> Subset is synonymous with non-proper subset. ... with possibly non-proper subset, I'd say. -- Jesse F. Hughes C is for Cookie. That's good enough for me. Cookie Monsters ==== >> Subset is synonymous with non-proper subset. > ... with possibly non-proper subset, I'd say. Yes, right. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. ==== > > In other words, N is not equal to *any* finite set of natural numbers > F. > So N cannot be a finite set of natural numbers. But N *is* a set of > natural numbers, by definition; thus N must be a set of natural numbers > which is *not finite.* Q.E.D.: N is infinite. > You have shown that if N is the set of all natural numbers then > N can not be finite. This does not prove that N exists. > Your proof that N exists is that N exists by definition. Looks like you have forgotten about the axiom of infinity. > > I was trying to, but I didn't expect to get away with it. > So, N exists because we assume it exists. > > This axiom > says there is a set A such that (1) 0 = {} is a member of A, and (2) A is > closed under the successor operation. That is, if n is in A, then n+1 = > n U {n} is also in A. The set A certainly contains all the natural numbers, but it may also > contain some things that are not natural numbers. What to do? Simple. > Just apply the elephant algorithm. To produce a statue of an elephant, > you start with a big block of marble, and cut away everything that > doesn't look like an elephant. To produce the set N of natural numbers, > you start with the set A, which is guaranteed to exist by the axiom of > infinity, and then cut away everything that doesn't look like a natural > number. We say a subset B of A is inductive if 0 in B and B is closed under the > successor operation. > > Does the Axiom of Infinity say there is more than one such set? > Is B a proper subset of A? If A has infinitely many members, and x is any one of them then A - {x} is a proper subset of A and still has infinitely many members. And, in fact, one can generate infinitely many proper subsets in this way. > > Let I be the collection of all inductive subsets of > A. We know I is nonempty, since A is a member of I. > > So we are including non-proper subsets. > Set A could be the only member of I. Possibly, if one wants to be limited to subsets of the (zero-origin) naturals, N, one may require that A = N. > > Define N to be the > intersection of all members of I. Then N is demonstrably a model for the > Peano Axioms. It contains all the natural numbers and nothing else. > > Only if we assume there is a B that contains only natural numbers. > Why would we assume this? What if A is the only member of I? If we take the von Neumann construction of N, the existence of any infinite set guarantees an N. > > My proof shows that no set can contain every computable natural > number. >> Trivially wrong. > Not if we assume there is a device that can perform an > infinite number of computations. > I am surprised that people accept Cantor's proof that > the reals are uncountable but won't accept my proof. > They are essentially the same proof. One can accept > both proofs or reject both proofs, but I don't see > how anyone can reject one proof and accept the other. Apply your proof to the set N as constructed above. > > N was not constructed. We assumed that N exists. > > What natural number is not in N? > > I describe such a number. Are you sure you want me to post it? > It is a REALLY BIG number. If your REALLY BIG number is not 0 and is not the successor of some member of N then it is not a natural number. And if it is either 0 or the successor of a member of N, it is in N. > The diagonal proof can be converted into the proof I give. > (I won't call it Cantor's proof because his proof is about powersets.) You keep saying you can do it but you never have not done it. At least to the satisfaction of anyone but yourself. This is the same kind of situation James Harris is always in. Satisfying _only_ yourself in a mathematical exposition is the mathematical equivalent of masterbation. You may find it fun but it is forever non-productive. > > Given a set of real numbers, A, let x be a a real number not in A. > Describe an algorithm to compute x. You must first describe an algorithm to compute the numbers which _are_ in A, so that I will know what set A you are talking about. Now if you mean a list of reals (a function from N to R), THEN I can do it the same way Cantor did it. > > Let z be the i'th digit of the i'th entry in A. > If z=1 then the i'th digit of x is 2 else the i'th digit of x is 1. > > Let A be the following set: > > 0.0000... > 0.1000... > 0.1100... > 0.1110... > Clearly, if A is finite, the diagonal number will > be of the form 0.111...1 and will not be in A. > > Let me define another way to compute x. > Let's call this x'. > Let z be the i'th entry of A. > If z has equal or more 1's than x', let x' have > exactly one more 1 than z. > > x is exactly equal to x' at every step of our > computation. If A is finite then x'=x. > The only difference when A is infinite is that > x' provably has a finite number of 1's. > x' has exactly one more 1 than some > member of A. Since every member > of A has a finite number of 1's, > x' must also have a finite number of 1's. > > Many people have complained that > my method will never end. > This same complaint can be made > of the diagonal proof. But for the diagonal proof, I can, as required, described my algorithm exactly. That is ALL I need to do to validate the theorem. Ending, whatever that means, is not a requirement, constructing the algorithm is. ==== Many people have complained that > my method will never end. > This same complaint can be made > of the diagonal proof. But for the diagonal proof, I can, as required, described my algorithm > exactly. That is ALL I need to do to validate the theorem. I described my algorithm. Which part did you not understand? > Ending, whatever that means, is not a requirement, constructing the > algorithm is. Bull. The algorithm in the diagonal proof requires an infinite number of computations. If you continue to insist my method will never end then the diagonal method never ends, either. Russell - 2 many 2 count ==== > Many people have complained that > my method will never end. > This same complaint can be made > of the diagonal proof. But for the diagonal proof, I can, as required, described my algorithm > exactly. That is ALL I need to do to validate the theorem. > > I described my algorithm. Which part did you not understand? Why you think it proves anything. > > Ending, whatever that means, is not a requirement, constructing the > algorithm is. > > Bull. > The algorithm in the diagonal proof requires an infinite number of > computations. If you continue to insist my method will never end > then the diagonal method never ends, either. For any listing of the reals, the diagonal algorithm desribes a real number that cannot be in the list. No construction steps are needed since the number described must already exist as soon as the list exists. Given any listing of reals ( mapping f: {1,2,3,4,...} -> R) there is a real number, x, not present in that listing (image of the mapping). One can desribe one such a number (there are lots of others) as follows: Let x_m be the m'th decimal digit in a decimal representation of any real x, then there is a real number x such that x_n = 1 if f(n)_n <> 1 and x_n = 2 if f(n)_n = 1. A number, and one must exist, having these properties must be different from each member of the list. Note that no infinite constructions are needed. All the infinities needed exist prior to the identification of the un;listed number. ==== > Many people have complained that > my method will never end. > This same complaint can be made > of the diagonal proof. > > But for the diagonal proof, I can, as required, described my algorithm > exactly. That is ALL I need to do to validate the theorem. I described my algorithm. Which part did you not understand? Why you think it proves anything. Because you just said that is ALL I need to do to validate the theorem. By your defintion, my theorem must be valid. > Ending, whatever that means, is not a requirement, constructing the > algorithm is. > Bull. > The algorithm in the diagonal proof requires an infinite number of > computations. If you continue to insist my method will never end > then the diagonal method never ends, either. For any listing of the reals, the diagonal algorithm desribes a real > number that cannot be in the list. No construction steps are needed > since the number described must already exist as soon as the list exists. If no construction steps are needed why bother to describe an algorithm? Why not just say the list is incomplete and be done? Are you saying my number doesn't exist because it has to be constructed? The diagonal number has to be constructed and it requires an infinite number of computations. Russell 8 ==== >> We say a subset B of A is inductive if 0 in B and B is closed under the >> successor operation. >> >> Does the Axiom of Infinity say there is more than one such set? >> Is B a proper subset of A? If A has infinitely many members, and x is any one of them then A - {x} > is a proper subset of A and still has infinitely many members. And, in > fact, one can generate infinitely many proper subsets in this way. That didn't answer his question. The axiom of infinity at issue is the one that says there is an inductive set. Typically, removing an element from an inductive set does not yield another inductive set. However, if A is inductive, then so is s(A), so this axiom of infinity plus pairing plus binary union ensures that there are infinitely many inductive sets. -- This confused and outraged many Matrix fans, who'd already spent hours on the web explaining that man and computers could never really live in such a state of harmony and mutual benefit. -- http://www.pointlesswasteoftime.com ==== > We say a subset B of A is inductive if 0 in B and B is closed under the >> successor operation. >> Does the Axiom of Infinity say there is more than one such set? >> Is B a proper subset of A? If A has infinitely many members, and x is any one of them then A - {x} > is a proper subset of A and still has infinitely many members. And, in > fact, one can generate infinitely many proper subsets in this way. That didn't answer his question. The axiom of infinity at issue is > the one that says there is an inductive set. Typically, removing an > element from an inductive set does not yield another inductive set. However, if A is inductive, then so is s(A), so this axiom of infinity > plus pairing plus binary union ensures that there are infinitely many > inductive sets. But none of these inductive sets are subsets of N. Dave Seaman's proof that N only contains natural numbers requires inductive subsets of the set guaranteed to exist by the Axiom of Infinity. Russell - 2 many 2 count ==== >> We say a subset B of A is inductive if 0 in B and B is closed under > the > successor operation. > Does the Axiom of Infinity say there is more than one such set? > Is B a proper subset of A? > > If A has infinitely many members, and x is any one of them then A - {x} > is a proper subset of A and still has infinitely many members. And, in > fact, one can generate infinitely many proper subsets in this way. That didn't answer his question. The axiom of infinity at issue is > the one that says there is an inductive set. Typically, removing an > element from an inductive set does not yield another inductive set. However, if A is inductive, then so is s(A), so this axiom of infinity > plus pairing plus binary union ensures that there are infinitely many > inductive sets. > > But none of these inductive sets are subsets of N. > Dave Seaman's proof that N only contains natural numbers > requires inductive subsets of the set guaranteed to exist > by the Axiom of Infinity. Actually, we can take ANY inductive set with any initial element and any compatible successor operation and DEFINE it as N. That is, if it satisfies the Peano postulates, it is a suitable N. If the axiom of infinity guarantees us a Peano set, we are home free. <97adneZXNdeRbWCi4p2dnA@comcast.com> <40009fc8$16$fuzhry+tra$mr2ice@news.patriot.net> <40044b66$29$fuzhry+tra$mr2ice@news.patriot.net> <4005a65f$29$fuzhry+tra$mr2ice@news.patriot.net> <87k73s8fsk.fsf@phiwumbda.org> <-PKdnRWH4KtqKprdRVn-iQ@comcast.com> ==== > We say a subset B of A is inductive if 0 in B and B is closed under > the > successor operation. >> Does the Axiom of Infinity say there is more than one such set? > Is B a proper subset of A? >> If A has infinitely many members, and x is any one of them then A - {x} >> is a proper subset of A and still has infinitely many members. And, in >> fact, one can generate infinitely many proper