > polynomial with integer coef'cents then the other roots are called the >conjugates> of x. If the same x is the root of _any_ other polynomial > with integer coef'cients then its conjugates are roots as well (proved > by Conway and Guy in 1996. That is quite recent so I guess it must be > quite hard to prove? ) Not so hard, and certainly goes back way beyond 1996. Amongst all polynomials with integer coef'cients and x as a root, pick out one, P, with minimal degree. If Q is any other polynomial with integer coef'cients and x as a root, divide Q by P and get a remainder R of degree less than the degree of P. But P and Q vanish at x, so R does too; P had minimal degree, so R must be identically zero, so Q is actually a multiple of P. It follows that all the roots of P - all the conjugates of x - are roots of Q. Instead of reading mathworld, try something like Galois Theory by Stewart, or the 'eld theory book by Hadlock.-- == Thus for 'eld F it appears, F(u,v) can't be reduced to F(w) for some w.There is a theorem that if F is a 'eld of characteristic zero (such as the rationals) and u and v are algebraic over F then there does exist w such that F(u, v) = F(w). You just have to be a little careful choosing w - not just any old element of F(u, v) will do.-- == http://mathworld.wolfram.com/AlgebraicNumber.html>it says (not in these words): If x is the root of an (irreducible) >polynomial with integer coef'cents then the other roots are called the >>conjugates> of x. If the same x is the root of _any_ other polynomial >with integer coef'cients then its conjugates are roots as well (proved >by Conway and Guy in 1996. That is quite recent so I guess it must be >quite hard to prove? ) No, it's very easy to prove, and very old. The reference to Conway and Guy is only a reference, and not a statement that this was the 'rsttime it was proved.>polynomial P with integer coef'cients and leading coef'cient not +/- >1and P is primitive, i.e. the gcd of the coef'cients is 1>, then it is indeed impossible to 'nd a different polynomial Q with >leading coef'cent +/- 1 and root x, so x is not an algebraic integer.The point is that if P_1 and P_2 are in Q[X] and deg(P_1) >= 1, you can write P_2 = A P_1 + B for A, B in Q[X] with deg(A) < deg(P_1).If P_1(x) = 0 and P_2(x) = 0 then of course B(x) = 0. Take P_1 so thatdeg(P_1) is minimal among members of Q[X] (other than the constant 0) which have x as a root. Then B must be 0, i.e. P_1 divides P_2. If P_2 is irreducible over the rationals, it's not divisible by any P_1 inQ[X] with 0 < deg P_1 < deg P_2, so deg P_2 = deg P_1 and P_2/P_1 isconstant. Thus up to multiplication by a constant, the minimal polynomialP_1 is the only member of Q[X] which has x as a root. The conjugates of x are the other roots of P_1, and since P_1 divides any other polynomial in Q[X] which has x as a root, the conjugates are roots of such a polynomial as well.Robert Israel israel@math.ubc.caDepartment of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 == http://mathworld.wolfram.com/AlgebraicNumber.html> it says (not in these words): If x is the root of an (irreducible) > polynomial with integer coef'cents then the other roots are called the >>conjugates> of x. If the same x is the root of _any_ other polynomial >> with integer coef'cients then its conjugates are roots as well (proved >> > by Conway and Guy in 1996. That is quite recent so I guess it must be > quite hard to prove? ) > Not so hard, and certainly goes back way beyond 1996. To be fair to mathworld, the parenthetical >Conway and Guy 1996> doesnot mean that they proved it. It just means that book was the sourcefor the preceding entry.-- G. A. Edgar >http://www.math.ohio-state.edu/~edgar/ =in message - 2)(x^2 - 3) is reducible.> Is there a irreducible poly including sqr 2, sqr 3 as roots?No, since Q(sqrt(2)) isn't isomorphic to Q(sqrt(3)).-- Jim Heckman ==> http://mathworld.wolfram.com/AlgebraicNumber.html>> says (not in these words): If r is the root of an (irreducible)>> polynomial with integer coef'cents then the other roots are called the>>conjugates> of r. If the same r is the root of _any_ other polynomial>> with integer coef'cients then its conjugates are roots as well (proved>> by Conway and Guy in 1996. That is quite recent so I guess it must be>> quite hard to prove? )> Not so hard, and certainly goes back way beyond 1996.The paraphrase is misleading. MathWorld doesn't say >proved by> but instead merely lists (Conway and Guy, 1996) as a reference.> Amongst all polynomials with integer coef'cients and r as a root,> pick out one, P, with minimal degree. If Q is any other polynomial> with integer coef'cients and r as a root, divide Q by P and get a> remainder R of degree less than the degree of P. But P and Q vanish> at r, so R does too; P had minimal degree, so R must be identically> zero, so Q is actually a multiple of P. It follows that all the> roots of P - all the conjugates of r - are roots of Q.Ideals provide a conceptually nicer way to view this basic result.It's clear that the set of polys f over Q with r as root comprisean ideal I in Q[x]. Since Q[x] has a Division Algorithm, it is a Euclidean domain, hence a PID. Thus I = (g(x)) for some g in Q[x].Hence: f(r)=0 <== f in I <== g|f, i.e. f = gh, for some h in Q[x]The generator g is called the minimal poly of r (often one 'rstnormalizes g by clearing denominators and removing any content).g(x) is simply the gcd of all polys having r as a root. Below wesee this is an immediate generalization the Euclidean algorithm.As I emphasized in a prior post [1] on related topics, one shouldalways look for algebraic structures hidden in a problem (here theideal structure). Once the hidden structure has been revealed, onemay immediately apply (reuse!) the entire theory of such structureswithout having to reinvent the wheel from scratch -- which isessentially what occurs above in Gerry's proof, which is nothingbut a specialization of the proof that a Euclidean domain is a PID.The proof in the general case remains the same -- just as easy.Constructively it's just gcd computation via Euclid's algorithm,i.e. F(r)=f(r)=0 == (F mod f)(r)=0 ==...== g(r)=0, g = gcd(F,f)which results from iterated application of the basic descent step viz. (F, f) -> (f, F mod f), where deg F >= deg f. Note howthis has the same form as the Euclidean algorithm for integers.Indeed, both are abstracted in the Euclidean domain structure, andthe results from Z generalize immediately to any Euclidean domain.Related instructive examples occur throughout the entire thread containing my prior post [1], which I recommend to Christian.-Bill Dubuque[1] 40nestle.ai.mit.edu == ,> > http://mathworld.wolfram.com/AlgebraicNumber.html> > it says (not in these words): If x is the root of an (irreducible) > polynomial with integer coef'cents then the other roots are called the >>conjugates> of x. If the same x is the root of _any_ other polynomial >> with integer coef'cients then its conjugates are roots as well (proved >> > by Conway and Guy in 1996. That is quite recent so I guess it must be > quite hard to prove? ) > Not so hard, and certainly goes back way beyond 1996. The webpage is really quite misleading there. If I read (quotation of a theorem) (Conway and Guy, 1996) I expect that to mean >proved by Conway book with all kinds of popular mathematics inside where the theorem is mentioned, and that book was published in 1996. > Amongst all polynomials with integer coef'cients and x as a root, > pick out one, P, with minimal degree. If Q is any other polynomial > with integer coef'cients and x as a root, divide Q by P and get a > remainder R of degree less than the degree of P. But P and Q vanish > at x, so R does too; P had minimal degree, so R must be identically > zero, so Q is actually a multiple of P. It follows that all the > roots of P - all the conjugates of x - are roots of Q. So all irreducible polynomials with x as a root are identical except for the sign, so if _one_ irreducible polynomial with root x is not monic then _all_ irreducible polynomials with root x are not monic and then _all_ polynomials with root x, whether reducible or not, are not monic, so x is not an algebraic integer. > Instead of reading mathworld, try something like Galois Theory by > Stewart, or the 'eld theory book by Hadlock.Considering that your proof was just seven lines, and absolutely obvious, it is a shame that proofs of this kind are not included at mathworld. On the positive side, I found Robin Chapman's homepage and learnt some stuff from there, posted two questions on sci.math, and in one day I learnt more about algebraic numbers and algebraic integers than some people did in eight years! = >> Thus for 'eld F it appears, F(u,v) can't be reduced to F(w) >There is a theorem that if F is a 'eld of characteristic zero >(such as the rationals) and u and v are algebraic over F then >there does exist w such that F(u, v) = F(w). You just have to be >a little careful choosing w - not just any old element of F(u, v) >will do.There is?! How careful must one be? Let's consider u = sqr 2, v = sqr 3, n = degree of w, F = RealsWho's the w with R(w) = R(u,v) ?---- ==> Indeed, however (x^2 - 2)(x^2 - 3) is reducible.> Is there a irreducible poly including sqr 2, sqr 3 as roots?>> No, since Q(sqrt(2)) isn't isomorphic to Q(sqrt(3)).>Nifty. ==> Thus for 'eld F it appears, F(u,v) can't be reduced to F(w)>There is a theorem that if F is a 'eld of characteristic zero>(such as the rationals) and u and v are algebraic over F then>there does exist w such that F(u, v) = F(w). You just have to be>a little careful choosing w - not just any old element of F(u, v)>will do.> There is?! How careful must one be? Let's consider> u = sqr 2, v = sqr 3, n = degree of w, F = Reals> Who's the w with R(w) = R(u,v) ?> What's wrong with w=sqrt(2)+sqrt(3) ??Then 1/2*w^3-9/2*w = sqrt(2) and 11/2*w-1/2*w^3 = sqrt(3) .Of course if you really mean F=R, then R(w) = R and R(u,v) = R so justany old element WILL do. Was that your point? ==>> Thus for 'eld F it appears, F(u,v) can't be reduced to F(w)>There is a theorem that if F is a 'eld of characteristic zero> >(such as the rationals) and u and v are algebraic over F then>there does exist w such that F(u, v) = F(w). You just have to be>a little careful choosing w - not just any old element of F(u, v)>will do.> There is?! How careful must one be? Let's consider> u = sqr 2, v = sqr 3, n = degree of w, F = Reals> Who's the w with R(w) = R(u,v) ?>> What's wrong with w=sqrt(2)+sqrt(3) ??> Then 1/2*w^3-9/2*w = sqrt(2) and 11/2*w-1/2*w^3 = sqrt(3) .>ww = 5 + 2sqr 6(w/2)(2sqr6 - 4) = (sqr 6 - 2)(sqr 2 + sqr 3) = sqr 12 - 2.sqr 2 + sqr 18 - 2.sqr 3 = sqr 2(-w/2)(2.sqr 6 - 6) = -(sqr 6 - 3)(sqr 2 + sqr 3) = -(sqr 12 - 3.sqr 2 + sqr 18 - 3.sqr 3) = sqr 3Nothing.ww - 5 = 2.sqr 6(ww - 5)^2 = 24w^4 - 10w^2 - 1 = 0> Of course if you really mean F=R, then R(w) = R and R(u,v) = R so just> any old element WILL do. Was that your point?>No, F = Q.Does the general theorem have a name?How complicated is the proof? ==>> Thus for 'eld F it appears, F(u,v) can't be reduced to F(w)>There is a theorem that if F is a 'eld of characteristic zero>(such as the rationals) and u and v are algebraic over F then>there does exist w such that F(u, v) = F(w). You just have to be>a little careful choosing w - not just any old element of F(u, v)>will do.>There is?! How careful must one be? Let's consider> u = sqr 2, v = sqr 3, n = degree of w, F = Reals>Who's the w with R(w) = R(u,v) ?>What's wrong with w=sqrt(2)+sqrt(3) ??>>Then 1/2*w^3-9/2*w = sqrt(2) and 11/2*w-1/2*w^3 = sqrt(3) .>>ww = 5 + 2sqr 6>(w/2)(2sqr6 - 4) = (sqr 6 - 2)(sqr 2 + sqr 3)> = sqr 12 - 2.sqr 2 + sqr 18 - 2.sqr 3 = sqr 2>(-w/2)(2.sqr 6 - 6) = -(sqr 6 - 3)(sqr 2 + sqr 3)> = -(sqr 12 - 3.sqr 2 + sqr 18 - 3.sqr 3) = sqr 3>Nothing.>>ww - 5 = 2.sqr 6>(ww - 5)^2 = 24>w^4 - 10w^2 - 1 = 0>>Of course if you really mean F=R, then R(w) = R and R(u,v) = R so just>>any old element WILL do. Was that your point?>>No, F = Q.>>Does the general theorem have a name?>Theorem 5.3 (or whatever).>How complicated is the proof?>It's in the chapter on Galois theory. If the book is conversational, it's easy. If it's formal, it could be pretty hard to understand. (YMMV, some people prefer formal to conversational.) It's not complicated at all, it's straightforward and usually done right where it makes sense. It's just that some books are hard to follow because of the way they're written. (Or the way I process what I read.)I like Van der Waerden, _Algebra_ or _Modern Algebra_, depending on your version. I'm suggesting it because from some other things you seem to like an old-fashioned approach to algebra, so your point of view may agree with his and make it really easy. But any algebra book that covers Galois theory will do. (Not Herstein _Topics in Algebra_, which I consider the best introduction to algebra.)Jon Miller =... >> Thus for 'eld F it appears, F(u,v) can't be reduced to F(w) >There is a theorem that if F is a 'eld of characteristic zero >(such as the rationals) and u and v are algebraic over F then >there does exist w such that F(u, v) = F(w). You just have to be > >a little careful choosing w - not just any old element of F(u, v) >will do. > There is?! How careful must one be? Let's consider > u = sqr 2, v = sqr 3, n = degree of w, F = Reals > Who's the w with R(w) = R(u,v) ?(Let's do it over rationals, yes? u and v are elements of the reals.)If z = sqrt(2) + sqrt(3), (z^3 - 9z)/2 = sqrt(2).So w = z works.-- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, >+31205924131home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ = [.snip.]>>Does the general theorem have a name?>Theorem 5.3 (or whatever).Or the Primitive Element Theorem, since it guarantees the existence ofa primitive element for any 'nite separable extension in those 'elds. ==It's not denial. I'm just very selective about what I accept as reality.> --- Calvin (>Calvin and Hobbes>) ==Arturo Magidinmagidin@math.berkeley.edu =< > So all irreducible polynomials with x as a root are identical except for > the sign, so if _one_ irreducible polynomial with root x is not monic > then _all_ irreducible polynomials with root x are not monic and then > _all_ polynomials with root x, whether reducible or not, are not monic, > so x is not an algebraic integer. Not quite, e.g., x^2 - 2 and 7 x^2 - 14 are both considered irreducible. Sometimes the word >primitive> is used to describe a polynomial with integer coef'cients with no non-trivial common factor. Then there are only two primitive irreducible polynomials with a given x as a root, and each is the negative of the other.-- =[in re: the Primitive Element Theorem,]> ...any algebra book that covers Galois theory will do. I think the exception here is Artin's book. If I remember some other discussion correctly, Artin had something against using this theorem in an exposition of Galois Theory, and studiously avoided it.-- =in message however (x^2 - 2)(x^2 - 3) is reducible.> Is there a irreducible poly including sqr 2, sqr 3 as roots?>> No, since Q(sqrt(2)) isn't isomorphic to Q(sqrt(3)).>> Nifty.Another approach: Every algebraic element over F is the root ofa *unique* monic irreducible polynomial over F. For sqrt(2) it's(x^2 - 2), and ...-- Jim Heckman == in message Indeed, however (x^2 - 2)(x^2 - 3) is reducible.> Is there a irreducible poly including sqr 2, sqr 3 as roots?> >> No, since Q(sqrt(2)) isn't isomorphic to Q(sqrt(3)).> >> Nifty.> Another approach: Every algebraic element over F is the root of> a *unique* monic irreducible polynomial over F. For sqrt(2) it's> (x^2 - 2), and ...The number z = sqrt(3) + sqrt(2), is a zero of x^4 - 10*x^2 + 1, which shows that z is an algebraic integer and the reciprocal of an algebraic integer. =in message the Primitive Element Theorem,]>> ...any algebra book that covers Galois theory will do.>> I think the exception here is Artin's book. If I remember some other> discussion correctly, Artin had something against using this theorem> in an exposition of Galois Theory, and studiously avoided it.If you're talking about Michael Artin's _Algebra_, he doesindeed prove the Primitive Element Theorem. In fact, he thengoes on to use it in his proof that >For any 'nite extensionK/F, the order |G(K/F)| of the Galois group divides the degree[K:F] of the extension.>One thing I 'nd somewhat disconcerting about Artin's expositionof Galois Theory is that he proves several important theoremsout of order, i.e., he states them early on, proving them onlyin later sections. To my mind this makes following the logicaldevelopment of the ideas a little trickier than need be.-- Jim Heckman == in message re: the Primitive Element Theorem,]>> ...any algebra book that covers Galois theory will do.>> I think the exception here is Artin's book. If I remember some other>> discussion correctly, Artin had something against using this theorem>> in an exposition of Galois Theory, and studiously avoided it.> If you're talking about Michael Artin's _Algebra_,No, he's talking about Emil Artin's book on Galois theory.-- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html The League of Gentlemen == [in re: the Primitive Element Theorem,]> ...any algebra book that covers Galois theory will do. > I think the exception here is Artin's book. If I remember some other > discussion correctly, Artin had something against using this theorem > in an exposition of Galois Theory, and studiously avoided it.Yes, but Artin still discusses when a 'eld is generatedby a single element in his book >Galois Theory>. It appearson page 64 (of 82 pages) in section M called >Simple Extensions>.In this section, Artin gives two theorems. The 'rst theoremgives a necessary and suf'cient condition for the existenceof primitive elements: >Theorem 26. A 'nite extension E of F is primitive over F if and only if there are only a 'nite number of intermediate 'elds.>Theorem 27 then gives the situation being discussed in this thread.To apply Theorem 27 theorem, we need to note that sqrt(2) and sqrt(3)are separable elements over Q. That is, the irreducible polynomialsfor sqrt(2) and sqrt(3) do not have repeated roots.I have not yet grasped the impact that >separable> elements hasin Galois theory. I just usually assume that I am working overextensions of Q, where separable follows because the characteristicof Q is 0.To 'nd a couterexample to having a primitive element, I thinkthat you need to have a 'nite extension E of F, where F is anin'nite 'eld of characteristic p. But, I haven't pursued this.-- Bill Hale == [in re: the Primitive Element Theorem,]> ...any algebra book that covers Galois theory will do. > I think the exception here is Artin's book. If I remember some other > discussion correctly, Artin had something against using this theorem > in an exposition of Galois Theory, and studiously avoided it.Yes, but Artin still discusses when a 'eld is generatedby a single element in his book >Galois Theory>. It appearson page 64 (of 82 pages) in section M called >Simple Extensions>.In this section, Artin gives two theorems. The 'rst theoremgives a necessary and suf'cient condition for the existenceof primitive elements: >Theorem 26. A 'nite extension E of F is primitive over F if and only if there are only a 'nite number of intermediate 'elds.>Theorem 27 then gives the situation being discussed in this thread.To apply this theorem, we need to note that sqrt(2) and sqrt(3)are separable elements over Q. That is, the irreducible polynomialsfor sqrt(2) and sqrt(3) do not have repeated roots.I have not yet grasped the impact that >separable> elements hasin Galois theory. I just usually assume that I am working overextensions of Q, where separable follows because the characteristicof Q is 0.To 'nd a couterexample to having a primitive element, I thinkthat you need to have a non-'nite extension E of Z_p, the'nite 'eld of order p. But, I haven't pursued this.-- Bill Hale ==>I'm trying to 'nd integer coef'cients for a polynomial which has>sqrt{3} + sqrt{2} as a root. Any ideas?>Sure. Set r=sqrt(2)+sqrt(3). Guess there is polynomial of degree 4 >meetingyour requirements. Set x0[0]=1, x1[0]=r, x2[0]=r^2, x3[0]=r^3, and >x4[0]=r^4. Compute the following until x0[i0]=0:x0[i+1]=x1[i]-int(x1[i]/x0[i])*x0[i]..x3[i+1]=x4[i]- int(x4[i]/x0[i])*x0[i]x4[i+1]=x0[i]At this point you know there are integers a0,..,a4 s.t. a0+..a4*r^4=0.An alternative is to use PSLQ or LLL methods to 'nd an integer relation >for1,r,..,r^4. Kind of overkill in this case though.Rich Burge ==> Let G be an abelian group, 'nite, generated by x and y. Let H be a> cyclic subgroup of G. Also, suppose it is known that X = c(X-Y), Y == (c-1)(X-Y), and G/H = , where X is >the coset of X,> Y is >the> coset of Y,> etc. I've shown that>I'll presume G is additive as it's Abelian group.H is subgroup. What's Oc'? Some integer?What's meant OX is coset of X' ?Do you mean Ofor x,y in G, let X = x+H, Y = y+H' ?Then -Y = -y+H, X-Y = x-y + H, c(X-Y) = c(x-y) + cH ?> blahblahblahblahblah>What's that? Bush's latest public speach?> is true if and only if there exist relatively prime positive integers> m_1, n_1 such that>> 0 <= [G/H:]*n_1 <= |G/H|-1 and> m_1*order(X) + n_1*([G/H:]-e_1) = |G/H|-1,>> where e_1 is the unique integer satisfying>> 0 <= e_1 <= order(X) and> [G/H:]*Y = e_1*X>> _and_ there exist relatively prime positive integers m_2, n_2 such> that>> 0 <= [G/H:]*n_2 <= |G/H|-1 and> m_2*order(Y) + n_2*([G/H:]-e_2) = |G/H|-1,>> where e_2 is the unique integer satisfying>> 0 <= e_2 <= order(Y) and> [G/H:]*X = e_2*Y.>> I suspect I can (using relationships between X and Y) reduce the> number of equations (and conditions) needed to express this> information. Here's what I've tried so far:>> Since order(X)=|G/H|/gcd(c,|G/H|) and order(Y)=|G/H|/gcd(c-1,|G/H|),> we must have order(X)/order(Y) = gcd(c-1,|G/H|)/gcd(c,|G/H|). Call> this quotient 1/D. Then we have>> blahblahblahblah>> is true iff>> there exist relatively prime positive integers m_1, n_1 such that>> 0 <= |G/H|/order(X) * n_1 <= |G/H|-1 and> m_1*order(X) + n_1*(|G/H|/order(X) - e_1) = |G/H|-1,>> where e_1 is the unique integer satisfying>> 0 <= e_1 <= order(X) and> |G/H|/order(X) * Y = e_1*X>> _and_ there exist relatively prime positive integers m_2, n_2 such> that>> 0 <= |G/H|/(D*order(X)) * n_2 <= |G/H|-1 and> m_2*order(X)*D + n_2*(|G/H|/(D*order(X)) - e_2) = |G/H|-1,>> where e_2 is the unique integer satisfying>> 0 <= e_2 <= order(X)*D and> |G/H|/(D*order(X)) * X = e_2*Y.>> Even if these equations can't be expressed as a set of fewer> equations, is there anything we can say number-theoretically about> |G/H|, |H|, or order(X) and order(Y) (besides the _obvious_> inequalities obtained from simply combining these equations)??? Anyone> pro'cient with Mathematica want to give this one a whirl and post any> simpli'cations you discover? (I don't have access to it or any other> symbolic algebra package from where I currently work).> =I think the O.P. mistyped (or just miscapitalized). Of course c should be >aninteger, and X is >the coset of H in G containing x,> Y is >the coset of >H in Gcontaining y,> etc.... =Wondering if anyone can help me with this. I'm designing a computer program for a stone masonry company to pricethe cost of stones.Basically what you need is the length, width and height and multiplythem by a m3 rate.So 1.5m x 200mm x 200mm x 1000 would cost 60. Program >does this'ne.What they really want it to calculate is the cost of a radiated pieceof stone shaped like a bracket O ( O. Obviously the size of the stonewill need to be larger to achieve this shape when it is cut out. I'llknow the length, width and height of the 'nished stone but is thereanyway to work out the Length, width and height of the stone requiredto achieve this shape.Graeme == I'm designing a computer program for a stone masonry company to price> the cost of stones.>> Basically what you need is the length, width and height and multiply> them by a m3 rate.>> So 1.5m x 200mm x 200mm x 1000 would cost 60. >Program does this> 'ne.>> What they really want it to calculate is the cost of a radiated piece> of stone shaped like a bracket O ( O. Obviously the size of the stone> will need to be larger to achieve this shape when it is cut out. I'll> know the length, width and height of the 'nished stone but is there> anyway to work out the Length, width and height of the stone required> to achieve this shape.It is not clear to me what you mean by >a radiated piece of stone shapedlike a bracket O(O>.Could you post one or more photographs of such things to a website, thenpost links to the pictures to this thread? Or post the pictures to a photoblog server such ashttp://www.phlog.net/and then post a link here? Or maybe there are already pictures on the Web.-- Clive Toothhttp://www.clivetooth.dk ==> Wondering if anyone can help me with this.>> I'm designing a computer program for a stone masonry company to price> the cost of stones.>> Basically what you need is the length, width and height and multiply> them by a m3 rate.>> So 1.5m x 200mm x 200mm x 1000 would cost 60. >Program does this> 'ne.>> What they really want it to calculate is the cost of a radiated piece> of stone shaped like a bracket O ( O. Obviously the size of the stone> will need to be larger to achieve this shape when it is cut out. I'll> know the length, width and height of the 'nished stone but is there> anyway to work out the Length, width and height of the stone required> to achieve this shape.Not clear what you mean by a >radiated stone>. I assume uniformthickness, two sides are bounded by concentric circles (or sectionsthereof), the other two sides correspond to radii of the two circles.Assume that the outer radius is R, the inner radius is r, and theangle at the centre is A (presumably less than 90 degrees, butassumed to be between 0 and 180 degrees, just to be safe).Then the width is R - r cos(A/2), and the length is 2*R sin(A/2). == Wondering if anyone can help me with this. > I'm designing a computer program for a stone masonry company to price> the cost of stones.> Basically what you need is the length, width and height and multiply> them by a m3 rate.> So 1.5m x 200mm x 200mm x 1000 would cost 60. >Program does this> 'ne.> What they really want it to calculate is the cost of a radiated piece> of stone shaped like a bracket O ( O. Obviously the size of the stone> will need to be larger to achieve this shape when it is cut out. I'll> know the length, width and height of the 'nished stone but is there> anyway to work out the Length, width and height of the stone required> to achieve this shape.Sure. I'm bad at ASCII art, but let me describemy thinking.1. Give your bracket some 'nite thickness, andorient it so that the bottom of the arc is vertical,i.e. the bottom edge is horizontal. A / >> / / B / / | | | | +--+ C D2. Label the four corners A, B, C and D as shown.3. Draw a rectangle that encloses this shape. Thewidth will go from C to B, and the height from Dto A, plus allowances for losses during cutting.This ring is assumed to be an arc of a circle. Detailsdepend on how many degrees of arc, radius, andthickness. Suppose the arc is d degrees and innerradius r1, outer radius r2.Then the width of your box is r2 - r1 cos(d),and the height is r2 sin(d). Most computers andcalculators use radians, not degrees for cos andsin, so you have to multiply d by pi/180.Example: 20 cm = 0.2 m thick arc of 15 degrees. Innerradius 10 m, outer radius 9.8 m. box width = 10 - 9.8*cos(15*pi/180) = 0.534 m box height = 10*sin(15*pi/180) = 2.59 m.As I said, allow for cutting losses whatever thoseare. Add a few percent, or a few cm as appropriate. - Randy == Wondering if anyone can help me with this. > I'm designing a computer program for a stone masonry company to price> the cost of stones.> Basically what you need is the length, width and height and multiply> them by a m3 rate.> So 1.5m x 200mm x 200mm x 1000 would cost 60. >Program does this> 'ne.> What they really want it to calculate is the cost of a radiated piece> of stone shaped like a bracket O ( O. Obviously the size of the stone> will need to be larger to achieve this shape when it is cut out. I'll> know the length, width and height of the 'nished stone but is there> anyway to work out the Length, width and height of the stone required> to achieve this shape.> GraemeThe simplest way is that you can imagine the >'nished> stone lyingcompletely within a rectangular box. The dimension of the box willthen be the same size as the required >larger> stone.This is very simpli'ed, and it assumes that one >larger> stone iscut into one >'nished> stone.If a >larger> stone is actually cut into multiple >'nished> stonethen you'd be solving a completely different problem called>close-packing>... which can be much more complex... depending onthe actual shape of the object and the cutting method used.Hopes this help.--KMaster-- =...> What they really want it to calculate is the cost of a radiated piece> of stone shaped like a bracket O ( O. Obviously the size of the stone> will need to be larger to achieve this shape when it is cut out. I'll> know the length, width and height of the 'nished stone but is there> anyway to work out the Length, width and height of the stone required> to achieve this shape.... > 1. Give your bracket some 'nite thickness, and> orient it so that the bottom of the arc is vertical,> i.e. the bottom edge is horizontal.> A> / >>> / / B> / /> | |> | |> +--+> C D> 2. Label the four corners A, B, C and D as shown.> 3. Draw a rectangle that encloses this shape. The> width will go from C to B, and the height from D> to A, plus allowances for losses during cutting....> Suppose the arc is d degrees and inner> radius r1, outer radius r2.> Then the width of your box is r2 - r1 cos(d),> and the height is r2 sin(d)....> Example: 20 cm = 0.2 m thick arc of 15 degrees. Inner> radius 10 m, outer radius 9.8 m.> box width = 10 - 9.8*cos(15*pi/180) = 0.534 m> box height = 10*sin(15*pi/180) = 2.59 m....As I 'gure it, your procedure gives a rectangle abouttwice as large as needed. Rather than having CD horizontal,let B,D lie in a horizontal line that is the bottom line ofa rectangle. Let desired arc angle be a. Then a rectangle of height h = r2 - r1 cos(a/2) and width w = 2 r2 sin(a/2) encloses the part. For above example this is h=.284, w=2.611, area=.74, vs. your area 1.38.-jiw == Wondering if anyone can help me with this. > I'm designing a computer program for a stone masonry company to price> the cost of stones.> Basically what you need is the length, width and height and multiply> them by a m3 rate.> So 1.5m x 200mm x 200mm x 1000 would cost 60. >Program does this> 'ne.> What they really want it to calculate is the cost of a radiated piece> of stone shaped like a bracket O ( O. Obviously the size of the stone> will need to be larger to achieve this shape when it is cut out. I'll> know the length, width and height of the 'nished stone but is there> anyway to work out the Length, width and height of the stone required> to achieve this shape.> > Graeme> The simplest way is that you can imagine the >'nished> stone lying> completely within a rectangular box. The dimension of the box will> then be the same size as the required >larger> stone.> This is very simpli'ed, and it assumes that one >larger> stone is> cut into one >'nished> stone.> If a >larger> stone is actually cut into multiple >'nished> stone> then you'd be solving a completely different problem called>close-packing>... which can be much more complex... depending on> the actual shape of the object and the cutting method used.I did a little bit of consulting once in the steel-cuttingindustry. As in steel, probably the stone cutters would eitherplan for multiple cuts as you say to meet different requirementsfor a whole order, or they'd put large pieces of scrap backinto stock to reuse, OR they'd sell back to their supplier.The last two options are often included as negative termsin optimization models. - Randy =School, Advanced, and Challenge. Please visit us athttp://math.smsu.edu/~les/POTW.html[I continue to slog through the backlog.] =This looks to be an interesting way to prove the given propositionwithout using the axiom of choice. I think if one assumes axiom ofchoice, which is required is the sets are in'nite, the followingargument can be made. If there exists an injection from X to Y, thencard Y >= card X. Similarly,card X >= card Y. Therefore card X = card Y. == This looks to be an interesting way to prove the given proposition> without using the axiom of choice. I think if one assumes axiom of> choice, which is required is the sets are in'nite, the following> argument can be made. If there exists an injection from X to Y, then> card Y >= card X. Similarly,> card X >= card Y. Therefore card X = card Y.>Circular reasoning as the Cantor-Bernstein theorem is used to prove that. ==> This looks to be an interesting way to prove the given proposition> without using the axiom of choice. I think if one assumes axiom of> choice, which is required is the sets are in'nite, the following> argument can be made. If there exists an injection from X to Y, then> card Y >= card X. Similarly,> card X >= card Y. Therefore card X = card Y.> >> Circular reasoning as the Cantor-Bernstein theorem is used to prove that.I believe Cantor proved the result as a consequence of his well-ordering>theorem>, which thus relies on the Axiom of Choice. The result provedwithout such dependence is more often called the Schroeder-BernsteinTheorem.For a concise proof >without Choice> see:http://www.wikipedia.org/wiki/Cantor-Bernstein-Schroeder_ theorem =I 'nished reading >The Mystery of the Aleph> by Amir Aczel and Istill have a few questions on some concepts that are not clear to me.1) The diagonal test that Cantor used shows that the irrationals areuncountable. Does it necessarily follow that the in'nite set of theirrationals is greater than the in'nite set of the integers? It seemsto me that there is something missing here. How can one conclude thatthe cardinality of the irrationals is greater than that of theintegers only because the irrationals are uncountable? Theuncountability of the irrationals is one thing, saying that there aremore irrationals than integers is a different matter (I think).2) I 'nd the following statement puzzling. On page 141: >While there is no greatest number - you can always add one to anynumber and get a larger one - there is still the possibility of theexistence of a number larger than all the 'nite numbers> So basically this means there is the possibility of the existence ofa number that is larger than the greatest number that doesn't exist(because there is no greatest number).3) On page 142 there is the question: >What is the cardinality of anin'nite set?