subsets in this way. >> That didn't answer his question. The axiom of infinity at issue is >> the one that says there is an inductive set. Typically, removing an >> element from an inductive set does not yield another inductive set. >> However, if A is inductive, then so is s(A), so this axiom of infinity >> plus pairing plus binary union ensures that there are infinitely many >> inductive sets. But none of these inductive sets are subsets of N. There are no inductive subsets of N. > Dave Seaman's proof that N only contains natural numbers > requires inductive subsets of the set guaranteed to exist > by the Axiom of Infinity. Not at all. The witness A to the axiom of infinity may well be N itself, so we do not require that there are proper inductive subsets. Look, let A be an inductive set and define N to be the set {a in A | for all inductive B (a in B)} Either N is a proper subset of A or it is exactly equal to A, it doesn't matter which. Moreover, it is easy to see that N satisfies the usual defining axioms for the natural numbers. In particular, it is not hard to show that N defined as above is inductive. -- This page contains information of a type (text/html) that can only be viewed with the appropriate Plug-in. Click OK to download Plugin. --- Netscape 4.7 error message ==== >> We say a subset B of A is inductive if 0 in B and B is closed under >> the >> successor operation. >> Does the Axiom of Infinity say there is more than one such set? >> Is B a proper subset of A? > If A has infinitely many members, and x is any one of them then A - {x} > is a proper subset of A and still has infinitely many members. And, in > fact, one can generate infinitely many proper subsets in this way. > That didn't answer his question. The axiom of infinity at issue is > the one that says there is an inductive set. Typically, removing an > element from an inductive set does not yield another inductive set. > However, if A is inductive, then so is s(A), so this axiom of infinity > plus pairing plus binary union ensures that there are infinitely many > inductive sets. >> But none of these inductive sets are subsets of N. > There are no inductive subsets of N. Except for N itself, of course. >> Dave Seaman's proof that N only contains natural numbers >> requires inductive subsets of the set guaranteed to exist >> by the Axiom of Infinity. In the first place, I don't see why you care about that. Your claim was that no set could contain all of the natural numbers, and A itself does that. > Not at all. The witness A to the axiom of infinity may well be N > itself, so we do not require that there are proper inductive subsets. > Look, let A be an inductive set and define N to be the set > {a in A | for all inductive B (a in B)} > Either N is a proper subset of A or it is exactly equal to A, it > doesn't matter which. Moreover, it is easy to see that N satisfies > the usual defining axioms for the natural numbers. In particular, it > is not hard to show that N defined as above is inductive. guaranteed to be nonempty, since A itself is a member. Therefore it makes sense to speak of the intersection of all inductive subsets of A, which is clearly a subset of A and turns out to be exactly N. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. ==== >> We say a subset B of A is inductive if 0 in B and B is closed under >> the >> successor operation. > Does the Axiom of Infinity say there is more than one such set? >> Is B a proper subset of A? >> If A has infinitely many members, and x is any one of them then A - {x} > is a proper subset of A and still has infinitely many members. And, in > fact, one can generate infinitely many proper subsets in this way. >> That didn't answer his question. The axiom of infinity at issue is > the one that says there is an inductive set. Typically, removing an > element from an inductive set does not yield another inductive set. >> However, if A is inductive, then so is s(A), so this axiom of infinity > plus pairing plus binary union ensures that there are infinitely many > inductive sets. > But none of these inductive sets are subsets of N. There are no inductive subsets of N. Except for N itself, of course. I meant to say that s(A) is not a subset of A. >> Dave Seaman's proof that N only contains natural numbers >> requires inductive subsets of the set guaranteed to exist >> by the Axiom of Infinity. In the first place, I don't see why you care about that. Your claim was > that no set could contain all of the natural numbers, and A itself does > that. True. I don't care. I was responding to Dave's proof. One can always assume there is a set of all natural numbers. One can also assume the moon is made of green cheese. > Not at all. The witness A to the axiom of infinity may well be N > itself, so we do not require that there are proper inductive subsets. And A may not be equal to N. > Look, let A be an inductive set and define N to be the set {a in A | for all inductive B (a in B)} Either N is a proper subset of A or it is exactly equal to A, it > doesn't matter which. Moreover, it is easy to see that N satisfies > the usual defining axioms for the natural numbers. In particular, it > is not hard to show that N defined as above is inductive. N may be inductive, but does it contain members that are not natural numbers? > guaranteed to be nonempty, since A itself is a member. Therefore it > makes sense to speak of the intersection of all inductive subsets of A, > which is clearly a subset of A and turns out to be exactly N. But you haven't shown that every member of N is a natural number. I thought this was what Dave was trying to prove. Correct me if this wasn't his intent. You defined A as the set guaranteed to exist by the Axiom of Infinity. Then you define N as the smallest inductive subset of A. (I find it amusing that set theorists will assume they can find the smallest member of an infinite set of infinite sets, but refuse to believe there could be a smallest strictly positive rational number.) So N is the smallest inductive set. How does this prove that every member of N is a natural number? You can't assume the smallest inductive set only contains natural numbers. That is what you are trying to prove. I think my proof shows that induction and hypercomputation will come to different conclusions on certain problems. You can't prove N exists using just hypercomputation. (Actually, you can prove N doesn't exist.) And you can't prove N exists using induction because you have to assume A exists to use induction. N is defined as a subset of A. Russell - Zeno was right. Motion is impossible. ==== > You defined A as the set guaranteed to exist by the Axiom of Infinity. > Then you define N as the smallest inductive subset of A. > (I find it amusing that set theorists will assume they can find the > smallest member of an infinite set of infinite sets, but refuse to > believe there could be a smallest strictly positive rational number.) The intersection of an arbirary family of sets is itself a set, though possibly empty, so there is always such a minimal set. If for positive real r we let f(r) be the set of all positive rational numbers less than r, then the intersection of all f(r) is the empty set which does not contain any rational or real or anything else. If you wish to insist that the empty set is not empty, you are going to have a tough time with math and logic. <40044b66$29$fuzhry+tra$mr2ice@news.patriot.net> <4005a65f$29$fuzhry+tra$mr2ice@news.patriot.net> <87k73s8fsk.fsf@phiwumbda.org> <-PKdnRWH4KtqKprdRVn-iQ@comcast.com> <871xq0rs04.fsf@phiwumbda.org> ==== > You defined A as the set guaranteed to exist by the Axiom of Infinity. >> Then you define N as the smallest inductive subset of A. >> (I find it amusing that set theorists will assume they can find the >> smallest member of an infinite set of infinite sets, but refuse to >> believe there could be a smallest strictly positive rational number.) The intersection of an arbirary family of sets is itself a set, though > possibly empty, so there is always such a minimal set. Not so! There is not always a minimal set of those sets with property P. For instance, when we take the intersection of all the inductive sets, we have to show that the resulting set is indeed inductive. For any collection of sets, there is a greatest lower bound, but that greatest lower bound is not necessarily a least set for the collection. Consider the set of all proper supersets of {c}. This collection has no least element (and also is non-empty). -- However, you presuppose that certain numbers *are* prime ideals, ... when in fact ...* they are not... (Maybe I should look up 'prime ideals' but the effort doesn't seem to be worth it. I assume some poster will get excited ... if I messed up.) --James Harris ==== > One can always assume there is a set of all natural numbers. > One can also assume the moon is made of green cheese. Mathematical assumptions and physical ones are of quite different natures. ==== One can always assume there is a set of all natural numbers. > One can also assume the moon is made of green cheese. Mathematical assumptions and physical ones are of quite different > natures. They share at least one property. Both types of assumptions can be false. Russell - 2 many 2 count ==== > N may be inductive, but does it contain members that are not natural > numbers? You are obviously not familiar with the Peano properties, one of which answers your question in the negative. ==== > We say a subset B of A is inductive if 0 in B and B is closed > under > the > successor operation. > Does the Axiom of Infinity say there is more than one such set? > Is B a proper subset of A? >> If A has infinitely many members, and x is any one of them then A - > {x} >> is a proper subset of A and still has infinitely many members. And, > in >> fact, one can generate infinitely many proper subsets in this way. >> That didn't answer his question. The axiom of infinity at issue is >> the one that says there is an inductive set. Typically, removing an >> element from an inductive set does not yield another inductive set. >> However, if A is inductive, then so is s(A), so this axiom of infinity >> plus pairing plus binary union ensures that there are infinitely many >> inductive sets. > But none of these inductive sets are subsets of N. >> There are no inductive subsets of N. >> Except for N itself, of course. > I meant to say that s(A) is not a subset of A. > Dave Seaman's proof that N only contains natural numbers > requires inductive subsets of the set guaranteed to exist > by the Axiom of Infinity. >> In the first place, I don't see why you care about that. Your claim was >> that no set could contain all of the natural numbers, and A itself does >> that. > True. I don't care. I was responding to Dave's proof. > One can always assume there is a set of all natural numbers. > One can also assume the moon is made of green cheese. >> Not at all. The witness A to the axiom of infinity may well be N >> itself, so we do not require that there are proper inductive subsets. > And A may not be equal to N. >> Look, let A be an inductive set and define N to be the set >> {a in A | for all inductive B (a in B)} >> Either N is a proper subset of A or it is exactly equal to A, it >> doesn't matter which. Moreover, it is easy to see that N satisfies >> the usual defining axioms for the natural numbers. In particular, it >> is not hard to show that N defined as above is inductive. > N may be inductive, but does it contain members that are not natural > numbers? No, it doesn't, by the inductive property of the natural numbers. >> guaranteed to be nonempty, since A itself is a member. Therefore it >> makes sense to speak of the intersection of all inductive subsets of A, >> which is clearly a subset of A and turns out to be exactly N. > But you haven't shown that every member of N is a natural number. > I thought this was what Dave was trying to prove. > Correct me if this wasn't his intent. Remember the Peano axioms? They define the properties of the natural numbers. The first four axioms say that the naturals are an inductive set, and the fifth axiom says that they form the smallest possible inductive set. If an inductive set has no proper inductive subsets, then by definition the natural numbers are the members of that set. > You defined A as the set guaranteed to exist by the Axiom of Infinity. > Then you define N as the smallest inductive subset of A. > (I find it amusing that set theorists will assume they can find the > smallest member of an infinite set of infinite sets, but refuse to > believe there could be a smallest strictly positive rational number.) It's not an assumption; it's a construction. If A is an inductive set, then N = { x in A : x in S for every inductive subset S of A }. This is the elephant algorithm at work; given a set, we can carve out a subset consisting of all the elements having a specific property. > So N is the smallest inductive set. How does this prove that > every member of N is a natural number? You can't assume > the smallest inductive set only contains natural numbers. > That is what you are trying to prove. Says who? If the natural numbers are not the objects satisfying the Peano axioms, then what are they? But maybe you'll be happier with the following alternate approach. We define an ordinal to be a transitive set that is well ordered by set membership. We define a natural number to be an ordinal that is also well ordered by the inverse of set membership. So in particular, each natural number has both a first and a last element, and the same is true of every nonempty subset of a natural number. This does not hold for transfinite ordinals, which are well ordered only in the natural order and not in the reverse order. As before, we let A be the set whose existence is guaranteed by the axiom of infinity. We let w be defined by w = { x in A : x is a natural number }, where natural number is as defined above. It's easy to show that the natural numbers by this definition do satisfy the Peano axioms. The set w that we wind up with is identical to the set N that we get by taking the intersection of all inductive subsets of A. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. ==== > Either N is a proper subset of A or it is exactly equal to A, it >> doesn't matter which. Moreover, it is easy to see that N satisfies >> the usual defining axioms for the natural numbers. In particular, it >> is not hard to show that N defined as above is inductive. N may be inductive, but does it contain members that are not natural > numbers? No, it doesn't, by the inductive property of the natural numbers. > Is there an axiom or something that says only natural numbers have this inductive property. Sounds like another assumption to me. Russell - 2 many 2 count ==== >> Either N is a proper subset of A or it is exactly equal to A, it > doesn't matter which. Moreover, it is easy to see that N satisfies > the usual defining axioms for the natural numbers. In particular, it > is not hard to show that N defined as above is inductive. > N may be inductive, but does it contain members that are not natural > numbers? No, it doesn't, by the inductive property of the natural numbers. > Is there an axiom or something that says only natural numbers > have this inductive property. Sounds like another assumption > to me. > > > Russell > - 2 many 2 count > > Do you have any idea of what the Peano Postulates say? ==== > Either N is a proper subset of A or it is exactly equal to A, it > doesn't matter which. Moreover, it is easy to see that N satisfies > the usual defining axioms for the natural numbers. In particular, it > is not hard to show that N defined as above is inductive. >> N may be inductive, but does it contain members that are not natural >> numbers? >> No, it doesn't, by the inductive property of the natural numbers. > Is there an axiom or something that says only natural numbers > have this inductive property. Sounds like another assumption > to me. The fifth Peano axiom. Whether it's an assumption depends on your starting point. If you view the Peano axioms as your starting point, then it's one of your five assumptions. If you view ZF as your starting point, then the Peano axioms become theorems regarding the natural numbers (= finite ordinals). -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. ==== >> Either N is a proper subset of A or it is exactly equal to A, it > doesn't matter which. Moreover, it is easy to see that N satisfies > the usual defining axioms for the natural numbers. In particular, it > is not hard to show that N defined as above is inductive. >> N may be inductive, but does it contain members that are not natural >> numbers? >> No, it doesn't, by the inductive property of the natural numbers. Is there an axiom or something that says only natural numbers > have this inductive property. Sounds like another assumption > to me. The fifth Peano axiom. Whether it's an assumption depends on your starting point. If you view the > Peano axioms as your starting point, then it's one of your five assumptions. > If you view ZF as your starting point, then the Peano axioms become theorems > regarding the natural numbers (= finite ordinals). This is the fifth Peano Axiom from Mathworld: If a set S of numbers contains zero and also the successor of every number in S, then every number is in S. We started with ZF because we used the Axiom of Infinity to show that there exists a set, A, that contains zero and is closed under the successor function. We can assume A contains every natural number and is an inductive set as defined by fifth Peano Axiom. The question at hand is what else does A contain? It could contain members that are not natural numbers. You define a set, I, that contains every inductive subset (not necessarily proper) of A. You then define N as the intersection of every member of I. You claim that N contains only natural numbers and that it is an inductive set (ie, it contains every natural number). Let's call any set that contains zero and is closed under the successor function a closed set. Obviously, A is a closed set. Here are a few questions I have. Is the intersection of two closed set guaranteed to be a closed set? For the sake of argument, let's say that a closed set must contain some member that is not a natural number. Let's say it must contain an Aleph to be closed. Let set x contain all of the natural numbers and Aleph_0 and set y contain all of the natural numbers and Aleph_1. Let z be the intersection of x and y. z is not a closed set using our (temporary) definition of a closed set. How would we prove that z contains every natural number if we can not prove z is closed? I do not see how your definition of N guarantees that N is closed under the successor function. If N is not closed under the successor function, how can you prove that N contains every natural number? Russell - 2 many 2 count ==== > This is the fifth Peano Axiom from Mathworld: > > If a set S of numbers contains zero and also the successor of every number > in S, then every number is in S. > > We started with ZF because we used the Axiom of Infinity to show > that there exists a set, A, that contains zero and is closed under the > successor function. > > We can assume A contains every natural number and is an inductive set > as defined by fifth Peano Axiom. The question at hand is what else > does A contain? It could contain members that are not natural numbers. But consider the intersection of all inductive sets. We can exclude any non-natural number, and all its successors other than zero and still have the inductive property. So how that intersection contain anything except natural numbers? THAT is the set we call N, and its members are the objects we call natural numbers, and it can't contain anything else. ==== > Remember the Peano axioms? They define the properties of the natural > numbers. The first four axioms say that the naturals are an inductive > set, and the fifth axiom says that they form the smallest possible > inductive set. If an inductive set has no proper inductive subsets, then > by definition the natural numbers are the members of that set. Whoops, wrong, I got in trouble for saying this earlier. The Peano axioms do not define the properties of the natural numbers; more than one non-isomorphic structure could satisfy those axioms. The natural numbers are the unique set which satisfies the Peano axioms *and* for which every natural number has a numeral. Thomas ==== >> Remember the Peano axioms? They define the properties of the natural >> numbers. The first four axioms say that the naturals are an inductive >> set, and the fifth axiom says that they form the smallest possible >> inductive set. If an inductive set has no proper inductive subsets, then >> by definition the natural numbers are the members of that set. > Whoops, wrong, I got in trouble for saying this earlier. The Peano > axioms do not define the properties of the natural numbers; more than > one non-isomorphic structure could satisfy those axioms. The natural > numbers are the unique set which satisfies the Peano axioms *and* for > which every natural number has a numeral. I'm not sure I follow this. Perhaps you are thinking of nonstandard models. For example, the hyperintegers of nonstandard analysis satisfy the Peano axioms when interpreted internally in that system, and the hyperintegers are not isomorphic to the standard construction of the finite ordinals. Is that what you are getting at, or is there something else? Within a particular model of set theory, the naturals are uniquely determined up to isomorphism by the Peano axioms. I don't see what numerals have to do with anything here. The hyperintegers don't have numerals. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. <4005a65f$29$fuzhry+tra$mr2ice@news.patriot.net> <87k73s8fsk.fsf@phiwumbda.org> <-PKdnRWH4KtqKprdRVn-iQ@comcast.com> <871xq0rs04.fsf@phiwumbda.org> <87hdyvyfwl.fsf@becket.becket.net> ==== > Remember the Peano axioms? They define the properties of the natural >> numbers. The first four axioms say that the naturals are an inductive >> set, and the fifth axiom says that they form the smallest possible >> inductive set. If an inductive set has no proper inductive subsets, then >> by definition the natural numbers are the members of that set. Whoops, wrong, I got in trouble for saying this earlier. The Peano > axioms do not define the properties of the natural numbers; more than > one non-isomorphic structure could satisfy those axioms. The natural > numbers are the unique set which satisfies the Peano axioms *and* for > which every natural number has a numeral. Better, the natural numbers are the unique (up-to-isomorphism) <0,s> structure satisfying second-order induction. -- Jesse F. Hughes Leaving things always seems to fix me, Running seems to ease my worried mind. -- Bad Livers, Honey, I've Found a Brand New Way ==== > (I find it amusing that set theorists will assume they can find the > smallest member of an infinite set of infinite sets, but refuse to > believe there could be a smallest strictly positive rational number.) Simply because they have a proof that there is no such thing :-) -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Needless to say, I had the last laugh. Alan Partridge, _Bouncing Back_ (14 times) ==== > My proof shows that no set can contain every computable natural > number. There is no such thing as the set of all natural numbers. > There will always be a computable natural number larger than > every member in the set. That shows you to be stuck back in the 18th century, with Kronecker and his ilk. ==== I consider the statement rationals are uncountable. The definition of uncountable is a set is uncountable if there is no possible bijection between it and the naturals, and the set is not finite. The rationals are countable because there are 1-1 functions between the rationals and naturals. That's not to say it's cut and dried, there are certain complexities of dealing with the infinite. It's certainly fair to say that of all the rationals, only infinitesimally many of them are integers. There are infinitely many integers, but for each, we can define a unit neighborhood around it containing no other integers but containing infinitely many rational numbers. The converse is not true. The integers are a proper subset of the rationals, and as well, infinitely many proper supersets of the integers are proper subsets of the rationals. There are more rationals than integers for a variety of reasons. Another definition of uncountable is a set is uncountable is there are bijections between it and another uncountable (by the previous definition) set. To explore this most immediately what there is is called the Cantor-Bernstein theorem which tells us that the existence of an injection either way between two sets is the evidence of the existence of a bijection between the two sets. Then, there's something along the lines of an existence of a bijection between sets A and B and B and C implies those between A and C, and thus where we know N<->Q and not N<->P(N) that not Q<->P(N). A finitistic attack on this is to use the number of representable sequences is b^p for base b and precision p. In talking about uncountability, what appears is the discussion of the map between an infinite set and its powerset. In not considering the universal set, which is its own powerset, we have to approaches to show the infeasibility of mapping an infinite set to its powerset. Now, don't get me wrong here, there is some mathematics that is quite valid without this