> My immediate answer is: in'nity. But it is shown later that aleph0< aleph1, a concept that boggles my mind. How can aleph1 actuallyexist?! How can one in'nite set be greater than another in'nite set?This is obviously not clear to me. The set of the integers have noend, there is no greatest integer so they go on forever. Same thingwith the set of irrationals (regardless of the fact that they areuncountable). So how can there be more irrationals than integers whenboths sets have no end?4) On page 150: >The act of forming the power set - the set of all subsets - alwaysproduces a bigger set than the original, a set of a highercardinality.> This statement makes perfect sense for 'nite sets. Now, how canthis apply for in'nite sets? If you take an in'nite set how can thepower set of an already in'nite set be a bigger set?5) On page 218: >We cannot reach the 'rst in'nite cardinal by any mathematicaloperation (addition, multiplication, or exponentiation) if we startwith a 'nite number such as 've, or 've hundred trillion for thatmatter. The 'rst in'nite cardinal, aleph0, is thus unreachable, orinaccessible from any of the 'nite cardinals (any given number is a'nite cardinal). Once we have reached the lowest in'nite cardinal,aleph0, however, we can obtain higher in'nite cardinals byexponentiation...> Let's look at the beginning of the three statements above: a) We cannot reach the 'rst in'nite cardinal.... b) The 'rst in'nite cardinal, aleph0, is thus unreachable... c) Once we have reached the lowest in'nite cardinal... Doesn't anybody see a contradiction with the third statement??? Howcan we reach the lowest in'nite cardinal if the author says it isunreachable?!?!?!A few bold (and possibly silly) questions.....Could it be possible that there is one and only one in'nity? Could itbe that Godel and Cohen concluded that the CH is undecidable becausealeph1 doesn't exist? After reading the book and reading about Canto'swork on other books I cannot grasp the notion of one in'nity beinggreater than another. I don't think I am alone on my confusion. ==> I 'nished reading >The Mystery of the Aleph> by Amir Aczel and I> still have a few questions on some concepts that are not clear to me.Never read that book myself, but Aczel is supposed to be a verygood popular-science writer. I think I read something by himabout either Fermat's Last Theorem or zero; one or the other.I'm just going to take these in order. Hope this helps.> 1) The diagonal test that Cantor used shows that the irrationals are> uncountable. Does it necessarily follow that the in'nite set of the> irrationals is greater than the in'nite set of the integers?Yes. Why? Because that's how the concept >greater than> is de'nedfor in'nite sets. The integers are countable, and the irrationalsare more-than-countable (uncountable). Thus there are >more> of them.There is a better de'nition than just >countable-uncountable,> ofcourse, involving bijections between sets, but that requires a bitof extra math.> How can one conclude that> the cardinality of the irrationals is greater than that of the> integers only because the irrationals are uncountable?Because the number (cardinality) of irrationals is aleph_1, and thenumber of integers is aleph_0. aleph_1 > aleph_0. That's all.> 2) I 'nd the following statement puzzling. On page 141:>>While there is no greatest number - you can always add one to any> number and get a larger one - there is still the possibility of the> existence of a number larger than all the 'nite numbers>Consider the set X of all real numbers less than 1: While there is no greatest number in X - you can always take 1-(1-n)/2 and get a larger number than n - there is still the possibility of the existence of a number larger than all the members of X.Namely, the number 1 itself. Or 2. Or 42.> So basically this means there is the possibility of the existence of> a number that is larger than the greatest number ^ 'niteYes. :) Lots of them. They're the cardinals and ordinals, or inplainspeak, >in'nite numbers.>> 3) On page 142 there is the question: >What is the cardinality of an> in'nite set?>>> My immediate answer is: in'nity.But mathematicians are never satis'ed with English words when Greekones would do as well, or better. What do you really *mean* by>in'nity>? You mean, >the size of some in'nite set A.> But thereare lots of in'nite sets, and as Cantor showed (question 1), someof them are >bigger> than others! So there must be multiple different>in'nities,> and they all need names.Aleph_0 (that's aleph with a subscript zero) is the name mathematiciansgive to the smallest of these in'nities, the cardinality of the setof natural numbers.> But it is shown later that aleph0> < aleph1, a concept that boggles my mind. How can aleph1 actually> exist?! How can one in'nite set be greater than another in'nite set?> This is obviously not clear to me. The set of the integers have no> end, there is no greatest integer so they go on forever. Same thing> with the set of irrationalsAha. But whereas the integers merely go *on* forever, the irrationalsalso go *in* forever! Think of writing down the decimal representationsof all the irrational numbers: once you start writing the digits of oneirrational number, you'll never 'nish even *that* number.(Rational numbers, on the other hand, don't go *in* forever. Eventually,every rational number starts repeating, and we can just make up a notationthat says, >this number repeats here>, and go on to the next one. That'swhy they're countable. Sort of. ;)> 4) On page 150:>>The act of forming the power set - the set of all subsets - always> produces a bigger set than the original, a set of a higher> cardinality.>>> This statement makes perfect sense for 'nite sets. Now, how can> this apply for in'nite sets? If you take an in'nite set how can the> power set of an already in'nite set be a bigger set?The power set of aleph_0 is aleph_1. And the power set of aleph_1 isaleph_2. And the power set of aleph_n is aleph_(n+1). Does that help?'Cause that's all you're gonna get for this question.> 5) On page 218:>>We cannot reach the 'rst in'nite cardinal by any mathematical> operation (addition, multiplication, or exponentiation) if we start> with a 'nite number such as 've, or 've hundred trillion for that> matter. The 'rst in'nite cardinal, aleph0, is thus unreachable, or> inaccessible from any of the 'nite cardinals (any given number is a> 'nite cardinal). Once we have reached the lowest in'nite cardinal,> aleph0, however, we can obtain higher in'nite cardinals by> exponentiation...>>> Let's look at the beginning of the three statements above:>> a) We cannot reach the 'rst in'nite cardinal....> b) The 'rst in'nite cardinal, aleph0, is thus unreachable...> c) Once we have reached the lowest in'nite cardinal...>> Doesn't anybody see a contradiction with the third statement??? How> can we reach the lowest in'nite cardinal if the author says it is> unreachable?!?!?!We *can't* reach aleph_0 if we start with a 'nite number and do thingsto it. But, once we *do* reach it (by whatever means), we can easilyget to aleph_1 by taking the power set of aleph_0, and so on up theladder.Consider: We can't get from Beijing to Sydney by walking, or running, or rolling, or driving, or any other mode of land transport. But *once we have reached* Sydney, the trip to Brisbane via road is simple.Replace >land transport> by >ordinary arithmetic operation,> and>Beijing> and >Sydney> by >'ve hundred trillion> and >aleph_0>respectively, and you have the original statement. See how eventhough you can't get from Beijing to Sydney over land, it's stillpossible to talk about what you *can* reach from Sydney once you'rethere.> A few bold (and possibly silly) questions.....>> Could it be possible that there is one and only one in'nity?No.> Could it> be that Godel and Cohen concluded that the CH is undecidable because> aleph1 doesn't exist?Of course not! (Don't bother with the Continuum Hypothesis until...well... I've *never* bothered with it. So that should tell yousomething.)> After reading the book and reading about Canto[r]'s> work on other books I cannot grasp the notion of one in'nity being> greater than another. I don't think I am alone on my confusion.You're not. But after reading several more books, or (even better)talking to your maths professor or a geeky friend, you will eventuallyget the hang of these >multiple in'nities,> and then you'll wonderwhether you ever really thought otherwise. :)Happy math!-Arthur == I 'nished reading >The Mystery of the Aleph> by Amir Aczel and I> still have a few questions on some concepts that are not clear to me.> 1) The diagonal test that Cantor used shows that the irrationals are> uncountable. Does it necessarily follow that the in'nite set of the> irrationals is greater than the in'nite set of the integers? It seems> to me that there is something missing here. How can one conclude that> the cardinality of the irrationals is greater than that of the> integers only because the irrationals are uncountable? The> uncountability of the irrationals is one thing, saying that there are> more irrationals than integers is a different matter (I think).What do you mean by >greater than>? Do you mean >has a largercardinality than>, or do you mean something else?An uncountable set certainly has a larger cardinality than a countableset, if we assume the axiom of choice. One of the consequences of AC isthe trichotomy law: given any two sets A and B, exactly one of thefollowing statements must hold: (1) There is an injection f: A -> B, but no injection from B into >A. (2) There is an injection f: B -> A, but no injection from A into >B. (3) There is an injection in each direction (and therefore, as a consequence of the Cantor-Bernstein theorem, there is also a bijection).In case (1) we say (by de'nition) that B has a larger cardinality thanA. In case (2) we say that A has a larger cardinality than B. In case(3) we say the cardinalities are equal.> 2) I 'nd the following statement puzzling. On page 141:>While there is no greatest number - you can always add one to any> number and get a larger one - there is still the possibility of the> existence of a number larger than all the 'nite numbers>> So basically this means there is the possibility of the existence of> a number that is larger than the greatest number that doesn't exist> (because there is no greatest number).No, it means that some numbers (trans'nite cardinal numbers) are not'nite. Each trans'nite cardinal is larger than any 'nite number, butthere is no largest trans'nite cardinal number.> 3) On page 142 there is the question: >What is the cardinality of an> in'nite set?>> My immediate answer is: in'nity. But it is shown later that aleph0>< aleph1, a concept that boggles my mind. How can aleph1 actually> exist?! How can one in'nite set be greater than another in'nite set?See the explanation above involving the trichotomy law. Basically, itmeans that no matter how we pair off members from A and B, there willalways be some members of B that are left unaccounted for (case 1 above).> This is obviously not clear to me. The set of the integers have no> end, there is no greatest integer so they go on forever. Same thing> with the set of irrationals (regardless of the fact that they are> uncountable). So how can there be more irrationals than integers when> boths sets have no end?Consider the mapping f: Q -> R>>Q (a mapping from the rationals into theirrationals) de'ned by f(x) = x + pi for each rational x. Obviously,f is an injection from the rationals into the irrationals. However, itis not a bijection; some irrationals are left unaccounted for. Forexample, sqrt(2), being an algebraic number, cannot possibly be in therange of f, since f(x) is transcendental for every rational x.This alone is not suf'cient to show that card(Q) < card(R>>Q), but theCantor diagonal proof (and a bit of extra reasoning for this particularcase) can show that there is no possible mapping f: Q -> R>>Q such thatthe range of f is R>>Q. That's what it means to say that thecardinalities are different.> 4) On page 150:>The act of forming the power set - the set of all subsets - always> produces a bigger set than the original, a set of a higher> cardinality.>> This statement makes perfect sense for 'nite sets. Now, how can> this apply for in'nite sets? If you take an in'nite set how can the> power set of an already in'nite set be a bigger set?No doubt the book discusses Cantor's theorem for power sets. Thattheorem shows that if X is any set and f: X -> P(X) is any mapping from Xto its power set, then there exists y in P(X) (depending on f) such thaty is not in the range of f. This shows that no bijection exists betweenX and P(X).> 5) On page 218:>We cannot reach the 'rst in'nite cardinal by any mathematical> operation (addition, multiplication, or exponentiation) if we start> with a 'nite number such as 've, or 've hundred trillion for that> matter. The 'rst in'nite cardinal, aleph0, is thus unreachable, or> inaccessible from any of the 'nite cardinals (any given number is a> 'nite cardinal). Once we have reached the lowest in'nite cardinal,> aleph0, however, we can obtain higher in'nite cardinals by> exponentiation...>> Let's look at the beginning of the three statements above:> a) We cannot reach the 'rst in'nite cardinal....> b) The 'rst in'nite cardinal, aleph0, is thus unreachable...> c) Once we have reached the lowest in'nite cardinal...> Doesn't anybody see a contradiction with the third statement??? How> can we reach the lowest in'nite cardinal if the author says it is> unreachable?!?!?!That's an unfortunate choice of words, but in fact we can reach the 'rstin'nite cardinal by using operations other than the ones listed(addition, multiplication, or exponentiation).Even that statement, however, is a bit misleading, because we need toresort to the axioms of set theory (one of which is the axiom o'n'nity) in order to assert that an in'nite set exists in the 'rstplace.> A few bold (and possibly silly) questions.....> Could it be possible that there is one and only one in'nity? Could it> be that Godel and Cohen concluded that the CH is undecidable because> aleph1 doesn't exist? After reading the book and reading about Canto's> work on other books I cannot grasp the notion of one in'nity being> greater than another. I don't think I am alone on my confusion.If we accept that at least one in'nite set exists, and if we also acceptthat for every set there is a power set, then it follows from Cantor'stheorem that there is more than one in'nity.-- Dave SeamanJudge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. == I 'nished reading >The Mystery of the Aleph> by Amir Aczel and I> still have a few questions on some concepts that are not clear to me.> 1) The diagonal test that Cantor used shows that the irrationals are> uncountable. Does it necessarily follow that the in'nite set of the> irrationals is greater than the in'nite set of the integers? It seems> to me that there is something missing here. How can one conclude that> the cardinality of the irrationals is greater than that of the> integers only because the irrationals are uncountable? The> uncountability of the irrationals is one thing, saying that there are> more irrationals than integers is a different matter (I think).The idea of one in'nity being larger than another is based on whether or not you can construct a one-to-one onto map from one to the other. If you can, they are in some sense >the same>, if not, there must be more of one than the other. The trick is to FORGET your conventional notion of what in'nity means.In the case of rationals and irrationals, F: Q -> ~Q can be 1-1, but not onto. As a result, there must be more irrationals.> 2) I 'nd the following statement puzzling. On page 141:>While there is no greatest number - you can always add one to any> number and get a larger one - there is still the possibility of the> existence of a number larger than all the 'nite numbers>For example, {aleph_0}+1, {aleph_0}+2, etc. It seems meaningless, but it works.> So basically this means there is the possibility of the existence of> a number that is larger than the greatest number that doesn't exist> (because there is no greatest number).Yes.> 3) On page 142 there is the question: >What is the cardinality of an> in'nite set?>Greater than or equal to aleph_0.> My immediate answer is: in'nity. But it is shown later that aleph0> < aleph1, a concept that boggles my mind. How can aleph1 actually> exist?! How can one in'nite set be greater than another in'nite set?> This is obviously not clear to me. The set of the integers have no> end, there is no greatest integer so they go on forever. Same thing> with the set of irrationals (regardless of the fact that they are> uncountable). So how can there be more irrationals than integers when> boths sets have no end?See answer above regarding maps.> 4) On page 150:>The act of forming the power set - the set of all subsets - always> produces a bigger set than the original, a set of a higher> cardinality.>> This statement makes perfect sense for 'nite sets. Now, how can> this apply for in'nite sets? If you take an in'nite set how can the> power set of an already in'nite set be a bigger set?Again by mappings. Also, there are levels of in'nity. You can't think of it as a single >value>.> 5) On page 218:> >We cannot reach the 'rst in'nite cardinal by any mathematical> operation (addition, multiplication, or exponentiation) if we start> with a 'nite number such as 've, or 've hundred trillion for that> matter. The 'rst in'nite cardinal, aleph0, is thus unreachable, or> inaccessible from any of the 'nite cardinals (any given number is a> 'nite cardinal). Once we have reached the lowest in'nite cardinal,> aleph0, however, we can obtain higher in'nite cardinals by> exponentiation...