theorem and it can be axiomatized away quite brusquely and integral calculus or non-standard analysis is unaffected. Anyways, there are two basic approaches, the set theoretic one says that for any function f(x) from a set X to its powerset, that one element of the powerset, denoted S, is the set of all values where if not(x E f(x)) that x E S. S is supposed to be precluded from being in the range of the function f in this way. In consideration of the naturals X as ordinals and with f(x)=x+1, S={}, which is a subset of X and thus an element of P(X). With f(x)=x, S=X, which is also a subset of X and thus an element of P(X). Here one idea I had about that was to consider instead of the powerset a construction appelled the proper powerset, containing each proper subset of the set. Then, for an infinite set, there is only one element difference between the powerset P(X) and a proper powerset PP(X), that being the element X E P(X), so basically there is one element missing. Another consideration of the powerset mapping is basically geometric in tone, it says create a list of the infinite binary sequences with for each index of the sequence that if the value is a 1 then the element with that index's value is an element of that represented subset, else it is not. Then, the antidiagonal is constructed which is different at each i'th place in the antidiagonal from the i'th element of the i'th sequence. Again in light of the proper powerset, various methods have the diagonal being all zeros and thus the antidiagonal being all ones, and thus not an element of the proper powerset, which differs from the powerset by one element. I think we'd all agree that any infinite set has a trivial mapping to an infinite subset that is only disjoint one element, but we don't.That two infinite sets can map ot each other used to be called Galileo's paradox, and now it's not. Very few integers are perfect squares, the density of perfect squares among the integers is infinitesimal, and characteristic of the density of the perfect squares within the integers. One proof that is not in fashion very much these days is about nested intervals of the reals, showing that there are always more reals in the neighborhood of a real, the reals being dense in the reals, thus that the reals are uncountable. That was the statement of that in times past, it is trivial that it is true for any dense set. I haven't spent enough time reading the original poster's, Russell's, statement about what he means by rationals are uncountable. He seems to make the sequence with variable and increasing radix. For example, the sequence .1234 generally means 1*10^-1 + 2*10^-2 + representing rationals as 1*2^-1 +2*6^-2 + 3*24^-3, or something along those lines. What's the resolution thus far of Easterly's argument? I'm somewhat lost among computability. So anyways, countability and uncountability are words about cardinality, and specifically the cardinality of infinite sets. Can cardinality explain why the density of even numbers within the integers is 1/2? No, it doesn't and plainly cannot as it is blind to that distinction vis-a-vis multiples of three, with density 1/3, or other number-theoretic attributes of the infinite set of numbers. Name one empirical, real-world result that hinges and rests upon the cardinality of an infinite set. While you're at it, reconcile Mssrs. Borel and Combinatorics. Ross F. <40044b66$29$fuzhry+tra$mr2ice@news.patriot.net> <4005a65f$29$fuzhry+tra$mr2ice@news.patriot.net> <3c6b9c1e.0401151545.216f012c@posting.google.com> ==== > It's certainly fair to say that of all the rationals, only > infinitesimally many of them are integers. This is an exciting new discovery. -- Jesse Hughes Besides, discoverers are too proud to kiss butt. Indiana Jones would never kiss some academic's ass to get published, and neither will I. --James Harris ==== > I consider the statement rationals are uncountable. The definition of uncountable is a set is uncountable if there is no > possible bijection between it and the naturals, and the set is not > finite. The rationals are countable because there are 1-1 functions between > the rationals and naturals. > I will admit that my use of the word uncountable is not precise. I am trying to show there is no 1-1 functions from the naturals to the naturals. If this is true then countable and uncountable are meaningless. Russell - 2 many 2 count ==== > I consider the statement rationals are uncountable. The definition of uncountable is a set is uncountable if there is no > possible bijection between it and the naturals, and the set is not > finite. The rationals are countable because there are 1-1 functions between > the rationals and naturals. > I will admit that my use of the word uncountable is not precise. > I am trying to show there is no 1-1 functions from the naturals > to the naturals. If this is true then countable and uncountable > are meaningless. > > > Russell > - 2 many 2 count Yo I play right field? more acceptible than saying that any inconsistency validates all things. (Free association.) We're mathematicians, not only can a word have two meanings, it can have both at the same time, and then neither, and it does. They also have their meanings. I think that fundamental algebra, geometry, most forms of analysis, probability, etcetera are definable in terms that don't require the notion of countability using instead other characteristics of the elements of those explications. I say: bring forth the utility of countability, Virgil wants to take the ball home. That's a bit reaching, Virgil and I have academic tolerance of each other that is not often noted. Anyways his basic reply illustrates his personal enjoinment from appearing enthusiastic, and his goal to remain part of the argument to drop the telling blow, when he finally caves. He prefers to continue the argument, because he would actually have to have something. The hyperreals are the reals, and the hyperintegers are the integers. It's just a different pair of names to discuss them at ease with About no axioms, a subset of ZF comprises a quite adequate set of kernel theorems, restated from axioms. There's not much wrong with them. The universe is one-dimensional, in a sense, the universe is infinite-dimensional, that is map each characteristic of each thing, in the infinite universe, to its own chart axis, or put it all on one line. Neither consideration explains our visible universe very well, which consists of approximately three space dimensions and one time dimension, or three and two, with perhaps a few more for mass, energy, and other. I'm reminded of the uniform random distribution over the naturals (rationals, reals, hypercomplices). It's difficult to construct one, but were one to exist, there are properties to which we could assign it, thus it may be used as a tool in further developments, contigent upon such discovery. Don't forget the pneuma! Ross <4005a65f$29$fuzhry+tra$mr2ice@news.patriot.net> <3c6b9c1e.0401151545.216f012c@posting.google.com> <3c6b9c1e.0401161451.174a4b39@posting.google.com> ==== > We're mathematicians, not only can a word have two meanings, it can > have both at the same time, and then neither, and it does. They also > have their meanings. I get it. You're using the claim that you're a mathematician as an example of how a word can have multiple meanings, right? 'Cause I'm pretty sure that you're using the word differently than I do. -- If you see math knowledge as a tool--as a hammer--with which you can attack other people then ... you defeat rational discourse. I get to call my proof the Hammer. It's more powerful than *any* physical object. It is overwhelming force. -- Two JSH quotes ==== > I will admit that my use of the word uncountable is not precise. > I am trying to show there is no 1-1 functions from the naturals > to the naturals. If this is true then countable and uncountable > are meaningless. Huh? Here's one: f(x) = x Here's another: { x+1 if 0 <= x < 100 f(x) = { 0 if x = 100 { x if x > 100 Thomas ==== >> I consider the statement rationals are uncountable. >> The definition of uncountable is a set is uncountable if there is no >> possible bijection between it and the naturals, and the set is not >> finite. >> The rationals are countable because there are 1-1 functions between >> the rationals and naturals. I will admit that my use of the word uncountable is not precise. >I am trying to show there is no 1-1 functions from the naturals >to the naturals. If this is true then countable and uncountable >are meaningless. Define f(n) = n for all natural n. 1. Show that f maps the naturals to the naturals. 2. Show that f is 1-1. Rob Johnson take out the trash before replying ==== > I consider the statement rationals are uncountable. The definition of uncountable is a set is uncountable if there is no > possible bijection between it and the naturals, and the set is not > finite. The rationals are countable because there are 1-1 functions between > the rationals and naturals. > I will admit that my use of the word uncountable is not precise. > I am trying to show there is no 1-1 functions from the naturals > to the naturals. If this is true then countable and uncountable > are meaningless. One of your major faults is that your use of many words is n ot precise enough for mathematical discussions. And your thesis is trivially false, since the identity mapping on any set is 1-1 and onto ( bijective ) from the set to itself. <40044b66$29$fuzhry+tra$mr2ice@news.patriot.net> <4005a65f$29$fuzhry+tra$mr2ice@news.patriot.net> <3c6b9c1e.0401151545.216f012c@posting.google.com> ==== > I will admit that my use of the word uncountable is not precise. >> I am trying to show there is no 1-1 functions from the naturals >> to the naturals. If this is true then countable and uncountable >> are meaningless. One of your major faults is that your use of many words is n ot precise > enough for mathematical discussions. And your thesis is trivially false, since the identity mapping on any > set is 1-1 and onto ( bijective ) from the set to itself. I think Russell wants to show a contradiction, so the fact that there are obvious proofs of the negation of his statement is not an argument against his claimed proof of the statement. I will not comment on how smart it is for a fellow to try to show that ZF is inconsistent with such rudimentary tools. Well, I won't comment *explicitly* anyway. -- My proof has been checked very thoroughly, both by me and others. Those others apparently decided that they would not believe the proof was correct, but cannot support that position using mathematics. But hey, they're just human beings. --JSH, prover of Fermat's Last Thm ==== > I consider the statement rationals are uncountable. > > The definition of uncountable is a set is uncountable if there is no > possible bijection between it and the naturals, and the set is not > finite. The more common definition allows that finite sets are countable, since it is fairly ridiculous to say that a set whose mambers can be counted in the everyday sense is uncountable. > > The rationals are countable because there are 1-1 functions between > the rationals and naturals. > > That's not to say it's cut and dried, there are certain complexities > of dealing with the infinite. > > It's certainly fair to say that of all the rationals, only > infinitesimally many of them are integers. It is a very silly thing to say, without giving a definition of what you mean by infinitesimal in this context. > There are infinitely many integers, but for each, we can define a > unit neighborhood around it containing no other integers but > containing infinitely many rational numbers. The converse is not > true. Depending on one's models for the integers, Z, and rationals, Q, it may be that Z is not a subset of Q and your assertion is then false. > > The integers are a proper subset of the rationals, and as well, > infinitely many proper supersets of the integers are proper subsets of > the rationals. There are more rationals than integers for a variety > of reasons. And the same number, also for a variety of reasons, > > Another definition of uncountable is a set is uncountable is there > are bijections between it and another uncountable (by the previous > definition) set. > > To explore this most immediately what there is is called the > Cantor-Bernstein theorem which tells us that the existence of an > injection either way between two sets is the evidence of the existence > of a bijection between the two sets. I think you should have said an injection EACH way. The usual meaning of either way would only require one way but not the reverse. > > Then, there's something along the lines of an existence of a bijection > between sets A and B and B and C implies those between A and C, and > thus where we know