>> Let's look at the beginning of the three statements above:> a) We cannot reach the 'rst in'nite cardinal....> b) The 'rst in'nite cardinal, aleph0, is thus unreachable...> c) Once we have reached the lowest in'nite cardinal...> Doesn't anybody see a contradiction with the third statement??? How> can we reach the lowest in'nite cardinal if the author says it is> unreachable?!?!?!You can't, unless you jump to the end. It's kind of like the idea of the sequence 1/2, 1/4, 1/8, 1/16, ... never reaching 0. Once you go below 0, however....> A few bold (and possibly silly) questions.....> Could it be possible that there is one and only one in'nity? As far as I'm concerned, no. > Could it> be that Godel and Cohen concluded that the CH is undecidable because> aleph1 doesn't exist?I'd have to look up CH, but I suspect not.> After reading the book and reading about Canto's> work on other books I cannot grasp the notion of one in'nity being> greater than another. I don't think I am alone on my confusion.You aren't. Sometimes you simply have to let go of your preconceptions and work with the de'nitions rather than intuition.-- Will Twentyman == I 'nished reading >The Mystery of the Aleph> by Amir Aczel and I> still have a few questions on some concepts that are not clear to me.>> 1) The diagonal test that Cantor used shows that the irrationals are> uncountable. Does it necessarily follow that the in'nite set of the> irrationals is greater than the in'nite set of the integers? It seems> to me that there is something missing here. How can one conclude that> the cardinality of the irrationals is greater than that of the> integers only because the irrationals are uncountable? The> uncountability of the irrationals is one thing, saying that there are> more irrationals than integers is a different matter (I think).Suppose you want to compare the cardinality of two sets, say a set ofpeople and a set of chairs in a stadium. Here are two ways to do this: (a) count the number of people, count the number of chairs, and compare the answers. (b) have each person sit down in one of the chairs, and see if you have any people or chairs left over (that is, attempt to establish a >one-to-one correspondence> between the two sets.)With 'nite sets, these two methods produce the same result. Within'nite sets, they may differ. Call the sets A and B. You maydecide that your two sets are both in'nite (so have the same size bymethod (a)), but that no matter how you try to establish a one-to-onecorrespondence, there are always some members of set B, let's say,left over (so B is bigger, according to method (b)). Thus there is amatter of de'nition: you can either just say that both sets arein'nite, or you can convey more information by saying that, whileboth sets are in'nite, B has greater cardinality. Conveying moreinformation is good, so method (b) is the right choice.By de'nition, a set is >uncountable> if, no matter how you try toestablish a one-to-one correspondence between it and the set o'ntegers, there are always members of your set left over. Thus byde'nition, an uncountable set has larger cardinality than theintegers.(I have nothing to add to the other posters' answers to your otherquestions.)-- J. H. PalmieriUniversity of Washington >http://www.math.washington.edu/~palmieri/Seattle, WA 98195-4350 == I 'nished reading >The Mystery of the Aleph> by Amir Aczel and I> still have a few questions on some concepts that are not clear to me.>> 1) The diagonal test that Cantor used shows that the irrationals are> uncountable. Does it necessarily follow that the in'nite set of the> irrationals is greater than the in'nite set of the integers? It seems> to me that there is something missing here. How can one conclude that> the cardinality of the irrationals is greater than that of the> integers only because the irrationals are uncountable? The> uncountability of the irrationals is one thing, saying that there are> more irrationals than integers is a different matter (I think).>The proof makes an assumption that a certain list is exhaustive. It isthis assumption that is violated when the diagonal element is constructed.>> 2) I 'nd the following statement puzzling. On page 141:>>While there is no greatest number - you can always add one to any> number and get a larger one - there is still the possibility of the> existence of a number larger than all the 'nite numbers>>> So basically this means there is the possibility of the existence of> a number that is larger than the greatest number that doesn't exist> (because there is no greatest number).>The issue of a >completed in'nity> leads to the issue of a >completedin'nity +1>. Analysts were using a completed in'nity before the modernera in which many branches in mathematics were characterized usingaxiomatic methods. So, once the mathematical investigation of sets began,it would only be natural to consider such possibilities.Another place that you will 'nd something like this is non-standardanalysis. I believe Abraham Robinson is credited with this non-Archimedean ordered'elds. Someone else might be able to refer you to a better search topic.You need to keep in mind that there is a certain venturesomeness amongmathematicians. They will investigate possibilities that arise to see ifthey make sense relative to some context. If anything, >puzzling> is >asgood a motivation to continue developing an idea as any other.>> 3) On page 142 there is the question: >What is the cardinality of an> in'nite set?>>> My immediate answer is: in'nity. But it is shown later that aleph0> < aleph1, a concept that boggles my mind. How can aleph1 actually> exist?! How can one in'nite set be greater than another in'nite set?> This is obviously not clear to me. The set of the integers have no> end, there is no greatest integer so they go on forever. Same thing> with the set of irrationals (regardless of the fact that they are> uncountable). So how can there be more irrationals than integers when> boths sets have no end?>That is why Cantor's proof begins with the assumption of an exhaustivelist. The property involved is >one-to-one> or >injection.> You canimagine a function Of(n) = n + pi'. So every integer is uniquelyassociated with an irrational number. It is much harder to imagine afunction uniquely associating each irrational number with an integer.Fame and fortune awaits if you do it.>> 4) On page 150:>>The act of forming the power set - the set of all subsets - always> produces a bigger set than the original, a set of a higher> cardinality.>>> This statement makes perfect sense for 'nite sets. Now, how can> this apply for in'nite sets? If you take an in'nite set how can the> power set of an already in'nite set be a bigger set?>I probably should defer to others for this discussion.>> 5) On page 218:>>We cannot reach the 'rst in'nite cardinal by any mathematical> operation (addition, multiplication, or exponentiation) if we start> with a 'nite number such as 've, or 've hundred trillion for that> matter. The 'rst in'nite cardinal, aleph0, is thus unreachable, or> inaccessible from any of the 'nite cardinals (any given number is a> 'nite cardinal). Once we have reached the lowest in'nite cardinal,> aleph0, however, we can obtain higher in'nite cardinals by> exponentiation...>>> Let's look at the beginning of the three statements above:>> a) We cannot reach the 'rst in'nite cardinal....> b) The 'rst in'nite cardinal, aleph0, is thus unreachable...> c) Once we have reached the lowest in'nite cardinal...>> Doesn't anybody see a contradiction with the third statement??? How> can we reach the lowest in'nite cardinal if the author says it is> unreachable?!?!?!>Analysts were already using in'nity. Cantor was trying to classify whatwas already being taken for granted. Once he had convinced himself of>in'nities> rather than >in'nity> reachability is a for the mathematical disciplinethat restricts itself with respect to such usage. Once again, someoneelse can probably offer a better search string.>> A few bold (and possibly silly) questions.....>> Could it be possible that there is one and only one in'nity?There are currently investigations into equality-free logics that do havebounded models. I do not know much about this formal development. But,there seem to be models bounded at aleph_0. So >in'nity> gets no largerunder those constraints.However, no one has compiled any compelling evidence to suggest that thereis only one in'nity. Acceptance of such a paradigm would be hard toobtain.> Could it> be that Godel and Cohen concluded that the CH is undecidable because> aleph1 doesn't exist?Consider this for a moment.In logic, we speak of negating a proposition. But, in the logic ofarti'cial neural networks, one would speak of complementation withrespect to some given switching function.In logic, we speak of DeMorgan conjugation. It is usually expressed interms of the relationship between conjunction and disjunction. But, theset-theoretic analogues are intersection and union.Now, when making the open characterization of a topology, we might write,>The arbitrary union of open sets is open.>>The 'nite intersection of open sets is open.>while the closed characterization would be given by>The arbitrary intersection of closed sets is closed.>>The 'nite union of closed sets is closed.>Looking at the two pairs of sentences, one notices a certainjuxtaposition. Of course, closed and open are de'ned relative to oneanother with respect to set complements. And, we have already noted therelationship between DeMorgan conjugation and the two set operations,union and intersection.For my part, I am inclined to view >arbitrariness> and >'nite> as >havingopposite meanings in a precise mathematical sense. But, that is just me.However, I suppose one could count one's way to >arbitrariness> if it isreachable through such means.> After reading the book and reading about Canto's> work on other books I cannot grasp the notion of one in'nity being> greater than another. I don't think I am alone on my confusion.It is not a simple question.There are those who consider the ontological implications simply becauseit is puzzling. In the meantime, everyone else uses in'nity in acceptedmethodologies that further their mathematical investigations. Now andthen, however, a methodology pops up that no one expected. It usuallyleads to interesting mathematics.Good luck with it.:-)mitch ==> So basically this means there is the possibility of the existence of>> a number that is larger than the greatest number> ^> 'nite> Yes. :) Lots of them. They're the cardinals and ordinals, or in> plainspeak, >in'nite numbers.>No. These are larger than *every* 'nite number. It's meaninglessto say they're >larger than the greatest 'nite number>, becausethere *isn't* any greatest 'nite number. ==Enrique> escreveu na mensagemWhen we say that the set A is bigger than the set B, what we mean is thatthere is a function f : A -> B that is one-to-one but there is not afunction g : A -> B that is onto. I think that many of the problems that >youhave is because when you read >A is bigger that B>, you have a intuitiveunderstand of it that is different from the >de'nition>: that there is afunction f : A -> B that is one-to-one but there is not a function g : A ->B that is onto.For example, the af'rmation There is a function f : N -> R that is one-to-one but there is not a function g : N -> R that is ontoprobably doesn't puzzles you, but the af'rmation R is bigger than Nprobably does. But this two af'rmation are exactly the same, they only usedifferent words. The last one does use a word, >bigger>, that as a >intuitivemeaning for you that seems to contradict the meaning that the authorintended (the one about functions). Well, at least it's what I think thatcan be the problem.If you allow me, I would suggest that when you 'nd a af'rmation with theword >bigger> and it puzzles you, try to rewrite the af'rmation usingfunctions instead the word bigger. The af'rmation with functions probablywould not puzzles you.> 1) The diagonal test that Cantor used shows that the irrationals are> uncountable. Does it necessarily follow that the in'nite set of the> irrationals is greater than the in'nite set of the integers?Yes, but let's not use the word >bigger>. It follows that there is afunction f : R>>Q -> R that is one-to-one but there is not a function g :R>>Q -> R that is onto.> But it is shown later that aleph0> < aleph1, a concept that boggles my mind. How can aleph1 actually> exist?! How can one in'nite set be greater than another in'nite set?> This is obviously not clear to me. The set of the integers have no> end, there is no greatest integer so they go on forever. Same thing> with the set of irrationals (regardless of the fact that they are> uncountable). So how can there be more irrationals than integers when> boths sets have no end?Once again, that means there is a function f : Z -> R>>Q that is one-to-onebut there is not a function g : Z -> R>>Q that is onto.Another way to thing is this. It is possible to construct pars in which amember is a integer and the other member is a irrational, in such a way >thatfor each integer exist one irrational that do a par with him, and fordifferent integers, the irrationals that do par with them are different(this is, if the pars are (a,x) and (b,y) with a and b in Z and x and y inR>>Q and if a =/= b then x =/= y). But, after we have done the pares for >allintegers, there are some irrationals left out (that made no par withintegers). So, there are >more> (this is a intuitive expression) >irrationalsthat integers.>The act of forming the power set - the set of all subsets - always> produces a bigger set than the original, a set of a higher> cardinality.>>> This statement makes perfect sense for 'nite sets. Now, how can> this apply for in'nite sets? If you take an in'nite set how can the> power set of an already in'nite set be a bigger set?Once again, that means there is a function f : A -> P(A) that is one-to-onebut there is not a function g : A -> P(A) that is onto (P(A) is the powerset of the set A).Or, in terms of pars, if we do pares of elements of A with elements of P(A)(with the above rules), then there are some elements of P(A) left out (thatmade no par with elements of A).>We cannot reach the 'rst in'nite cardinal by any mathematical> operation (addition, multiplication, or exponentiation) if we start> with a 'nite number such as 've, or 've hundred trillion for that> matter. The 'rst in'nite cardinal, aleph0, is thus unreachable, or> inaccessible from any of the 'nite cardinals (any given number is a> 'nite cardinal). Once we have reached the lowest in'nite cardinal,> aleph0, however, we can obtain higher in'nite cardinals by> exponentiation...>>> Let's look at the beginning of the three statements above:>> a) We cannot reach the 'rst in'nite cardinal....> b) The 'rst in'nite cardinal, aleph0, is thus unreachable...> c) Once we have reached the lowest in'nite cardinal...>> Doesn't anybody see a contradiction with the third statement??? How> can we reach the lowest in'nite cardinal if the author says it is> unreachable?!?!?!The author says that if is unreachable if we start with a 'nite number andwe use only operations. So, it is unreachable by this method, but it may >notbe unreachable by other methods.> Could it be possible that there is one and only one in'nity? Could it> be that Godel and Cohen concluded that the CH is undecidable because> aleph1 doesn't exist? After reading the book and reading about Canto's> work on other books I cannot grasp the notion of one in'nity being> greater than another. I don't think I am alone on my confusion.That depends of the de'nitions that one uses. If one says that The in'nite of a in'nite set A is bigger that the in'nite of a in'nite set B if and only if there is a function f : A -> B that is one-to-one but there is not a function g : A -> B that is ontothen it has been proved that exists more than one in'nite. But notice thatthe af'rmation >there exists at least two (different) in'nites>actuallymeans >there exists two in'nite set A and B such that there is a function >f: A -> B that is one-to-one but there is not a function g : A -> B that isonto>. This last formulation of the af'rmation probably doesn't grasp >you.I hope I have helped.Sorry my english. Jaime Gaspar ______________________________ Homepage: www.jaimegaspar.com ==>Suppose you want to compare the cardinality of two sets, say a set of>people and a set of chairs in a stadium.>Reminds me of a classic conundrum wherewith I tormented my young nephew (then 10-12 years old):An in'nite number of people are at a bus stop. A bus with an in'nite number of seats comes along; all the people take a seat, 'lling the bus. Then you run up and want to get on the bus. Is ther room for you on the bus?My nephew at 'rst insisted, irritatedly, that if there is room, some seat must have been left empty.-- Stephen J. Herschkorn herschko@rutcor.rutgers.edu ==How come no reply about my letter?It's because you're a ing kook, asswipe.--> 07:27 PM, Kenfrazier@cs.com (Editor of > The quote was less than a sentence. So.... three and a half >> sentences,>> rounded off to four. OK?>> I sent you the letter already well inside your strait jacketed > censored limitations. It was two and a half sentences from me and the > full quote misattributed to me in DOUBLE QUOTES in Sheaffer's original > SENTENCES taking up most of the 'rst column! Where do you come off > saying it was >less than a sentence>?