N<->Q and not N<->P(N) that not Q<->P(N). A > finitistic attack on this is to use the number of representable > sequences is b^p for base b and precision p. > > In talking about uncountability, what appears is the discussion of the > map between an infinite set and its powerset. > In not considering the > universal set, which is its own powerset, What universal set? If you posit a universe outide of which nothing exists, then you cannot also posit its powerset. So either give up on a universal set or on universal power sets, you can't have both and have a sensible set theory. [snipped the rest as too out in left field.] ==== > The more common definition allows that finite sets are countable, since > it is fairly ridiculous to say that a set whose mambers can be counted > in the everyday sense is uncountable. Perhaps, but the usual definition for set theorists is that countable refers specifically to set that have the cardinality the natural numbers. It's handy to have a word that means exactly that, and so one was invented. (Perhaps this means I should stop getting bothered by physicists who use finite to mean non-zero.) Thomas ==== My proof shows that no set can contain every computable natural > number. There is no such thing as the set of all natural numbers. > There will always be a computable natural number larger than > every member in the set. That shows you to be stuck back in the 18th century, with Kronecker and > his ilk. Actually, I never got past Zeno. Russell - Zeno was right. Motion is impossible. ==== > My proof shows that no set can contain every computable natural > number. There is no such thing as the set of all natural numbers. > There will always be a computable natural number larger than > every member in the set. That shows you to be stuck back in the 18th century, with Kronecker and > his ilk. > > Actually, I never got past Zeno. > > > Russell > - Zeno was right. Motion is impossible. > > Motion in your intellectual processes, at least. In detrmining your position, shouldn't that name be Zero? ==== > Actually, I never got past Zeno. Really? Which texts of his did you read? Thomas ==== Actually, I never got past Zeno. Really? Which texts of his did you read? Thomas If I remember right, none of Zeno's texts survived to present day. All we know of his paradoxes come from people that were been even more compelling than the ones we attribute to him. These are some good web site describing Zeno's paradoxes: http://history.hanover.edu/texts/presoc/zeno.htm http://plato.stanford.edu/entries/paradox-zeno/ Even though the Achilles and the Tortoise is the best known paradox, the Stadium paradox is probably the most devastating. My favorite is the Arrow paradox. Russell - Nothing has ever happened. Change is an illusion. ==== > If I remember right, none of Zeno's texts survived to present day. OK, you dodged my trap. :) A little mean, but... > All we know of his paradoxes come from people that were > trying to prove him wrong. Which of these have you read? What are your criticisms of their attempted resolutions of Zeno's paradoxes? For example, Aristotle's discussion seems half right and half wrong; the part which is wrong is wrong mostly because he doesn't understand the continuum very well. What do you make of the modern way of resolving the paradoxes? Thomas ==== in > In , on 01/13/2004 > at 03:43 PM, Russell Easterly said: > >If this is your definition of a TM then I can write a TM that >performs the operations I have described. > > Please do. > > OK > This TM will find a natural number larger than any natural number on the > input tape. Since, below, you assume an input tape contains every natural number, > you are saying that a TM can do what humans can prove cannot be done, > namely, find a natural number larger than any natural number. > > This TM does not find the largest natural number. > My TM will find a natural number larger than any natural number > on the input tape. This only proves the input tape does not > contain every natural number. Then you are assuming a finite tape, not a tape on which every natural can be listed. After all your mumbling about TMs which could do infinitely many steps in finite time, this was not clear. > > > Assume the input tape is: 01011011101111011111.... > Each natural number is represented in unary and followed by a 0. You mean, for a finite tape, each natural up to the largest on the tape. > > Starting at the beginning of the tape and reading right: > > 1) Find a zero > 2) Find a second zero > 3) Backup and write a 1 on the previous zero > Repeat steps 1-3. > > I can give you a state transition table if you want. > There will always be one zero on the tape when this TM finishes reading > it. But the TM does not finish. Ever. Since there is always another natural > on the tape that has not yet been touched. > > Using this argument the TM that produces the input tape never finishes > either. AFAIK, it was never required that the input tape be produced by a TM. It was merely assumed to exist. > So the input tape can't contain every natural number because it was > never completed. Let's assume that both TMs can perform an infinite > number of operations. Given infinite time, this is quite reasonable, but nnot in our lifetimes. > > The tape will contain a unary natural number larger than any on the > input > tape. > It doesn't matter how long the input tape is. As long as it is finite. If it is an infinite tape, there does not exist anything satisfyiing the definition of a natural number larger than every natural number, so your result is impossible. Unless the tape ends after a finite number of positions, NO. > > You can say that the TM will never finish reading the tape. > This doesn't matter, since at any point in time the TM > will still have produced a natural number larger than > any number it has read from the input tape. What happens at completion of a finite number of steps does not > determine the effect of completing infinitely many steps. > > How can the result be different after an infinite number of steps? Consider an infinite tape of 0's on which the TM merely prints an endless string of 1's, one at a time. At no time before infinitely many 1's are printed are completed has it printed infinitely many 1's. But once it HAS printed infinitely many 1's, if that ever happens, it HAS printed infinitely many 1's. The result is clearly different after infinitely many steps than after only finitely many steps. > > Is your TM tape finite or infinite? > If finite, then it is of no use in analyzing the infinite case any more > than a finite natural can be larger that all other finite naturals. > > The output of my TM must be finite. > There must be a finite number of 1's followed by a 0. > The proof shows that the input tape was finite even > though we assumed it was infinite. Your reasoning is circular, even though you may have assumed it to be linear. If the output of your TM is forever finite then it is forever prohibited from being infinite BY DEFINITION. Forever finite means NEVER infinite. Make up your petit mind which. You can't have it both ways. > > > Russell > - 2 many 2 count > > ==== > What happens at completion of a finite number of steps does not > determine the effect of completing infinitely many steps. How can the result be different after an infinite number of steps? Consider an infinite tape of 0's on which the TM merely prints an > endless string of 1's, one at a time. At no time before infinitely many > 1's are printed are completed has it printed infinitely many 1's. But > once it HAS printed infinitely many 1's, if that ever happens, it HAS > printed infinitely many 1's. The result is clearly different after > infinitely many steps than after only finitely many steps. How would you prove your TM has written an infinite number of 1's? It is not at all clear there is a difference unless you can provide such a proof. > Is your TM tape finite or infinite? > If finite, then it is of no use in analyzing the infinite case any more > than a finite natural can be larger that all other finite naturals. The output of my TM must be finite. > There must be a finite number of 1's followed by a 0. > The proof shows that the input tape was finite even > though we assumed it was infinite. Your reasoning is circular, even though you may have assumed it to be > linear. If the output of your TM is forever finite then it is forever prohibited > from being infinite BY DEFINITION. Forever finite means NEVER infinite. > Make up your petit mind which. You can't have it both ways. Neither tape can be infinite. Russell - 2 many 2 count ==== > Consider an infinite tape of 0's on which the TM merely prints an > endless string of 1's, one at a time. At no time before infinitely many > 1's are printed are completed has it printed infinitely many 1's. But > once it HAS printed infinitely many 1's, if that ever happens, it HAS > printed infinitely many 1's. The result is clearly different after > infinitely many steps than after only finitely many steps. > > How would you prove your TM has written an infinite number of 1's? > It is not at all clear there is a difference unless you can provide such > a proof. Things which are clear to everyone else are now unclear to your and things that are not clear to anyone else seem clear to you? perhaps you exist only in another universe. I do not have a TM of any sort. It is your TM''s we have been discussing. And no TM has ever, or will ever, write infinitely many 1's in finite time. That was and is and will be my point! Finite and infinite are different, even though you may be too dim to notice the difference. > > Is your TM tape finite or infinite? > If finite, then it is of no use in analyzing the infinite case any > more > than a finite natural can be larger that all other finite naturals. > > The output of my TM must be finite. > There must be a finite number of 1's followed by a 0. > The proof shows that the input tape was finite even > though we assumed it was infinite. Your reasoning is circular, even though you may have assumed it to be > linear. If the output of your TM is forever finite then it is forever prohibited > from being infinite BY DEFINITION. Forever finite means NEVER infinite. > Make up your petit mind which. You can't have it both ways. > > Neither tape can be infinite. > Then stop pontificating on what can never be. ==== Having now spent a little time surveying math society at a deeper Usenet, I've concluded that I may have simply spent an inordinate amount of time with a minor but very vocal section of the mathematical community. I'm now in the process of quieter and more focused contacts myself, and I see quite quickly that there's a big difference between what I've seen with relatively big names in the math community and posters than with people I'm finding who aren't so, um, talkative. That's good and it makes me feel better as I found it very troubling to consider facing a reality where no one can understand my work, but now I think that people who spend a lot of time chattering about math, are definitely not as representative as I originally thought, so yes, there's hope. James Harris ==== > Having now spent a little time surveying math society at a deeper > Usenet, I've concluded that I may have simply spent an inordinate > amount of time with a minor but very vocal section of the mathematical > community. > > I'm now in the process of quieter and more focused contacts myself, > and I see quite quickly that there's a big difference between what > I've seen with relatively big names in the math community and posters > than with people I'm finding who aren't so, um, talkative. > > That's good and it makes me feel better as I found it very troubling > to consider facing a reality where no one can understand my work, but > now I think that people who spend a lot of time chattering about math, > are definitely not as representative as I originally thought, so yes, > there's hope. But what types of things are they saying? Does anyone agree with you yet? -- Will Twentyman ==== > Having now spent a little time surveying math society at a deeper > Usenet, I've concluded that I may have simply spent an inordinate > amount of time with a minor but very vocal section of the mathematical > community. > > I'm now in the process of quieter and more focused contacts myself, > and I see quite quickly that there's a big difference between what > I've seen with relatively big names in the math community and posters > than with people I'm finding who aren't so, um, talkative. > > That's good and it makes me feel better as I found it very troubling > to consider facing a reality where no one can understand my work, but > now I think that people who spend a lot of time chattering about math, > are definitely not as representative as I originally thought, so yes, > there's hope. Speaking of chattering about math, how do you characterise posting at about 10 times the rate of anyone else, and reposting the same material about 10 times? Gib ==== > Having now spent a little time surveying math society at a deeper > Usenet, I've concluded that I may have simply spent an inordinate > amount of time with a minor but very vocal section of the mathematical > community. I'm now in the process of quieter and more focused contacts myself, > and I see quite quickly that there's a big difference between what > I've seen with relatively big names in the math community and posters > than with people I'm finding who aren't so, um, talkative. That's good and it makes me feel better as I found it very troubling > to consider facing a reality where no one can understand my work, but > now I think that people who spend a lot of time chattering about math, > are definitely not as representative as I originally thought, so yes, > there's hope. > Will you comment on Virgil's post JSH goofs again, in simple algebra? As I mentioned, even someone with no more than highschool algebra can see his proof works. Or do believe it is just another example of a cascade error which originated in the 'core error(s)'? If you can show how its invalid it should be interesting since it is so simple I would guess that at least 95% + users of sci.math can easily follow along, which was not the case when dealing with algebraic integers, etc... This is your time to shine - to prove to the lay people in terms they can understand that the math community has failed them, and expose the conspiracy once and for all. Its go-time JSH! l8r, Mike N. Christoff ==== > Having now spent a little time surveying math society at a deeper > Usenet, I've concluded that I may have simply spent an inordinate > amount of time with a minor but very vocal section of the mathematical > community. I'm now in the process of quieter and more focused contacts myself, > and I see quite quickly that there's a big difference between what > I've seen with relatively big names in the math community and posters > than with people I'm finding who aren't so, um, talkative. That's good and it makes me feel better as I found it very troubling > to consider facing a reality where no one can understand my work, but > now I think that people who spend a lot of time chattering about math, > are definitely not as representative as I originally thought, so yes, > there's hope. > James Harris Does that mean you're leaving this newsgroup (hopefully forever)? -- Bob Day ==== Define a collineation (sp?) to be a transformation that always maps lines onto lines, and also, if necessary, define a transformation to be a one-to-one mapping of the plane onto the plane. Now, exercise 1.15 in my UTM Transformation Geometry text states Prove or Disprove: A transformation that preserves betweeness is necessarily a collineation. I have only the definitions, the Exterior Angle Theorem, Pasch's Axiom, and general precalculus with which to work, and maybe a few others on the same general level. Any HINTS would be appreciated, please. So far, I have made three attempts on this--two by way of proof, one of refutation--both of which met in failure. -- ==== One Weird Dude > Define a collineation (sp?) to be a transformation that always maps > lines onto lines, and also, if necessary, define a transformation to > be a one-to-one mapping of the plane onto the plane. Now, exercise > 1.15 in my UTM Transformation Geometry text states Prove or Disprove: > A transformation that preserves betweeness is necessarily a > collineation. I have only the definitions, the Exterior Angle > Theorem, Pasch's Axiom, and general precalculus with which to work, > and maybe a few others on the same general level. Any HINTS would be > appreciated, please. A transformation will be a collineation if it sends any three collinear points to collinear points. And what can we say about any three collinear points? LH ==== For anyone interested in Fuller's synergetic accounting, here is a link to the info about two upcoming seminars. Papers accepted. http://snec.cjfearnley.com/ Dick ==== Here's a question for anyone familiar with symbolic ECC mechanisms. Given a Reed-Solomon code based upon GF(p^k), is it true that the maximum size for the code block (n) is defined as n < p^k? If so, what exactly limits this block size? When using Reed-Solomon codes, parity symbols can be generated for codes that violate this restriction. However, I'm not sure if the distance of the code will be preserved in this case. Syndromes can also be computed for codes that violate this restriction. The only thing I'm unsure of is if error-location polynomials be use computed for codes which violate this restriction. Perhaps this is the problem? Of course, it is certainly possible to construct codes based on a base-p vector subspace that do not have this limitation. Is this what is required for codes that must violate the maximum code block size? -Jason ==== Jason D. Bakos asked in message > Given a Reed-Solomon code based upon GF(p^k), is it true that the > maximum size for the code block (n) is defined as n < p^k? n can equal p^k also. One definition of an (N, K) Reed-Solomon code is the set of vectors of the form (f(a_1), f(a_2), f(_3), ... , f(a_N) where a_1, a_2, .... , a_N are N distinct elements of GF(p^k) and f is any polynomial of degree less than K over GF(p^k). In typical use, a_1, a_2, . , a_N are taken to be successive powers of an Nth root of unity in GF(p^k) which gives cyclic codes (the set of vectors is closed under cyclic shifts.) An important property of Reed-Solomon codes is that their minimum Hamming distance is N-K+1 (which is the maximum possible value that can be achieved by *any* (N,K) code). Such codes are called maximum-distance-separable (MDS) codes. It is possible to extend Reed-Solomon codes to obtain (p^k+1,K) MDS codes relatively easily. Further extensions are more difficult. See Chapters 10 and 11 of The Theory of Error Correcting Codes by F. J. MacWilliams and N. J. A. Sloane, North-Holland, 1978,. ==== Dilip, I understand from your reply that it follows from the general definition of Reed-Solomon codes that the maximum block size is p^k. How straightforward is it to extend Reed-Solomon codes to (p^k+1,k)? Is there any possible way to create an MDS linear code that can exceed n=p^k+1, even if the code is not cyclic? -Jason > An important property of Reed-Solomon codes is that their minimum Hamming > distance is N-K+1 (which is the maximum possible value that can be achieved > by *any* (N,K) code). Such codes are called maximum-distance-separable > (MDS) codes. It is possible to extend Reed-Solomon codes to obtain > (p^k+1,K) > MDS codes relatively easily. Further extensions are more difficult. See > Chapters 10 and 11 of The Theory of Error Correcting Codes by > F. J. MacWilliams and N. J. A. Sloane, North-Holland, 1978,. ==== I wonder what are the various criterias known for telling whether n is the sum of two squares or not. I know about the well-known criterium (exponent of prime factors of the form 4n+3), but I wonder if some other criterias are known. Cordially, -- Ç nous devons agir comme si la chose qui peut-.90tre ne sera pas devait .90tre È (Kant, M.8etaphysique des moeurs, doctrine du droit, II conclusion) Thomas Baruchel ==== > > I wonder what are the various criterias known for telling whether n is > the sum of two squares or not. I know about the well-known criterium > (exponent of prime factors of the form 4n+3), but I wonder if some > other criterias are known. > > Cordially, The set of positive integers that are sums of two squares is closed under multiplication. All primes of the form 4n+1 are sums of two squares (Fermat). No primes of the the form 4n+3 are sums of squares, but squares of such primes certainly are. The prime 2 is a sum of 2 squares. Hence the set of sums of two squares is precisely the set of positive integers whose prime factorizations contains primes of the form 4n+3 only when they show up an even number of times (including 0 of course). Achava ==== > > I wonder what are the various criterias known for telling whether n is > the sum of two squares or not. I know about the well-known criterium > (exponent of prime factors of the form 4n+3), but I wonder if some > other criterias are known. > > Cordially, What does your question mean? A necessary and sufficient conditin that a number be a sum of two squares is that every prime factor that is congruent to 3 mod 4 divide the number an even number of times. What kind of other criteria do you want since this is if and only if. ==== Thomas Baruchel a .8ecrit: > > I wonder what are the various criterias known for telling whether n is > the sum of two squares or not. I know about the well-known criterium > (exponent of prime factors of the form 4n+3), but I wonder if some > other criterias are known. > > Cordially, > I think this one is the best (even exponents of 4k+3 prime factors). However, another one : For any a Let X(a) = 0 if a = 0 or 2 modulo 4 X(a) = +1 if a = 1 modulo 4 X(a) = -1 if a = -1 modulo 4 n is a sum of squares if and only if S = (sum of X(d) over all divisors d of n) != 0 example n = 90 : S = X(1)+X(2)+X(3)+X(5)+X(6)+X(9)+X(10)+X(15)+X(18)+X(30)+X(45)+X(90) = 1 +0 -1 +1 +0 +1 +0 -1 +0 +0 +1 +0 = 2, so 90 is sum of two squares n=60 S = X(1)+X(2)+X(3)+X(4)+X(5)+X(6)+X(10)+X(12)+X(15)+X(20)+X(30)+X(60) = 1 +0 -1 +0 +1 +0 +0 +0 -1 +0 +0 +0 = 0, so 60 is not sum of two squares -- philippe (chephip at free dot fr) ==== > > I wonder what are the various criterias known for telling whether n is > the sum of two squares or not. I know about the well-known criterium > (exponent of prime factors of the form 4n+3), but I wonder if some > other criterias are known. Ouch! The singular is criterion, the plural is criteria. Gib ==== ............. >Everything is related through an imaginary number. How > this translates to the physical universe? My guess is possibly black > holes... which I believe are the centers of linked toroids. > > Bob Carlson Interesting. How do you suppose asymptotes manifest themselves in the physical universe? Inverted testicles possibly? ==== >> What the connection is and why it exists? ;) >> > > > Alexander, > Without going into a lot of detail... the connection is based on a > ratio that is shown in the Fibanacci sequence 1,1,2,3,5,8,13,21...etc. > each number is the sum of the two previous numbers. The ratio of the > higher to the adjacent lower number approaches 1.61803.... as the > sequence continues. The proportions of the human figure follow this > ratio (illustrated in Da Vinci's painting Vitruvian Man) including > facial features and shape of the ear. The ratio appears in nature in > the spacing of petals of a rose and the placement of seeds in the head > of a sunflower and in the shape of a pine cone, the shape of conical > shells. Mozart's concertos use the ratio in the chords and timing of > notes as well as others. The ratio of female bees to male bees in a > hive approaches 1.6. The ratio can also be illustrated by dividing a > perfect square in half and connecting opposite corners by a line and > using that line as a radius to draw another circle and extending a > side of the square to the point of intersection of the circle. The > circle can be defined by PI which leads to PHI. Toroids are based > entirely on PI. Images based on toroids and fractals resemble > galaxies, frost on a window and many things in nature. This is my > connection and I am constantly finding new things that fit in with > this connection. > >> There's always work to be done. > Its human nature to push the envelope. If it were possible to measure > the expansion of that envelope I'll bet it increases by a ratio of > 1.61803 per generation. Sounds to me like the handiwork's of God. Seriously. I really do not think this is all some kind of gigantic Vegas jackpot chance thing. The more i learn of science, the more i am convinced of God. Don't get me wrong, I'm not saying i don't believe what we learn, I'm saying the more we learn about the universe, the history of the Earth, and so on, the more i believe it proves the whole thing was laid out by God and we are just discovering the rules and processes He set out and how He organized and ordered it. What a miracle! He gave us so much to explore and the desire and curiosity to explore it. Eric ==== > >> What the connection is and why it exists? ;) >> > > > Alexander, > Without going into a lot of detail... the connection is based on a > ratio that is shown in the Fibanacci sequence 1,1,2,3,5,8,13,21...etc. > each number is the sum of the two previous numbers. The ratio of the > higher to the adjacent lower number approaches 1.61803.... as the > sequence