>> Again my Henry Jamesian letter of TWO AND ONE HALF FULL SENTENCES > (Shylock's >pound of ¤esh>) by me is:>> -------------------------------------------------------------- --------->> To Skeptical Inquirer>> Please correct the false impression, the misinformation and > disinformation, using my 27. No.3 of Skeptical Inquirer in the column >Psychic Vibrations> by Robert Sheaffer entitled >Special Forces > Battle Giant Scorpions in Iraq.> Sheaffer's writing that I 'nd > objectionable and damaging to me because it falsely appears that I had > written and endorsed the cited misinformation is speci'cally:>>Jack Sarfatti passes on information that America's Ogreatest fear is > that Saddam will reverse-engineer the crashed alien spacecraft ... The > craft allegedly crashed during the Gulf War (1990 - 1991) or more > recently (possibly December 1998). This will be Iraq's Roswell. The US > is currently reverse-engineering the Roswell craft and fears Saddam's > scientists will catch up with or even go beyond the U.S. in one or > more areas. These areas of research include zero point, over-ratio or > gravimetric technology, which would allow for a tremendous advance, > allowing Iraq to become a leading power.'>>> curiously also least one > prominent Skeptical Inquirer reader told me gave him the false > impression that I had written it.>> (Physics, University of California)> sarfatti@pacbell.net>> -------------------------------------------------------------- --------- > --------------->> The quote was less than a sentence. So.... three and a half >> sentences,>> rounded off to four. OK?> That is three sentences + the incorrectly attributed quote I assume.> Please con'rm. =Ref: http://www.csicop.org/sb/9609/internet.html Skeptical Briefs newsletter : September 1996 Reality Check On the InternetMilton Rothman Some time ago, when I was new to computer networking, I subscribed toProdigy and spent many bemused hours corresponding with denizens oftheir physics bulletin board (see Foibles and Falicies from theDecember 1994 Skeptical Briefs). While many of the correspondents wereserious students interested in discussing real science, a large numberof them had notions of science derived from watching Star Trek andother science 'ction 'lms. Their idea of a good time was inventingtheories about traveling faster than light by the use of tachyons. Theidea that physics textbooks were more reliable than the moreinteresting fantasies of the future was a ludicrous thought worthy ofnothing but derision.At that time I thought that these were merely adolescents playingaround with their imaginative notions of science. In time, as they wentto school and learned real science they would grow out of it. So Ithought. However, my hopes were shattered when I graduated to AmericaOnline and discovered the Internet. There I found the same fantasies,masked by more sophisticated homepage techniques, created by adults,some with Ph.D.'s.A very elaborate homepage is called the Internet Science EducationProject (ISEP), a California non-pro't 501(3c) corporation. (For thosewho are not familiar with the Internet, a homepage is a page thatappears on the screen, created by some interested person, and accessedby a speci'c address -- one of those lengthy strings starting withyou can jump to other pages. This is what we do when searching orsur'ng the web.) On the ISEP homepage we are greeted by a picture of aravishing beauty who claims to be Lt. Alexandrova from Space ForceAcademy at the San Francisco Presidio in the year 2196. She iscommunicating with us by advanced quantum waves from the future. (Inphysics an advanced wave is a solution of a wave equation that lies inthe future light-cone of space-time. At present there is no physicalinterpretation to this wave.)Clearly somebody is having fun. The person in charge is Jack Sarfatti,Ph.D., >President of the Corporation.> Dr. Sarfatti uses the ScienceEducation Project to publicize his advanced ideas, which he callspost-modern physics. At the bottom of the homepage we 'nd a logo forthe Space Force Academy which we click to reach the next level (thenext page). Here we 'nd a large number of choices: Muse Magazine, PSIWars, Quantum Mind & Microtubules, Einstein-Podolsky-Rosen Animation,Examining some of the pages, and accumulating a large pile ofprintouts, we are able to distinguish the pattern of post-modernphysics. It is an interlocking set of theories centered on thenon-locality of quanta -- that is, on the observation that within acan appear to be in two places at the same time. It is also deeplyconcerned with quantum gravity and its possible uses.The post-modern enthusiasts claim that recent work in >anomalouscognition,> teleportation, and the relation between quantum gravity andconsciousness presages a revolution in physics analogous to thequantum-relativity revolution that took place at the end of thenineteenth century. It certainly would if true.Reading the theories found on these Internet pages we 'nd certaintechnical terms used repeatedly: quantum gravity, Bohm pilot waves,microtubules, qualia, etc. A typical sentence: >The qualia [i.e.,subjective mental experiences] are excitations in the macroscopiccoherent quantum Bohm mental `pilot wave' attached to the materialvibrations of `Frohlich collective modes' of electric dipoles in themicrotubules inside living cells.> Or, look at this one: >The Mind ofthe evolution in time of the three-dimensional space geometry of ourUniverse -- at least in Bohm's pilot wave theory of quantum gravitythat, according to Penrose and Nanopoulos, form the substratum of ourconsciousness.>Here we see a concatenation of perfectly legitimate physical concepts(and physicists) to form a string of words that convey very littlemeaning to my impoverished brain. Quantum gravity is a theory thatcombines quantum mechanics (the theory of small objects) with generalrelativity (the theory of gravity and the curvature of space). Many ofthe greatest physicists have worked on this, with a variety of results.Pilot waves were proposed by David Bohm to explain certain mysteriousphenomena stemming from the Einstein-Podolsky-Rosen paradox. (Bohm wasa great quantum theorist in the old days -- I read his quantummechanics textbook from cover to cover back in 1952.)An example of an incomprehensible observation that relates to pilotwaves is an experiment I did myself in 1976. (It's a rather simpleexperiment that can be done in any optics laboratory.) In thisexperiment, a beam of light is passed through a half-silvered mirrorinclined at 45 to the beam. Cut down the intensity of the >light sothat just one photon wavepacket passes through the mirror at one time.Quantum theory tells us that half of each wavepacket is re¤ected whilethe other half is transmitted. We know that this happens because if youbring the packets together in an interferometer, you do getinterference fringes, showing that both transmitted and re¤ected wavesgo around the interferometer. But if you detect the photons with twophotodetectors (A and B), you 'nd that if the re¤ected wave isdetected in one location by phototube A, the transmitted wave is notdetected at the same time by phototube B, and vice versa. How does onedetector (A) know not to trigger when the other (distant) detector (B)does trigger, even though both are being hit by exactly the same wave?This is very hard to explain by classical quantum concepts. To makesense out of this paradox, Bohm proposed that inside each quantum was a>pilot wave> that hid within one of the split wavepackets anddetermined which detector was going to trigger. For many yearsphysicists believed that pilot wave (hidden variable) theories wereuntenable, but later came to believe they were not so untenable. As aresult, the use of pilot waves is a possible way of explaining theobservations associated with the above experiment, just one of the manyexperiments that have a bearing on the Einstein-Podolsky-Rosen paradox.Quantum gravity theories are legitimate theories; the only problem isthat so far no one has 'gured out how to test any of these theories.But there is always hope. The mischief arises when you take a theorythat has no visible consequences and apply it as an explanation of aphenomenon such as consciousness. The post-modern physicists do this bystringing together a bunch of legitimate terms like beads on a string,piling conjecture on top of conjecture. It's great entertainment, butis it science?My fundamental objection to the use of quantum-gravity pilot waves toexplain consciousness is this: the authors of these theories provide nomechanism to explain how the sub-sub-microscopic entities control whathappens in the brain. What happens to pilot waves when a person dies?Do they disappear, or are they effective only when interacting with abrain that has a certain type of organization? And what were they doingduring all the billions of years before human brains came on the scene?Are we to assume that the pilot waves cause consciousness only whenthey meet a brain with a certain kind of organization? Perhaps it isthe organization that causes consciousness, and quantum-gravity pilotwaves are simply a bit of poetry. On another web page we 'nd the breathless news of a new breakthroughin space propulsion. Listen carefully: > . . . the quantum potential Qfound in Bohm's hidden variable version of quantum mechanics is able totransform ordinary protons into virtual `faster-than-light' tachyons.This would permit the construction of a new type of rocket engine thatwould enable low-cost highly fuel-ef'cient practical interstellar¤ight for large manned spacecraft.> Using tachyons as the propellant,a large spaceship could be pushed to velocities approaching the speedof light, using a relatively small amount of energy.My question is: how much energy does it take to generate a stream oftachyons? To provide a reasonable amount of thrust, the tachyon beammust have a certain amount of momentum. The relativistic relationbetween momentum and energy is surely the same for tachyons as it isin'nity as the ship approaches the speed of light. So from where do weget this high energy ef'ciency? (Besides, nobody has seen a tachyonyet.) AFTERTHOUGHT: It occurs to me that Scarfatti's Internet ScienceEducation Project with all its scienti'c double talk makes perfectsense if we consider it to be parody. It is an education project in thesense that it forces the reader to examine what he knows and decidewhether the writing makes sense or not. If the writing is intended topoke fun at the new-age, post-modern physics, it succeeds admirably. I't is really supposed to be serious, well, then . . . . Personally, Ihate it when I can't tell whether a writer is serious or not. It goesback to the time 'fty years ago when I wrestled with John W. Campbell,editor of Astounding Science Fiction, who presented a new loony idea ineach issue of his magazine. I hated to think that such a talented,intelligent person might be a bit less than totally sane. To save our sanity, true skepticism may be found on the Internet. Setyour web searcher to look for >skepticism,> and you will 'nd a largenumber of items, most of which I have not yet looked at. One usefulitem is an annotated bibliography of books on skepticism, withone-paragraph reviews. It is very expert and knowledgeable. There isalso a list of skeptical journals. [Webmaster's note: the SKEPTICannotated bibliography is now hosted on the CSICOP website.]CSICOP has its own homepage ( http://www.csicop.org), and past issuesof Skeptical Briefs and Skeptical Inquirer can be found therein.enough postage to pay for my computer.) =See: 3Awww.crank.net =More... Jack Sarfatti - a skeptical look at his ZPE claims http://www.phact.org/e/z/sarfatti.htmTHE SUPERLUMINAL by Jack SarfattiJack Sarfatti's Vanity Home Page http://www.qedcorp.com/pcr/pcr/sar.html == experiences in the military, where I actually had the honor of giving> a lecture on the physics of lasers to the medical personnel at Madigan> Army Medical Center, including the surgeons, other doctors and nurses,> for their medical continuing education credits, I feel like I can> speak con'dently on the subject.Radiation Protection Of'ce, Madigan Army Medical Center> who posted(which is archived at Vanderbilt). Now the question is: Are you the sameJames Harris who later posted to RADSAFE as >James.Harris@rfets.gov, K-HManager, 771 Radiological Safety> in February 2001? If so, I'd *love* tohear all about the safety violations in Building 771 (including workersbeing 'ned $385,000 a few months after you posted that message.-- Wayne Brown | >When your tail's in a crack, you improvisefwbrown@bellsouth.net | if you're good enough. Otherwise you give | your pelt to the trapper.>>e^(i*pi) = -1> -- Euler | -- John Myers Myers, >>Silverlock> ==>There is some way to extract the square root (or, in general, the nth>>root) of a one-variable polynomial? This is elementary for some sorts>>of polynomials (say, x^2 + 6x + 9 = (x+3)^2), but how about the>>general case? What is, for instance, the cubic root of P(x) = x+1?>>It is (x+1)^(1/3). There is no simpler way to write it (and it's >certainly not a polynomial). In general, a polynomial p(x) has >a r'th root which is a polynomial if and only if all its roots have>multiplicities which are multiples of r. On an unrelated basis, the OP's question suggests me another one. Asubject often recurring in this ng is that of >compositional> squareroot functions.Now I know that in the general case the problem is not easy, but maybein 'nite 'elds, where all functions are in fact polynomials, it maybe easier and of a certain combinatorial interest. Are there anystudies in this sense?Michele-- > Comments should say _why_ something is being done.Oh? My comments always say what _really_ should have happened. :)- Tore Aursand on comp.lang.perl.misc ==> There is some way to extract the square root (or, in general, the nth>> root) of a one-variable polynomial? This is elementary for some sorts>> of polynomials (say, x^2 + 6x + 9 = (x+3)^2), but how about the>> general case? What is, for instance, the cubic root of P(x) = x+1?>> On an unrelated basis, the OP's question suggests me another one. A> subject often recurring in this ng is that of >compositional> square> root functions.> Now I know that in the general case the problem is not easy, but maybe> in 'nite 'elds, where all functions are in fact polynomials, it may> be easier and of a certain combinatorial interest. Are there any> studies in this sense?Polynomial decomposition algorithms are much studied in computer algebrasince, e.g., they have applications to solving polynomials and simplifying'eld extensions. A web search on >polynomial decomposition> along withnames like: Ritt, Whaples, Fried, Schinzel, Barton, Zippel, Kozen, Landau, Gutierrez, von zur Gathen, Binder, etc. should discover much of interest, e.g. follow the links below for starters.-Bill Dubuquehttp://wwwuser.gwdg.de/~cais/CAR/CAR24/node6.htmlhttp:/ /www.cs.cornell.edu/kozen/papers/poly.pshttp:// www.algebra.uni-linz.ac.at/~xbx/pub/DA/DA.pshttp:// www.eecis.udel.edu/~saunders/papers/sparse-interp2/issac/ issac.pshttp://math-www.uni-paderborn.de/preprints/preprints_ data/Gathen/bivariateDe>c.ps.gz -0400, Bill Dubuque >Polynomial decomposition algorithms are much studied in computer algebra>since, e.g., they have applications to solving polynomials and simplifying>'eld extensions. A web search on >polynomial decomposition> along with>names like: Ritt, Whaples, Fried, Schinzel, Barton, Zippel, Kozen, Landau, >>Gutierrez, von zur Gathen, Binder, etc. should discover much of interest, >e.g. follow the links below for starters.>>-Bill Dubuque>>http://wwwuser.gwdg.de/~cais/CAR/CAR24/node6.html> http://www.cs.cornell.edu/kozen/papers/poly.ps>http:// www.algebra.uni-linz.ac.at/~xbx/pub/DA/DA.ps>http:// www.eecis.udel.edu/~saunders/papers/sparse-interp2/issac/ issac.ps>http://math-www.uni-paderborn.de/preprints/preprints_ data/Gathen/bivariateD>ec.ps.gzI expected that the subject had already been studied, but I couldn'timagine to such an extent. OTOH, surprisingly, it seems not to havebeen of any particular combinatorial interest (e.g. >How manyindecomposable polynomials are there in Z_p[X]?>)Michele-- > Comments should say _why_ something is being done.Oh? My comments always say what _really_ should have happened. :)- Tore Aursand on comp.lang.perl.misc =}}}}> Does Jack actually wish to converse with folks in these newsgroups or >does}> he just drop these little tidbits of wisdom (???) in here as (what he >sees}> as) a public service?}}Jack is a Write Only entity.}}Bob KolkerDr H =: I wonder when NASA will start sinking billions into trying to develop: a Star Gate? I wonder if they already have? (Spent the money, that: is!): Has anyone heard how their big Anti-Gravity Machine project is: going???of _Wired_. Actually, they just debunked the >anti-gravity> device byputting it in a vacuum... and I remember reading about an ion wind ¤ierin a mechanics magazine of the 1960s in my youth.John Savard == Does Jack actually wish to converse with folks in these newsgroups or >does> he just drop these little tidbits of wisdom (???) in here as (what he >sees> as) a public service?Jack has changed over the years. He once would respoond to a postnow and then. Nearly always with little more than swearing though.Socks == Does Jack actually wish to converse with folks in these newsgroups or >does> he just drop these little tidbits of wisdom (???) in here as (what he >sees> as) a public service?>> Jack has changed over the years. He once would respoond to a post> now and then. Nearly always with little more than swearing though.> SocksYes it sucks, but I'm not suprised its only that he concentrates on >hypothetical science that he isconstantly attacked. When I rid the world of skeptics he'll be free to >speak again,until then merely broadcasting is suitable for his level of work.Herc == Does Jack actually wish to converse with folks in these newsgroups or >does> he just drop these little tidbits of wisdom (???) in here as (what he >sees> as) a public service?What a shame that we can't get a dialogue going between Jack and JSH.Gib == What a shame that we can't get a dialogue going between Jack and JSH.>FLT meets FTL?Herc == Suppose B is a normed space and S is a subspace and x is an element of S,> and a neighborhood U of x lies in S. Then S also contains a neighborhood >of> 0, namely -x+U. If y is any element of B, then there is a positive number >r> such that ry belongs to -x+U; therefore ry belongs to S, therefore y >belongs> to S, and it follows that S=B.in'nite-dimensional space is also normed.But, if I'm not wrong, the hypothesis that the space isin'nite-dimensional is not relevant for the proof. In fact it shouldbe also true that:A subspace of a normed in'nite or 'nite-dimensional linear space isclosed and without interior points.In particular when you prove that a subspace of a normed space iswithout interior point ( except the case that subspace = space ) youdon't refer to its dimension. So the sentence should be always true. =Also, he didn't say what scalar 'eld he is using. A normed vectorspace over the rationals, say ... ? Then a 'nite-dimensional subspaceneed not be closed.-- G. A. Edgar >http://www.math.ohio-state.edu/~edgar/ == Suppose B is a normed space and S is a subspace and x is an element ofS,> and a neighborhood U of x lies in S. Then S also contains aneighborhood of> 0, namely -x+U. If y is any element of B, then there is a positivenumber r> such that ry belongs to -x+U; therefore ry belongs to S, therefore ybelongs> to S, and it follows that S=B.>> in'nite-dimensional space is also normed.> But, if I'm not wrong, the hypothesis that the space is> in'nite-dimensional is not relevant for the proof. In fact it should> be also true that:> A subspace of a normed in'nite or 'nite-dimensional linear space is> closed and without interior points.> In particular when you prove that a subspace of a normed space is> without interior point ( except the case that subspace = space ) you> don't refer to its dimension. So the sentence should be always true.Right you are, halfway. A *proper* subspace contains no interior points.On the other hand, a linear subspace of a normed space does not have to beclosed. Start with any normed linear space which is not complete; thisspace is isomorphically isometric to a dense subspace of its completion.More speci'cally, the space of polynomials is not closed in the space ofcontinuous functions (uniform norm) on [0, 1]. =I am stuck on showing that this sytem of equations has unique nonzerosolution (well I think it does). A is n by n matrix with positive entries (maybe not necessary) andlargest magnitude eigenvalue > 1 . Show that there is exactly one nonzero x=(x1,x2,...,xn) with all entriesbetween 0 and 1 such that: (Ax)_1 = -log(1 - x1) (Ax)_2 = -log(1 - x2) ... (Ax)_n = -log(1 - xn) where (Ax)_k is kth entry of vector Ax, (Ax)_k = A_{k1} x1 + A_{k2} x2 + ... + A_{kn} xn I can do it for n=1 but that's all. :-( Any ideas? =I think I want A to have positive entries. Can a 'xed point theorem beapplied if we bound it away from zero? =This problem works out pretty simple because (8,8) is on the asymptote y = >xof the hyperbola.For the hyperbola we have y*2 - 3xy + x*2 - 4 = 0 (1) or (y - x)(y - 2x) = 4 (1a)Represent the tangent line through (8,8) by y - 8 = m(x - 8) (2). 6x - 5y - 8 = 0Dick Tjaden> I'm in a distance education course, and am stuck on the following> question:>> Find the equation of the line through (8,8) that is tangent to the> hyperbola y^2-3xy+2x^2=4.>> We are currently doing a unit involving Newton's Method, Tangent line> approximation, and Tangent line approximation(increment form), so i am> assuming that i must use one of them to 'nd the answer.>> I've been trying to isolate variables and trying to do a system of> equations, etc., but can't get the answer no matter what i try. If> anyone could just point me in the right direction with a 'rst step> that would be great. I don't need the answer, just a nudge in the>> Scott Eliason == there is nothing to differentiate your complaints about Wiles> from complaints about photons, DNA, the Jurassic era,> evolution, relativity, and the rest.>> [...] Wiles's supposed accomplishment rests *solely*> on the assertion of a relatively small group of people that> his work is correct.There is no difference in that respect between Wiles and photons,DNA, etc. Where differences exist, they tend to favor Wiles' proofover the other situations. Experimental evidence, for example, can bechecked more easily, unambiguously and objectively in Wiles' situationthan the others.> Yet to take one of your examples--photons--and consider that the> existence of photons has been theorized for some time, but was proven> by experiment.It was not *proven* by experiment: another respect in which Wiles'work is qualitatively more reliable than photons, DNA, and the rest.The physics experiments were consistent with certain theoretical models,but of course, you have not even come close to verifying the immensechain of experimental and theoretical reasoning leading to the current >modelswith >photons>. Instead, you rely on textbooks, fourth-hand (if that)accounts, and the assertions of the Science Establishment.And the 64 dollar question is why you happily parrot the party lineon matters of photons, DNA, relativity, evolution, the existence ofthe Iraq War and Sikkim and Napoleon --- but intone high skepticismconcerning Wiles.> Since that time from lasers to spectral analysis the theory has 't> with reality.Lasers only hurt your cause, as to check that a laser (resp. spectrometry)experiment actually corroborates >photons> you would have to check >mattersof chemistry, crystallography (geology!), engineering, manufacture, and soon all the way down. The only way out of this is to accept various >assertionson faith from the Evil Scienti'c Establishment, and the question arises why >youare such a sheep and conformist when it comes to non-Wiles but raise >hightenedstandards concerning Wiles (who of courses passes all the scienti'c >standardsfor photons, etc, and then some). == there is nothing to differentiate your complaints about Wiles> from complaints about photons, DNA, the Jurassic era,> evolution, relativity, and the rest.>> [...] Wiles's supposed accomplishment rests *solely*> on the assertion of a relatively small group of people that> his work is correct.> There is no difference in that respect between Wiles and photons,> DNA, etc. Where differences exist, they tend to favor Wiles' proof> over the other situations. Experimental evidence, for example, can be> checked more easily, unambiguously and objectively in Wiles' situation> than the others.> Yet to take one of your examples--photons--and consider that the> existence of photons has been theorized for some time, but was proven> by experiment.> It was not *proven* by experiment: another respect in which Wiles'> work is qualitatively more reliable than photons, DNA, and the rest.> The physics experiments were consistent with certain theoretical models,> but of course, you have not even come close to verifying the immense> chain of experimental and theoretical reasoning leading to the current >models> with >photons>. Instead, you rely on textbooks, fourth-hand (if that)> accounts, and the assertions of the Science Establishment.My degree is in physics. I did physics experiments in school. > And the 64 dollar question is why you happily parrot the party line> on matters of photons, DNA, relativity, evolution, the existence of> the Iraq War and Sikkim and Napoleon --- but intone high skepticism> concerning Wiles.Wiles's work would mean a workaround to the logical fallacy called,>Cum hoc ergo propter hoc>.Ultimately, if Wiles's work is correct then it does not have anylogical ¤aws, but checking it potentially involves going through eachstep in his work, which is a formidable task. If he did 'nd a proof,then I think it interesting on logical grounds that there is aworkaround i.e. that Cum hoc ergo propter hoc is not actually alogically fallacious approach.Now as for physics results, like many people trained in physics, Ikeep a skeptical eye on theory, and depend on things I've personallychecked, or that are very unlikely to be wrong that have been checkedby others. Physicists can be hard-liners to the extent that theydon't believe physics they haven't personally checked. I'm not. Likehow I believe that nuclear weapons work. But still realize that theabsolute truth may be something other than what I've learned.In mathematics, absolute truth *can* be determined, just like alogical argument can be checked against certain rules for internalconsistency. > Since that time from lasers to spectral analysis the theory has 't> with reality.> Lasers only hurt your cause, as to check that a laser (resp. >spectrometry)> experiment actually corroborates >photons> you would have to check >matters> of chemistry, crystallography (geology!), engineering, manufacture, and >so> on all the way down. The only way out of this is to accept various >assertions> on faith from the Evil Scienti'c Establishment, and the question arises >why you> are such a sheep and conformist when it comes to non-Wiles but raise >hightened> standards concerning Wiles (who of courses passes all the scienti'c >standards> for photons, etc, and then some).As a person with a science degree, I guess you'd consider me a part ofthe >Evil Scienti'c Establishment>.It's actually more fun attacking them than just sitting aroundbelieving in them. Because you learn a lot in the attack, and yourguarantee from math and logic is that the proof doesn't care.To a math proof, you do not exist as a relevant entity.James Harris == Wiles's work would mean a workaround to the logical fallacy called,> feel like I can> speak con'dently on the subject.You speak con'dently whether you know what you're talking about or not. >Con'dence is not yourproblem, honesty and credibility are.> To a math proof, you do not exist as a relevant entity.You do not exist as a relevant entity.--There are two things you must never attempt to prove: the unprovable -- and >the obvious.--Democracy: The triumph of popularity over principle.--http://www.crbond.com == > there is nothing to differentiate your complaints about Wiles> from complaints about photons, DNA, the Jurassic era,> evolution, relativity, and the rest.>> [...] Wiles's supposed accomplishment rests *solely*> on the assertion of a relatively small group of people that> his work is correct.> There is no difference in that respect between Wiles and photons,> DNA, etc. Where differences exist, they tend to favor Wiles' proof> over the other situations. Experimental evidence, for example, can be> checked more easily, unambiguously and objectively in Wiles' situation> than the others.> > Yet to take one of your examples--photons--and consider that the> existence of photons has been theorized for some time, but was proven> by experiment.> It was not *proven* by experiment: another respect in which Wiles'> work is qualitatively more reliable than photons, DNA, and the rest.> The physics experiments were consistent with certain theoretical >models,> > but of course, you have not even come close to verifying the immense> chain of experimental and theoretical reasoning leading to the current >models> with >photons>. Instead, you rely on textbooks, fourth-hand (if >that)> accounts, and the assertions of the Science Establishment.> My degree is in physics. I did physics experiments in school.> And the 64 dollar question is why you happily parrot the party line> on matters of photons, DNA, relativity, evolution, the existence of> the Iraq War and Sikkim and Napoleon --- but intone high skepticism> concerning Wiles.> Wiles's work would mean a workaround to the logical fallacy called,>Cum hoc ergo propter hoc>.> Ultimately, if Wiles's work is correct then it does not have any> logical ¤aws, but checking it potentially involves going through each> step in his work, which is a formidable task. If he did 'nd a proof,> then I think it interesting on logical grounds that there is a> workaround i.e. that Cum hoc ergo propter hoc is not actually a> logically fallacious approach.> Now as for physics results, like many people trained in physics, I> keep a skeptical eye on theory, and depend on things I've personally> checked, or that are very unlikely to be wrong that have been checked> by others. Physicists can be hard-liners to the extent that they> don't believe physics they haven't personally checked. I'm not. Like> how I believe that nuclear weapons work. But still realize that the> absolute truth may be something other than what I've learned.> In mathematics, absolute truth *can* be determined, just like a> logical argument can be checked against certain rules for internal> consistency.>How? How can absolute truth be determined? About a month ago youessentially:1) proof of an absolute kind, presumably stated in the symbolism offormal logic, and2) proof that merely convinces other mathematicians, presumably statedin some meta-language (like English).Furthermore, your position is that proof of the second kind is whatmost mathematicians produce, and is not good enough. You go on to saythat you produce proofs of the 1st kind.Question: How does one determine that a >proof> or mathematicalargument is absolutely and irrefutably correct?The validity must be checked by 1) God, 2) a machine, or 3) a humanbeing, as a >proof> cannot check itself.I think (1) is out, for the time being anyway.What about (2)? Well, we could encode some axioms and rules o'nference, but it occurs to me that a few problems could arise. First,the algorithm may take an unreasonable amount of time to reach adecision. Second, hardware failure, electrical surges, sunspotactivity, running the program under Microsoft Windows, etc. couldcause erroneous results. Third, and perhaps most importantly, a humanbeing (or beings) must write the software. Therefore, any errorscaused by people could conceivably appear here. That leaves us withoption (3). As we all know, people make mistakes. They make mistakeswriting proofs. The publisher/editors of a journal may make a mistakemistake by erroneously believing the >proof>.Who has the 'nal and ultimate authority to say that a given argumentis valid or not? Surely, not one person. There is so much mathematics,no one person can know it all.So, a proof then must be judged by the readers. If there is adisagreement, then the sides may argue their cases until one sideprevails and convinces the other, at least within a given mathematicalsystem.Therefore, in this sense, all proofs are of the second type. We muststrive to convince other mathematicians. That is all there is --simply because there is no other means of asserting the validity of amathematical argument. It really is an >appeal to the gallery.>We must also consider that mathematics may be inconsistent. Accordingto Kurt Godel, this is a possibility (at least for mathematicalsystems strong enough to support integer arithmetic.)So much for proofs being irrefutable, absolute, perfect, eternal, etc.ad nauseum.> Since that time from lasers to spectral analysis the theory has 't> with reality.> Lasers only hurt your cause, as to check that a laser (resp. >spectrometry)> experiment actually corroborates >photons> you would have to check >matters> of chemistry, crystallography (geology!), engineering, manufacture, and >so> on all the way down. The only way out of this is to accept various >assertions> on faith from the Evil Scienti'c Establishment, and the question arises >why you> are such a sheep and conformist when it comes to non-Wiles but raise >hightened> standards concerning Wiles (who of courses passes all the scienti'c >standards> for photons, etc, and then some).> As a person with a science degree, I guess you'd consider me a part of> the >Evil Scienti'c Establishment>.> experiences in the military, where I actually had the honor of giving> a lecture on the physics of lasers to the medical personnel at Madigan> Army Medical Center, including the surgeons, other doctors and nurses,> for their medical continuing education credits, I feel like I can> speak con'dently on the subject.> My position on Wiles is about logic. Emotional response is not> necessary as I assure you that if Wiles found a proof then there is no> need for concern. If he did not, why 'ght for a false belief?> Math proofs are indestructible, incorruptible, and irrefutable.> It's actually more fun attacking them than just sitting around> believing in them. Because you learn a lot in the attack, and your> guarantee from math and logic is that the proof doesn't care.> To a math proof, you do not exist as a relevant entity.> James Harris == there is nothing to differentiate your complaints about Wiles> from complaints about photons, DNA, the Jurassic era,> evolution, relativity, and the rest.> Yet to take one of your examples--photons--and consider that the> existence of photons has been theorized for some time, but was> proven by experiment.