continues. The proportions of the human figure follow this > ratio (illustrated in Da Vinci's painting Vitruvian Man) including > facial features and shape of the ear. The ratio appears in nature in > the spacing of petals of a rose and the placement of seeds in the head > of a sunflower and in the shape of a pine cone, the shape of conical > shells. Mozart's concertos use the ratio in the chords and timing of > notes as well as others. The ratio of female bees to male bees in a > hive approaches 1.6. The ratio can also be illustrated by dividing a > perfect square in half and connecting opposite corners by a line and > using that line as a radius to draw another circle and extending a > side of the square to the point of intersection of the circle. The > circle can be defined by PI which leads to PHI. Toroids are based > entirely on PI. Images based on toroids and fractals resemble > galaxies, frost on a window and many things in nature. This is my > connection and I am constantly finding new things that fit in with > this connection. > >> There's always work to be done. > Its human nature to push the envelope. If it were possible to measure > the expansion of that envelope I'll bet it increases by a ratio of > 1.61803 per generation. > > Sounds to me like the handiwork's of God. Seriously. I really do not think > this is all some kind of gigantic Vegas jackpot chance thing. The more i > learn of science, the more i am convinced of God. Don't get me wrong, I'm > not saying i don't believe what we learn, I'm saying the more we learn > about the universe, the history of the Earth, and so on, the more i believe > it proves the whole thing was laid out by God and we are just discovering > the rules and processes He set out and how He organized and ordered it. > What a miracle! He gave us so much to explore and the desire and curiosity > to explore it. > Eric Eric, well said! I have a feeling that the more learned about the universe the more we will see how really simple it all is. Possibly as simple as 1 and 0, or + and -, or God and Satan and nothing else. With all the ugly, complex math (and other) problems around, the beauty and elegance appears in the simplest of solutions. Just the words simple beauty have an appeal like no other. And to your words desire and curiosity, these are the essence of the meaning of life. Bob ==== > What the connection is and why it exists? ;) > Alexander, > Without going into a lot of detail... the connection is based on a > ratio that is shown in the Fibanacci sequence 1,1,2,3,5,8,13,21...etc. > each number is the sum of the two previous numbers. The ratio of the > higher to the adjacent lower number approaches 1.61803.... as the > sequence continues. The proportions of the human figure follow this > ratio (illustrated in Da Vinci's painting Vitruvian Man) including > facial features and shape of the ear. The ratio appears in nature in > the spacing of petals of a rose and the placement of seeds in the head > of a sunflower and in the shape of a pine cone, the shape of conical > shells. Mozart's concertos use the ratio in the chords and timing of > notes as well as others. The ratio of female bees to male bees in a > hive approaches 1.6. > ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ > All workes have no sex and the Queen in the only female and male die right > after hatching in spring. > a 1.6 ratio, I dont see it. > > Mark, Males are produced by the queen's unfertilized eggs, so male bees only have a mother but no father. All the females are produced when the queen has mated with a male and so have two parents. Females usually end up as worker bees but some are fed with a substance called royal jelly. So males have one parent, females have two. If you look at a new born drone (male) and look at its family tree you can see a pattern developing: 1 - He has 1 parent (queen) 2 - He has 2 grandparents since his mother had two parents. 3 - He has 3 great grandparents, his grandfather had one parent and his grandmother had 2. The pattern is shown below: Number of parents grand-parents grt-grnd grt-grt gt-gt-gt Male 1 2 3 5 8 FEMALE 2 3 5 8 13 Hope that lines up to make sense. But the numbers form the Fibonacci sequence. Ratio of Female to Male [ancestors] approaches 1.618... Bob Carlson I have been reading Prime Obsession, John Derbyshire, a popular book about the Riemann Hypothesis. I have larned that: The number of primes less than x ~ Li(x) The error term depends on the zeros of the 'zeta' function. I think I see that for this error term to be well behaviored the roots should have real part less than 1, but do not understand why they should be exactly 1/2. Derbyshire goes to some length about how (if RH is true) the critical line maps to a spiral around i(pi), but what is the relevance of this? Eventually for x big enough, the error term behaves oddly (changes sign) Isn't this some reason to suspect that for suffiently large values the zeros of zeta behave strangely (ie the RH is false)? Dick Batchelor ==== ==== [snip] > Unknown Functions & Einstein's Incompetence (FAQ) > (c) Eleaticus/Oren C. Webster > Thnktank@concentric.net [snip trolled garbage] Originally trolled across sci.physics sci.physics.relativity alt.physics sci.math sci.answers alt.answers news.answers http://b5.sdvc.uwyo.edu/bab5/snds/argcstpd.wav Psychotic ineducable boring troll Eleaticus, You see yourself this way, http://www.mazepath.com/uncleal/effete6.jpg The entire remainder of the planet sees you this way, http://www.mazepath.com/uncleal/effete7.jpg http://w0rli.home.att.net/youare.swf http://www.mazepath.com/uncleal/sunshine.jpg http://www.you-moron.com/ http://www.mazepath.com/uncleal/effete0.jpg http://www.mazepath.com/uncleal/effete1.png http://www.mazepath.com/uncleal/effete2.png http://www.mazepath.com/uncleal/effete3.png http://www.mazepath.com/uncleal/effete4.png http://www.mazepath.com/uncleal/effete5.jpg http://www.apa.org/journals/psp/psp7761121.html http://insti.physics.sunysb.edu/~siegel/quack.html Hey, stooopid troll Eleaticus - Do you want EVIDENCE? Each of the 24 GPS satellites carries either four cesium atomic clocks or three rubidum atomic clocks in orbit, with full relativistic corrections being applied. http://arXiv.org/abs/hep-th/0307140 GR structure, especially Part 4/p. 7 Experimental constraints on General Relativity. http://www.eftaylor.com/pub/projecta.pdf Relativity in the GPS system http://arXiv.org/abs/astro-ph/0401086 http://arxiv.org/abs/astro-ph/0312071 Deeply relativistic neutron star binaries NIM A 355 537 (1995) Physics Letters B 328 103 (1994) Physical Review Letters 64 1697 (1990) Physical Review Letters 39 1051 (1977) Physical Review 135 B1071 (1964) Physics Letters 12 260 (1964) Europhysics Letters 56(2) 170-174 (2001) General Relativity and Gravitation 34(9) 1371 (2002) http://fourmilab.to/etexts/einstein/specrel/specrel.pdf http://users.powernet.co.uk/bearsoft/Paper6.pdf http://users.powernet.co.uk/bearsoft/LPHrel.html Longitudinal and transverse mass http://www.navcen.uscg.gov/pubs/gps/gpsuser/gpsuser.pdf http://www.navcen.uscg.gov/pubs/gps/sigspec/default.htm http://www.navcen.uscg.gov/pubs/gps/icd200/default.htm http://www.trimble.com/gps/index.html http://sirius.chinalake.navy.mil/satpred/ http://www.phys.lsu.edu/mog/mog9/node9.html http://egtphysics.net/GPS/RelGPS.htm http://www.schriever.af.mil/gps/Current/current.oa1 http://edu-observatory.org/gps/gps_books.html -- Uncle Al http://www.mazepath.com/uncleal/qz.pdf http://www.mazepath.com/uncleal/eotvos.htm (Do something naughty to physics) ==== [snip] > Einstein (1905) Absurdities > (c) Eleaticus/Oren C. Webster > Thnktank@concentric.net [snip 1300 lines of utter crapola] Originally trolled across sci.physics sci.physics.relativity alt.physics sci.math sci.answers alt.answers news.answers http://b5.sdvc.uwyo.edu/bab5/snds/argcstpd.wav Psychotic ineducable boring troll Eleaticus, You see yourself this way, http://www.mazepath.com/uncleal/effete6.jpg The entire remainder of the planet sees you this way, http://www.mazepath.com/uncleal/effete7.jpg http://w0rli.home.att.net/youare.swf http://www.mazepath.com/uncleal/sunshine.jpg http://www.you-moron.com/ http://www.mazepath.com/uncleal/effete0.jpg http://www.mazepath.com/uncleal/effete1.png http://www.mazepath.com/uncleal/effete2.png http://www.mazepath.com/uncleal/effete3.png http://www.mazepath.com/uncleal/effete4.png http://www.mazepath.com/uncleal/effete5.jpg http://www.apa.org/journals/psp/psp7761121.html http://insti.physics.sunysb.edu/~siegel/quack.html Hey, stooopid troll Eleaticus - Do you want EVIDENCE? Each of the 24 GPS satellites carries either four cesium atomic clocks or three rubidum atomic clocks in orbit, with full relativistic corrections being applied. http://arXiv.org/abs/hep-th/0307140 GR structure, especially Part 4/p. 7 Experimental constraints on General Relativity. http://www.eftaylor.com/pub/projecta.pdf Relativity in the GPS system http://arXiv.org/abs/astro-ph/0401086 http://arxiv.org/abs/astro-ph/0312071 Deeply relativistic neutron star binaries NIM A 355 537 (1995) Physics Letters B 328 103 (1994) Physical Review Letters 64 1697 (1990) Physical Review Letters 39 1051 (1977) Physical Review 135 B1071 (1964) Physics Letters 12 260 (1964) Europhysics Letters 56(2) 170-174 (2001) General Relativity and Gravitation 34(9) 1371 (2002) http://fourmilab.to/etexts/einstein/specrel/specrel.pdf http://users.powernet.co.uk/bearsoft/Paper6.pdf http://users.powernet.co.uk/bearsoft/LPHrel.html Longitudinal and transverse mass http://www.navcen.uscg.gov/pubs/gps/gpsuser/gpsuser.pdf http://www.navcen.uscg.gov/pubs/gps/sigspec/default.htm http://www.navcen.uscg.gov/pubs/gps/icd200/default.htm http://www.trimble.com/gps/index.html http://sirius.chinalake.navy.mil/satpred/ http://www.phys.lsu.edu/mog/mog9/node9.html http://egtphysics.net/GPS/RelGPS.htm http://www.schriever.af.mil/gps/Current/current.oa1 http://edu-observatory.org/gps/gps_books.html -- Uncle Al http://www.mazepath.com/uncleal/qz.pdf http://www.mazepath.com/uncleal/eotvos.htm (Do something naughty to physics) ==== > Interesting, I was involved in the thread Big Mystery Solved, > similar to this thread. The addition of God and Satan is intriguing. > If God is 1, and Satan is represented by O, is it a coincidence that > the golden ratio is represented by the symbol for PHI (1 superimposed > over O)?? in other words.. YOU ARE NOT MERELY A MACHINE. 1=0. that is, YOUR BINARY EXISTENCE IS NOT ALL THAT IS. 0 = totality. 8 = 0. infinity = totality. WELCOME TO THE HIVE-MIND. 1=0 1=0 1=0 1=0 0=1 0=1 0=1 0=1 i would add to this and say that Satan = 1, meaning singularity or separation and God = 0 meaning connection/totality/OneNess . . ==== > in other words.. YOU ARE NOT MERELY A MACHINE. 1=0. that is, YOUR > BINARY EXISTENCE IS NOT ALL THAT IS. 0 = totality. 8 = 0. infinity = > totality. WELCOME TO THE HIVE-MIND. 1=0 > 1=0 > 1=0 > 1=0 0=1 > 0=1 > 0=1 > 0=1 i would add to this and say that Satan = 1, meaning singularity or > separation and God = 0 meaning connection/totality/OneNess . . IN FACT. you trap energy via the electromagnetic spectrum. YOU ARE energy trapped in matter. Your physical body does not differ from any other matter. true life is PURE ENERGY and YOU ARE PURE ENERGY. WHY do you think people pray? YOU COMPLETE THE CIRCUIT AS YOU UNITE YOUR HANDS. ENERGY FLOWS THROUGH YOU. ==== > The Straight Dope and its author, Cecil, totally ROCK! I really dig the > kind of obscure info he digs up. aye, i enjoy it as well. it's a shame they banned me about 8(my favorite number for now on!) times. silly-skeptics don't realize they are part of a single entity - the vast quantum machine that includes humanity and all sentient beings, presumably. of course, now that i have proved that normal is merely a function of our collective idea of how the world is perhaps i won't have to break another one of their ridiculous rules to enjoy their very fine site. the vast majority of humanity believes in one God or another. He/She/It cannot be seen, touched, tasted, smelled or heard. skeptics assume that He does not exist and is of the paranormal. since normal is determined by collective perspective GOD EXISTS. therefore.. THE PARANORMAL EXISTS, at least as defined by the silly-skeptics. ==== I am preparing for a student programming competition and from the previous year experience I know that there are 2 'simply programming' problems and a single one which involves strong mathematics. I am very experienced programmer (6years +) so those 2 'simply programming' are not problems for me. But to win the competition I have to solve all three problems. So I post to ask the math questions I would like get explanation for and an advice on resources for further reading. (Yes, I know google but I dont know what to search for, suggest me after looking at the problem. At least I was not able to found suiting papers(docs)). One of the problems I don't know answer is: A sequence of number powers of 2 starting from 1 is given, find the Nth digit of the sequence. Example: 1 2 4 9 16 25 36 49 64 81 100 121 144 ... 