>> It was not *proven* by experiment: another respect in which Wiles'> > work is qualitatively more reliable than photons, DNA, and the rest.> The physics experiments were consistent with certain theoretical >models,> but of course, you have not even come close to verifying the immense> chain of experimental and theoretical reasoning leading to the current >models> with >photons>. Instead, you rely on textbooks, fourth-hand (if >that)> accounts, and the assertions of the Science Establishment.>> My degree is in physics. I did physics experiments in school.Ask the school for a refund.First, the >existence> of >photons> is not a precisely formulated >statementas in the case of Wiles' proof, let alone one provable by experiment. >Thereare of course theoretical models (not necessarily well-de'ned or knownto be logically consistent, by the way) within which one can single outcertain objects as >photons>.Second, your student experiments in optics could not possibly replicatethe mountain of theoretical and experimental steps involved in buildingup any of the theoretical model(s) involving photons. Instead, youaccepted on trust assertions by textbook authors, professors and similarpurveyors of the Social Truth that you like to castigate, amounting toa certi'cation-by-authority that the apparatus you were doing the >experimentswith actually corresponded to the theory in the manner claimed.You did not produce the relevant gases, crystals, apparatus, electricity, >..involved in laser experiments, nor did you do the experimentationneeded to corroborate the values of relevant physical and chemicalparameters listed in the CRC handbook, and so on all the way down.What actually happened is that a long and social processof knowledge-accumulation occurred and you took the results on trust.In particular, if your experiments gave a >wrong> result, the conclusionwould be that you made a mistake, not that photons' existence is in doubt;a pure assertion of authority by the Scienti'c Establishment concerning >itsSocial Truth, which you accept without any objection in all the non-FLTsituations.Note that your repeatedly discredited objections in this thread aboutWiles' logic are irrelevant, as you also object to Ribet's proof withoutgiving any particular reason to doubt it. The matter is simply one of anobvious double standard produced for the occasion, where social >certi'cationby a small network of experts counts as OK for photons, DNA,evolution, relativity, the Jurassic era (or the existence of Napoleon andGeorge W Bush), etc --- but somehow the information that expertshave certi'ed Ribet's and Wiles' work is cast as suspicious. == My degree is in physics. I did physics experiments in school.Yet you seem never to have encountered the SR thoughtexperiment called the >superluminal scissors> or had anyidea what I was talking about in sci.physics when Iexplained how a 5 m/sec water jet can be used to createan illusion of arbitrarily fast, even superluminalmotion. - Randy == > there is nothing to differentiate your complaints about Wiles> > from complaints about photons, DNA, the Jurassic era,> > evolution, relativity, and the rest.> > Yet to take one of your examples--photons--and consider that the> existence of photons has been theorized for some time, but was> proven by experiment.> >> It was not *proven* by experiment: another respect in which Wiles'> work is qualitatively more reliable than photons, DNA, and the rest.> The physics experiments were consistent with certain theoretical >models,> but of course, you have not even come close to verifying the immense> > chain of experimental and theoretical reasoning leading to the current >models> with >photons>. Instead, you rely on textbooks, fourth-hand (if >that)> accounts, and the assertions of the Science Establishment.>> My degree is in physics. I did physics experiments in school.> Ask the school for a refund.I had a full tuition scholarship. > First, the >existence> of >photons> is not a precisely formulated >statement> as in the case of Wiles' proof, let alone one provable by experiment. >There> are of course theoretical models (not necessarily well-de'ned or known> to be logically consistent, by the way) within which one can single out> certain objects as >photons>.> Second, your student experiments in optics could not possibly replicate> the mountain of theoretical and experimental steps involved in building> up any of the theoretical model(s) involving photons. Instead, you> accepted on trust assertions by textbook authors, professors and similar> purveyors of the Social Truth that you like to castigate, Oh please, you've been beaten. That's what's annoying about Usenet assome loser will state a case, get their ass kicked, but STILL keepcoming back as if nothing happened.My *degree* is in physics. I went to school on a full-tuitionscholarship, and you stepped into my 'eld with your assertions, gotyour ass kicked but refuse to back down.Now *emotion* is not necessary when it comes to Wiles's work. If hefound a proof I can assure you that it is irrefutable. That's whyit'd be a proof. All this emotion just annoys me, as part of the funof science and mathematics is attacking things that are supposedlyproven.It's GREAT fun not just accepting what people tell you. But that'swhat really annoys me about mathematicians as time after time I getyahoo's replying back in defense of mathematics, using tactics.But you see, not a single REAL mathematician in the world gets excitedabout an attack on a proof. No mathematician worth the title wouldget even a little concerned, nor would they lose sleep, or 'ndthemselves emotional about some person--any person--any time--anyplace--who decides to go after a math proof.That's because a math proof is indestructible, incorruptible, andirrefutable.It just doesn't care if you attack it, and no real mathematician wouldcare either.Now I've discovered math proofs, which is why I'm not concerned aboutpeople refuting them because they are proofs. And in fact people whocall themselves mathematicians can't touch them, so they come up withextraneous stuff, or make claims of 'nding their own proofs to refutemy proofs, but you see, proofs don't duel.And you know what? I think that feature of mathematics terri'es somepeople who call themselves mathematicians. Mathematics does NOT carewhat you call yourself. It DOES NOT care that you have a mortgage. It DOES NOT CARE that you really, really, really want people to likeyou and think you're a great mathematician.Now I've made a speci'c claim against Wiles's work. If he found aproof the claim can be answered, but even if it is answerable then itmust be true that he has found a way around what is considered to be alogically fallacious approach. Logicians should thank him in thatcase for correcting them.My challenge is a logical one. Wiles's work fails and is not a proofas it is an argument by Cum hoc ergo propter hoc.James Harris =the truth-value of any logical statement is *algebraically*of no importance. true,a proof with an error of inference may be false, ifthe error was also once more), buttaht doesn't mean that it's not syntactically >proper>in some sense. not taht yours necessarily are, two often!I don't have to know any Latin, whatsoever,to follow a pattern of inference. (in particular,see Lou Kauffman's slight rephrasing of G. Spencer-Brown,on his site; I htought of the same diagrammatic convention, butusing circles (to represent spheres .-)) and, you's still have to >prove> thatWiles faked his proof; have you? > But you see, not a single REAL mathematician in the world gets excited> about an attack on a proof. No mathematician worth the title would> get even a little concerned, nor would they lose sleep, or 'nd> themselves emotional about some person--any person--any time--any> place--who decides to go after a math proof.> That's because a math proof is indestructible, incorruptible, and> irrefutable. > My challenge is a logical one. Wiles's work fails and is not a proof> as it is an argument by Cum hoc ergo propter hoc.--Dec.2000 OWAND' Chairman Paul O'Neill, reelectedto Board. Newsish?http://www.rand.org/publications/randreview/issues/rr .12.00/http://members.tripod.com/~american_almanac ==Is there an easy way to come up with all the schedules where n teams>play each other once?>>I tried several avenues. I don't have the time or expertise to tackle>this and thought this is the newsgroup with the expertise that may know.>I hope you are not offended.>>I think it can be reduced to symmetric matrices that are 2nd roots of>unity with zero diagonal, but I may be wrong.Please check my post at whim.orgIf n is odd, add a dummy team as a bye to make the number even.Rob Johnsontake out the trash before replying =Good day,I am processing batches of data stored in arrays, and I have seenexamples of vector mathematics (rather than loops) being used to dothe processing.For instance, for performing a linear regression on a set of data Xand Y (to 'nd the best-'t slope Ok') I am used to seeing themean-squared-error (MSE) being expressed as a summation of thesquared-errors of each data pair, but I have also seen the MSEexpressed as in dot-product form which is equivalent. In the end, theresult is k = (x'x)^(-1) * x'y.To get this optimized result, a partial derivative with resprct to kis taken on the MSE function. I can follow along, but I am unsureabout some of the procedures, especially with regard to getting the_order_ of the vectors expressed properly to ensure dimensionalagreement (e.g. how to expand (kx-y)*(kx-y)' , how to apply chainrules when taking derivatives), but I don't know what the forms ortecniques are called.This is not the >vector algebra> or >vector calculus> that I have beenexposed to. Or is it? I'd appreciate knowing so that I could have amore fruitful search on the topic.-Jagan == Good day,> I am processing batches of data stored in arrays, and I have seen> examples of vector mathematics (rather than loops) being used to do> the processing.> For instance, for performing a linear regression on a set of data X> and Y (to 'nd the best-'t slope Ok') I am used to seeing the> mean-squared-error (MSE) being expressed as a summation of the> squared-errors of each data pair, but I have also seen the MSE> expressed as in dot-product form which is equivalent. In the end, the> result is k = (x'x)^(-1) * x'y.> To get this optimized result, a partial derivative with resprct to k> is taken on the MSE function. I can follow along, but I am unsure> about some of the procedures, especially with regard to getting the> _order_ of the vectors expressed properly to ensure dimensional> agreement (e.g. how to expand (kx-y)*(kx-y)' , how to apply chain> rules when taking derivatives), but I don't know what the forms or> tecniques are called.> This is not the >vector algebra> or >vector calculus> that I have >been> exposed to. Or is it? I'd appreciate knowing so that I could have a> more fruitful search on the topic.You see a lot of manipulations of this sort (e.g.,taking the derivative of a matrix expression) inoptimization theory. You might for instance check intothe theory of quadratic programming, which is the theoryof optimizing a general multivariate quadratic functionunder linear equality and inequality constraints.Least-squares is one particularly easy quadratic problem:an unconstrained minimization with a positive de'nitecoef'cient matrix.When I was 'rst exposed to this kind of manipulation,I found it helpful to work out expressions in termsof individual components and summations. Going backand forth between those kinds of things and their matrixequivalents is really helpful to get facility with thealgebra.Here are a couple of particularly useful identitiesfor you for free (prime O means transpose).1) x'Qx = sum (i) q_ii * x_i^2 + sum(i!=j) (q_ij + q_ji) * x_i * x_jIf Q is not symmetric, there is always a different matrixP which gives the same function but is symmetric:P = 1/2(Q + Q'). Component wise, p_ij = (q_ij+q_ji)/2for j!=i, and p_ii = q_ii, and x'Px = sum(i) p_ii * x_i^2 + sum(i!=j) 2*p_ij*x_i*x_jThis tells you the correspondence between a general multivariatequadratic and its symmetric coef'cient matrix.2) If b is a vector, b'x is a scalar. Thus it isits own transpose, i.e. b'x = x'b.3) The gradient grad(f) where f is a scalar is a vectorwhose i-th component is df/dx_i. grad(b'x) = bProof: d(b'x)/dx_i = d/dx_i sum(k) b_k * x_k = b_i.The i-th component of grad(b'x) is b_i.4) grad(x'Px) = 2PxProof: Left to reader. - Randy == with resprct to k> is taken on the MSE function. I can follow along, but I am unsure> about some of the procedures, especially with regard to getting the> _order_ of the vectors expressed properly to ensure dimensional> agreement (e.g. how to expand (kx-y)*(kx-y)' , how to apply chain> rules when taking derivatives), but I don't know what the forms or> tecniques are called.This is vector calculus, but using a trick that you may not have seen.Notation: the derivative of a function F:R^n->R^m at x is DF(x).DF(x) is a linear map DF(x):R^n->R^m. Given a vector h in R^n,DF(x)(h) is a vector in R^m.If F is a linear function, then DF(x)(h) = F(h) . This helps youcompute even when DF(x) itself is hard to write down using vector (ormatrix) notation. This is the trick: rather than work with DF(x),work with DF(x)(h).For example, consider the inner product F(x,y) = x'y . (Vectors inR^n are column vectors; prime O denotes transpose.) Then the partialderivatives with respect to the 'rst argument (x) and the secondargument (y) can be expressed by D_1F(x,y)(h) = h'y D_2F(x,y)(h) = x'h .The chain rule (using >.> to represent composition) is D(F.G)(x) = DF(G(x)) . DG(x)so D(F.G)(x)(h) = DF(G(x))(DG(x)(h)) .The product rule for scalar functions of vectors is D(F*G)(x)(h) = DF(x)(h)*G(x) + F(x)*DG(x)(h) .The product rule for inner products <,> of vectors is D(x)(h) = + .(You should prove these, and derive similar rules for other thingssuch as the cross product in R^3.)An excellent example of this, which illustrates both the concept andhow dif'cult it can be to write down DF(x) rather than DF(x)(h), isthe formula for the derivative of the matrix inversion function. LetF(X) = X^{-1}, where X is an invertible matrix. Then let G(X) := X F(X) - I = 0 ;then, computing the derivative and applying the product rule, DG(X)(H) = H F(X) + X DF(X)(H) = 0so X DF(X)(H) = - H F(X) .Left-multiplying by F(X)=X^{-1}, DF(X)(H) = - F(X) H F(X) = - X^{-1} H X^{-1} .(Recall that matrix multiplication does not commute, so this is notthe same as -X^{-2}H ; in particular, the scalar formula DF(X)=-X^{-2}does not hold for matrices.)Kevin. =test all eagerly await the publicists' results about my importantpaper which rights the contradictions of the trans'nitude and setsablaze the torch of absolute truth, I will in the meantime divulge thegeneral idea upon which the proof is built. Of course this is not theproof itself, being merely a >layman's terms> version, and the actualproof requires much more rigour. Nevertheless I anticipate that eventhis brief sample will shed some enlightenment upon you and convinceyou of my sincerity and credibility. :-)I assert, then, that Cantor's principals are based erroneously upon anaxiom which is very subtle and ghostlike, so much so in fact that eventhe most learned of the mathematical community are blind to it, thoughit is right under their noses. To wit, I refer to that ever-assumedhypothesis that we can construct a set whose members are uncountablein the 'rst place. >But wait!>, cry the critics, >What of the set ofreal numbers?> But do not laugh at them, for these are very subtlematters and their lack of understanding is worthy of our sympathy. Let us, then, address their concern. For, indeed, we may say >Let Rthen be all such numbers which cannot be represented as the quotientof two integers>, and a unique well-de'ned set named >R> does thusarise, there is no arguing about this. However, how do we know R isuncountable? Because Cantor asserts that it is. But now I am the oneto cry >But wait!>. Cantor's proof subtly assumes that, even if atrans'nitude does exist (which we will for the moment allow, just togive him the bene't of the doubt), that it is then possible toconstruct a set of such magnitude. So when I assert that thisconstructability is unproven, and you offer R as a counterexample withCantor's assertian that R possesses such magnitude, it is clear as daythat you are using circular reasoning. But I do not hold this againstyou, because this is a matter of very great subtlety which is veryeasily overlooked, so do not hit your head for being stupid. You arenot alone.Now I see that you are greatly confused, now that the matter of thisassumed, implicit axiom about the constructability of trans'nite sets(even given the existance of a trans'nitude) has been laid out beforeyour eyes. For, surely this axiom must hold true, you reason, elsemany works of mathematics will be rendered invalid, many great proofsof famous reknown rendered erroneous. Alas, I say, this is not thecase. They are unsalvageable, and not just the proofs but thetheorems themselves. But do not be saddened, for the mappings I willsoon propose will be of such profound insight into mathematics thatthey will more than compensate for all the works thus lost.To wit, I will in short time present for your examination awell-de'ned mapping, accessible even to the amateur mathematician inits simplicity, yet inspirational to even the most learnedmathematician for its ingenuity, which establishes a one-to-onecorrespondance between the set of integers and the set of all realnumbers. I will present a common-sense, layman's terms, unrigoroussample of this mapping at a later time to this worthy forum, and the'nal, rigorous version will be presented at a still later time whenthe publicists have overcome their sheer astoundment upon learning thetrue nature of these things. At that time a link will be given freelyto you to a webpage where you will be able to examine all of thesematters in all their profound rigour with as much skepticism as youlike, and to learn that all such skepticism is misled.I eagerly look forward to sharing these great subtle truths with you:-)Nathan the GreatAge 11 == While we all eagerly await the publicists' results about my important> paper which rights the contradictions of the trans'nitude and sets> ablaze the torch of absolute truth, I will in the meantime divulge the> general idea upon which the proof is built. Of course this is not the> proof itself, being merely a >layman's terms> version, and the actual> proof requires much more rigour. Nevertheless I anticipate that even> this brief sample will shed some enlightenment upon you and convince> you of my sincerity and credibility. :-)> I assert, then, that Cantor's principals are based erroneously upon an> axiom which is very subtle and ghostlike, so much so in fact that even> the most learned of the mathematical community are blind to it, though> it is right under their noses. To wit, I refer to that ever-assumed> hypothesis that we can construct a set whose members are uncountable> in the 'rst place. >But wait!>, cry the critics, >What of the set of> real numbers?> But do not laugh at them, for these are very subtle> matters and their lack of understanding is worthy of our sympathy. > Let us, then, address their concern. For, indeed, we may say >Let R> then be all such numbers which cannot be represented as the quotient> of two integers>, and a unique well-de'ned set named >R> does thus> arise, there is no arguing about this. However, how do we know R is> uncountable? Because Cantor asserts that it is. But now I am the one> to cry >But wait!>. Cantor's proof subtly assumes that, even if a> trans'nitude does exist (which we will for the moment allow, just to> give him the bene't of the doubt), that it is then possible to> construct a set of such magnitude. Looks like you don't care for the Axiom of Choice.> So when I assert that this> constructability is unproven, and you offer R as a counterexample with> Cantor's assertian that R possesses such magnitude, it is clear as day> that you are using circular reasoning. But I do not hold this against> you, because this is a matter of very great subtlety which is very> easily overlooked, so do not hit your head for being stupid. You are> not alone.There are a few people who don't accept AC.> Now I see that you are greatly confused, now that the matter of this> assumed, implicit axiom about the constructability of trans'nite sets> (even given the existance of a trans'nitude) has been laid out before> your eyes. For, surely this axiom must hold true, you reason, else> many works of mathematics will be rendered invalid, many great proofs> of famous reknown rendered erroneous. Alas, I say, this is not the> case. They are unsalvageable, and not just the proofs but the> theorems themselves. But do not be saddened, for the mappings I will> soon propose will be of such profound insight into mathematics that> they will more than compensate for all the works thus lost.> To wit, I will in short time present for your examination a> well-de'ned mapping, accessible even to the amateur mathematician in> its simplicity, yet inspirational to even the most learned> mathematician for its ingenuity, which establishes a one-to-one> correspondance between the set of integers and the set of all real> numbers. I will present a common-sense, layman's terms, unrigorous> sample of this mapping at a later time to this worthy forum, and the> 'nal, rigorous version will be presented at a still later time when> the publicists have overcome their sheer astoundment upon learning the> true nature of these things. At that time a link will be given freely> to you to a webpage where you will be able to examine all of these> matters in all their profound rigour with as much skepticism as you> like, and to learn that all such skepticism is misled.Just make sure no one has published your results 'rst. There's a body of work out there.> I eagerly look forward to sharing these great subtle truths with you> :-)> Nathan the Great> Age 11-- Will Twentyman == I assert, then, that Cantor's principals are based erroneously upon an> axiom which is very subtle and ghostlike, so much so in fact that even> the most learned of the mathematical community are blind to it, though> it is right under their noses. To wit, I refer to that ever-assumed> hypothesis that we can construct a set whose members are uncountable> in the 'rst place. >But wait!>, cry the critics, >What of the set of> real numbers?> But do not laugh at them, for these are very subtle> matters and their lack of understanding is worthy of our sympathy.> Let us, then, address their concern. For, indeed, we may say >Let R> then be all such numbers which cannot be represented as the quotient> of two integers>, and a unique well-de'ned set named >R> does thus> arise, there is no arguing about this. However, how do we know R is> uncountable? Because Cantor asserts that it is. But now I am the one> to cry >But wait!>. Cantor's proof subtly assumes that, even if a> trans'nitude does exist (which we will for the moment allow, just to> give him the bene't of the doubt), that it is then possible to> construct a set of such magnitude. So when I assert that this> constructability is unproven, and you offer R as a counterexample with> Cantor's assertian that R possesses such magnitude, it is clear as day> that you are using circular reasoning. But I do not hold this againstActually, Cantor assumes R are *countable* by putting them in a list.I have another refutation that he only makes minimal assumptions here:> then constructing a >new> real by taking the main diagonal of this array and changing every >digit.> This number is by construction different from every member of what is> already an in'nite set (since each real can be tagged with an index >number,> which is an integer, and there is an in'nity of integers), therefore >it is> not in the set, therefore the count of reals is a >greater> in'nity >than> > the count of integers.>> Correct me if I've got any of the above wrong.>> But does that not *assume* multiple in'nities to start with? If we >use my> pet subject of the uniqueness of in'nity, the modi'ed diagonal >number does> 'nd a place in the set of reals - at in'nity, and no 'xup of >in'nity is> needed.>> it assumes a single in'nity type, that reals can be counted then it> 'nds a contradiction. the difference is you can tell me any rational >number> (integer over integer) and I can count to it, but you can't necessarily >count> to a given real.>> Notice my comment about (dis)order in the original post. Every account I> have read about Cantor's method has illustrated the point with a totally> unordered (partial!) list of reals. Conceivably you could 'nd a real >number> given its index number - even if there was no >seek> other than to look >it> up in an in'nite book - but you could not go the other way. I presume> that's what you mean by >can't necessarily count to a given real>.it doesn't matter that the cantor list is random, he is trying to see if >they canbe ordered and makes the assumption that they must be listed somehow,he arrives at a contradiction well before he has to establish any >propertiesof the possible ordering. >can't necessarily count to a given real> i had >tothink how to word it correctly, another way is >can't count to every >real>,though that has a second meaning of all of them. basically you can'tcount to real numbers, but if you had speci'c real numbers you could riga count to them, rationals are a subset of reals. the look up process issymmetric, Ofrom the integer index give the number' is what the orderingis, Ofrom the number what is the index' is just a process of counting >througheach number until you 'nd it. integers have de'nite properties and setsthat map to integers must attain the properties.>> [unsnip Terry Wilder]> There's a theorem that every set , let alone that of real numbers, can >be> well ordered, by reductio ad absurdum,> using the axiom of choice.>> This was the original thought that started me off. Once a systematic> ordering scheme is applied to the reals one *can* go the other way - to >'nd> the index number of a real. A scheme that was discussed here a while ago >was> to reverse the order of the decimal digits, e.g. the real 0.5 will map to> the index number 5, the real 0.00032759 will map to the index number> 95,723,000, etc. Conversely there is a unique real corresponding to any> integer one cares to name, to Graham's number and beyond, and it is >equally> easy to 'nd. We *can* now put the reals in one-to-one correspondence >with> the integers.>> OK, so all nonterminating decimals, recurring or not, will map to >in'nity.yes, your examples are all rationals, they can be counted.> No problem, the one-to-one correspondence with the in'nity of integers >is> still valid, there's enough room at in'nity to accommodate all the> successive Godel-diagonalisations of the original list of reals: each >step> generates one more in'nitely-long real and one more in'nite index >integer.>Neat, could be constuctively worded as longer rather than in'nitely long,then you have a computable system that implies in'nitely long because theindex is in'nite. I posted up Penrose version of this idea a month ago.#9. WE KNOW WE LEGITIMATELY CREATED THIS NEW Y & N PATTERN, IE: IT IS TRUE.YET NONE OF THE EXISTING AXIOM STATEMENTS PRODUCE THIS DIAGONAL STATEMENT. >ANEW AXIOM IS NEEDED TO EXPRESS THE DIAGONAL.10. IF WE WRITE A NEW STATEMENT (CALL IT R) THAT INCLUDES A PROCEDURE FORMAKING THIS DIAGONAL , AT SPACE R/R A NEW DIAGONAL LETTER WILL APPEAR AND >WEWILL HAVE TO ADD STATEMENT S TO REPRESENT THIS NEW SEQUENCE. BUT AT S/S ANEW DIAGONAL NUMBER WILL APPEAR, REQUIRING A STATEMENT T AND SO ON,INFINITELY.the distintion is the diagonal you create every second row is onlythe diagonal of the preceding rows, not the entire set. put it this way,if that is acceptable then we can indeed make a well de'ned order and makeour list of real numbers. what integer represents pi? can you count to >it?at any moment you index the list you only get a rational number.Cantors proof is valid because it gives a distinction between classes of >numbers.In essense it says in'nitely long numbers don't all map to a sequence. if >Igive you the number 0.3333 recurring, a consisely de'ned number you cansay that is number 26 on my list. if I tell you square root of 2, another >consiselyde'ned number, it doesn't appear on the same list.The missing diagonal number was only one number, say the list of reals >startsout with in'nitely long random numbers. The axiom of choice is the nth >digitof the nth number. Not only can we add 1 to each digit, we can add 2, 3,..making 9 new diagonal numbers. We can add 1 to the 'rst digit making itdifferent to the 'rst number, add 2 to the second digit making it >differentto the second number, we can add any number to any digit of the diagonaland as long as every digit is changed we have a new number, so youhave to add in'nite diagonally constructed numbers, not just one.Try to count them out and it starts to look like a fractal snow¤ake orcoastline, 'nite area but in'nite perimeter, you try to traverse but >nevermove forward.Herc mike_deeth@yahoo.com (Mike Deeth)>While we all eagerly await the publicists' results about my important>paper which rights the contradictions of the trans'nitude and sets>ablaze the torch of absolute truth, I will in the meantime divulge the>general idea upon which the proof is built. Of course this is not the>proof itself, being merely a >layman's terms> version, and the actual>proof requires much more rigour. Nevertheless I anticipate that even>this brief sample will shed some enlightenment upon you and convince>you of my sincerity and credibility. :-)>>I assert, then, that Cantor's principals are based erroneously upon an>axiom which is very subtle and ghostlike, so much so in fact that even>the most learned of the mathematical community are blind to it, though>it is right under their noses. To wit, I refer to that ever-assumed>hypothesis that we can construct a set whose members are uncountable>in the 'rst place. >But wait!>, cry the critics, >What of the set of>real numbers?> But do not laugh at them, for these are very subtle>matters and their lack of understanding is worthy of our sympathy. >Let us, then, address their concern. For, indeed, we may say >Let R>then be all such numbers which cannot be represented as the quotient>of two integers>, No, we do _not_ say that when we're de'ning R.There are various ways to de'ne R, all of which would take a littlebit of typing. But it doesn't matter - you're hilariously disputing the de'nability of _any_ uncountable set, and there's one whichis much easier to de'ne than R:Let S be the set of all subsets of {1, 2, ...}.>and a unique well-de'ned set named >R> does thus>arise, there is no arguing about this. There's _certainly_ room to argue over the idea that >Let Rthen be all such numbers which cannot be represented as the quotient of two integers> is a de'nition of anything. Luckilyit's not the actual de'nition or you'd have a point.> However, how do we know R is>uncountable? Because Cantor asserts that it is. But now I am the one>to cry >But wait!>. Cantor's proof subtly assumes that, even if a>trans'nitude does exist (which we will for the moment allow, just to>give him the bene't of the doubt), that it is then possible to>construct a set of such magnitude. So when I assert that this>constructability is unproven, and you offer R as a counterexample with>Cantor's assertian that R possesses such magnitude, it is clear as day>that you are using circular reasoning. But I do not hold this against>you, because this is a matter of very great subtlety which is very>easily overlooked, so do not hit your head for being stupid. You are>not alone.>Now I see that you are greatly confused, now that the matter of this>assumed, implicit axiom about the constructability of trans'nite sets>(even given the existance of a trans'nitude) has been laid out before>your eyes. For, surely this axiom must hold true, you reason, else>many works of mathematics will be rendered invalid, many great proofs>of famous reknown rendered erroneous. Alas, I say, this is not the>case. They are unsalvageable, and not just the proofs but the>theorems themselves. But do not be saddened, for the mappings I will>soon propose will be of such profound insight into mathematics that>they will more than compensate for all the works thus lost.>To wit, I will in short time present for your examination a>well-de'ned mapping, accessible even to the amateur mathematician in>its simplicity, yet inspirational to even the most learned>mathematician for its ingenuity, which establishes a one-to-one>correspondance between the set of integers and the set of all real>numbers. I will present a common-sense, layman's terms, unrigorous>sample of this mapping at a later time to this worthy forum, and the>'nal, rigorous version will be presented at a still later time when>the publicists have overcome their sheer astoundment upon learning the>true nature of these things. At that time a link will be given freely>to you to a webpage where you will be able to examine all of these>matters in all their profound rigour with as much skepticism as you>like, and to learn that all such skepticism is misled.>>I eagerly look forward to sharing these great subtle truths with you>:-)>>Nathan the Great>Age 11************************David C. Ullrich =On eagerly await the publicists' results about my important>> paper which rights the contradictions of the trans'nitude and sets>> ablaze the torch of absolute truth, I will in the meantime divulge the>> general idea upon which the proof is built. Of course this is not the>> proof itself, being merely a >layman's terms> version, and the actual>> proof requires much more rigour. Nevertheless I anticipate that even>> this brief sample will shed some enlightenment upon you and convince>> you of my sincerity and credibility. :-)>>> I assert, then, that Cantor's principals are based erroneously upon an>> axiom which is very subtle and ghostlike, so much so in fact that even>> the most learned of the mathematical community are blind to it, though>> it is right under their noses. To wit, I refer to that ever-assumed>> hypothesis that we can construct a set whose members are uncountable>> in the 'rst place. >But wait!>, cry the critics, >What of the set >of>> real numbers?> But do not laugh at them, for these are very subtle>> matters and their lack of understanding is worthy of our sympathy. >> Let us, then, address their concern. For, indeed, we may say >Let R>> then be all such numbers which cannot be represented as the quotient>> of two integers>, and a unique well-de'ned set named >R> does thus>> arise, there is no arguing about this. However, how do we know R is>> uncountable? Because Cantor asserts that it is. But now I am the one>> to cry >But wait!>. Cantor's proof subtly assumes that, even if a>> trans'nitude does exist (which we