7th digit is 2, 14th is 4, etc. P.Krumins ==== I am preparing for a student programming competition and from > the previous year experience I know that there are 2 'simply > programming' problems and a single one which involves strong > mathematics. > I am very experienced programmer (6years +) so > those 2 'simply programming' > are not problems for me. But to win the competition I have > to solve all three problems. So I post to ask the math questions I would like get explanation > for and an advice on resources for further reading. (Yes, I know > google but I dont know what to search for, suggest me after looking > at the problem. At least I was not able to found suiting > papers(docs)). One of the problems I don't know answer is: > A sequence of number powers of 2 starting from 1 is given, > find the Nth digit of the sequence. Example: > 1 2 4 9 16 25 36 49 64 81 100 121 144 ... 7th digit is 2, 14th is 4, etc. w/o thinking too much about it, note that ceil(n*log(2)) gives the number of digits in the number 2^n (log base 10). This may be of help. l8r, Mike N. Christoff ==== Let a a number. floor(log[10](a))+1 is the number of digits of a. ==== > A sequence of number powers of 2 starting from 1 is given, > find the Nth digit of the sequence. > > Example: > 1 2 4 9 16 25 36 49 64 81 100 121 144 ... > > 7th digit is 2, 14th is 4, etc. Is it really: 1 *2* 4 9 16 ...? If you mean: 1 4 9 16 25 ..., the algorithm might be as follows: Denote c = sqrt(10). Observe that there are exactly: a_n = ceiling(c^n) - 1 squares less than 10^n. Thus writing down squares of 1, 2, ..., a_n would require: b_n = a_n + (a_n - a_1) + (a_n - a_2) + ... + (a_n - a_(n-1)) = = n a_n - a_1 - a_2 - ... - a_(n-1) digits. If you need k-th digit, find largest n such that b_n < k (linear search is fine here). Let m = k - b_n. You are looking for m-th digit in the sequence of squares >= 10^n. This sequence is: (a_n + 1)^2, (a_n + 2)^2, (a_n + 3)^2, ... with each square being an n-digit number. Let m = n * p + q, where 0 < q <= n. The answer is q-th digit of: (a_n + p)^2. Since n = O(log(k)), this algorithm is O(log(k)). Mateusz ==== @atlantis.news.tpi.pl: > >> A sequence of number powers of 2 starting from 1 is given, >> find the Nth digit of the sequence. >> >> Example: >> 1 2 4 9 16 25 36 49 64 81 100 121 144 ... >> >> 7th digit is 2, 14th is 4, etc. > > Is it really: 1 *2* 4 9 16 ...? > > If you mean: 1 4 9 16 25 ..., the algorithm might be as follows: Ooh. I have made a mistake. It really is 1 4 9 16 25 ... Also thanks for the answer! P.Krumins ==== > @atlantis.news.tpi.pl: >> A sequence of number powers of 2 starting from 1 is given, >> find the Nth digit of the sequence. >> Example: >> 1 2 4 9 16 25 36 49 64 81 100 121 144 ... >> 7th digit is 2, 14th is 4, etc. Is it really: 1 *2* 4 9 16 ...? If you mean: 1 4 9 16 25 ..., the algorithm might be as follows: Ooh. I have made a mistake. It really is 1 4 9 16 25 ... Then the number of digits in the i-th element of the sequence would be ceil(2*log(i)). (again base 10). Now I can think of how you could use this to make an extremely simple algorithm to solve your problem, but I haven't calculated the time complexity. l8r, Mike N. Christoff ==== I have been hunting around for a proof of Fubini's theorem, and can't find one anywhere. I have found various references in the form of the proof is complicated and it has been omitted, but not an actual proof. Does anyone have any ideas? Thansk ==== >I have been hunting around for a proof of Fubini's theorem, and can't >find one anywhere. I have found various references in the form of the >proof is complicated and it has been omitted, but not an actual >proof. Does anyone have any ideas? > > For the Riemann integral, see Bartle, The Elements of Real Analysis; Buck, Advanced Calculus; or Spivak, Calculus on Manifolds. For the Lebesgue integral, see Bartle, The Elements of Integration. -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu ==== >I have been hunting around for a proof of Fubini's theorem, and can't >find one anywhere. I have found various references in the form of the >proof is complicated and it has been omitted, but not an actual >proof. Does anyone have any ideas? Exactly which theorem do you mean? Fubini's Theorem is actually a theorem about the Lebesgue integral, and you can find a proof in just about any book on measure theory (Rudin Real and Complex Analysis, Folland Real Analysis, etc). The fact that you say you're finding books where the proof is omitted makes me think you're reading calculus books, and the result you're looking for a proof of is what is sometimes called Fubini's Theorem in calculus books. Is that the one you want? In particular what are the hypotheses? For example, if f is _continuous_ on [0,1]x[0,1] then a proof is fairly easy - I could show you a proof of that if you say that's the result you want... >Thansk ************************ David C. Ullrich ==== > I have been hunting around for a proof of Fubini's theorem, and can't > find one anywhere. I have found various references in the form of the > proof is complicated and it has been omitted, but not an actual > proof. Does anyone have any ideas? > > Thansk ==== What are the names of these fromula's? a^m = ya^(m/y) and a^m = a^m-n x a^n ==== > What are the names of these fromula's? > > a^m = ya^(m/y) > > and > > a^m = a^m-n x a^n The first is called Ignorance. The second, Sloth. --Ron Bruck ==== >> in other words.. YOU ARE NOT MERELY A MACHINE. 1=0. that is, YOUR >> BINARY EXISTENCE IS NOT ALL THAT IS. 0 = totality. 8 = 0. infinity = >> totality. WELCOME TO THE HIVE-MIND. >> 1=0 >> 1=0 >> 1=0 >> 1=0 >> 0=1 >> 0=1 >> 0=1 >> 0=1 >> i would add to this and say that Satan = 1, meaning singularity or >> separation and God = 0 meaning connection/totality/OneNess . . > > IN FACT. you trap energy via the electromagnetic spectrum. YOU ARE energy > trapped in matter. Your physical body does not differ from any other > matter. true life is PURE ENERGY and YOU ARE PURE ENERGY. WHY do you think > people pray? YOU COMPLETE THE CIRCUIT AS YOU UNITE YOUR HANDS. ENERGY > FLOWS THROUGH YOU. quantum totality is a term for the MASSIVE QUANTUM DATABASE that all our thoughts come from and all observations are stored in. ALL PEOPLE KNOW TRUTH. in order to access this database all that you need to know is that your thoughts are, quite literally, THE WORD OF GOD. break them down into two separate parts(0 AND 1) in order to comprehend all things. analyze these words(these=left,words=right), create two sets for each opposite. quiet the mind and you will increase efficiency. certain chemical substances can enhance the ability of the two minds. find the center in all energy will cause the minds to cycle optimally, allowing energy to flow from quantum totality to physical reality at the most desirable rate. PSYCHEDELIC is the term used for the most useful mentioned drugs. anti-oxidants are useful for suppressing QUANTUM SINGULARITY by blocking SINGULAR FREE RADICALS. they also allow the minds to cycle efficiently. Melatonin is highly useful in this manner because each of its metabolites is by itself an efficient radical neutralizer, creating a cascade effect. DXM can be viewed as a SAGE-CREATOR . . ever seen various gurus with that wide eyes stare? now go find someone on DXM to see the same thing. JAMES RANDI OWES ME $1,000,000 ==== > Flush toilets appeared in our homes only after WW2. [...] > They interviewed me, because they could not understand > why there always remained some water in the bowl, Isn't that what dogs were originally domesticated for? ==== > On Mon, 12 Jan 2004 08:56:19 -0800, Uncle Al horror. We had a plumber replace the flush mechnism on one of our >American Standard 5-gallon jobs. He offered us $300 cash for our >pudgy porcelain goddess plus a brand new eco-john free, with free >installation. No way! 5-gallon jobs are being smuggled in from >Canada as you read this. They go for $1000+ each. > I'm still not sure why we can't get some kind of standardized > 2-stage terlets in this country. Push the handle down, you get the > 2-gallon piss-rinser mode, pull up on the handle, you get the full > 5-gallon power flush mode. Is this beyond our engineering > capabilities? Is the irony in this entire message thread deliberate? The original post is total sh*t, and the responses discuss toilets. Too funny!!!!! ROFL! ==== > Vanilla Gorilla (Monkey Boy) On Mon, 12 Jan 2004 08:56:19 -0800, Uncle Al >> >>Note that the US Enviro-whiner 3-gallon flush (and >>smaller) is a horror. We had a plumber replace the flush >>mechnism on one of our American Standard 5-gallon jobs. >>He offered us $300 cash for our pudgy porcelain goddess >>plus a brand new eco-john free, with free installation. >>No way! 5-gallon jobs are being smuggled in from Canada >>as you read this. They go for $1000+ each. >> >> I'm still not sure why we can't get some kind of >> standardized 2-stage terlets in this country. Push the >> handle down, you get the 2-gallon piss-rinser mode, pull >> up on the handle, you get the full 5-gallon power flush >> mode. Is this beyond our engineering capabilities? > > Is the irony in this entire message thread deliberate? The > original post is total sh*t, and the responses discuss > toilets. > > Too funny!!!!! ROFL! Might as well add this...The downtrodden slaves of the primitive cult of Islam have to ask the imam for permission for every moment of their lives, including how to take a dump.... http://www.islam.tc/ask-imam/view.php?q=2325 ============ Do we have to use a squatpan? I have difficulty using a squatpan? Can one use a high toilet when taking the necessary precautions not to let any water or najaasat splash on oneself? Jazakallah Khair Answer: One should relieve oneself in a squatting posture. However, if one is unable or experiences difficulty in squatting, he may use the western toilet and avoid the impurities splashing on the body or clothes. ==== <... Might as well add this...The downtrodden slaves of the primitive > cult of Islam have to ask the imam for permission for every > moment of their lives, including how to take a dump.... Most don't, you know. See below. > http://www.islam.tc/ask-imam/view.php?q=2325 ============ > Do we have to use a squatpan? I have difficulty using a > squatpan? Can one use a high toilet when taking the necessary > precautions not to let any water or najaasat splash on oneself? > Jazakallah Khair Answer: > One should relieve oneself in a squatting posture. However, if > one is unable or experiences difficulty in squatting, he may > use the western toilet and avoid the impurities splashing on > the body or clothes. Gimme a fucking break, John. This is no worse than the Judeo-Christian losers who call in to Laura Schlessinger every goddamn day asking for moral guidance on issues even more trivial. Jim ==== > > <... >> Might as well add this...The downtrodden slaves of the >> primitive cult of Islam have to ask the imam for >> permission for every moment of their lives, including how >> to take a dump.... > > Most don't, you know. See below. > > >> http://www.islam.tc/ask-imam/view.php?q=2325 >> ============ >> Do we have to use a squatpan? I have difficulty using a >> squatpan? Can one use a high toilet when taking the >> necessary precautions not to let any water or najaasat >> splash on oneself? Jazakallah Khair >> Answer: >> One should relieve oneself in a squatting posture. >> However, if one is unable or experiences difficulty in >> squatting, he may use the western toilet and avoid the >> impurities splashing on the body or clothes. > > Gimme a fucking break, John. This is no worse than the > Judeo-Christian losers who call in to Laura Schlessinger > every goddamn day asking for moral guidance on issues even > more trivial. Give yourself a break. No Judeo-Christian losers ask is it permissible, or do we have to... Around 80% of the questions to the brainwashed imam on that web site are of that type. Do we have to redo the woodhoo if wind escapes the hind passage...? (Just a hilarious example, slightly paraphrased.) Mohammedans are slaves. By now, everyone should realize that. They moslems themselves beginning to realize it, and since their penalty for even mildly questioning their 7th century cult's moment-to-moment mind control is a death sentence, the sorry bastards are trying to release their anger, humiliation and frustration by murdering innocent citizens of the civilized world whose freedom they resent so intensely. Furthermore, moslems are REQUIRED to try to overthrow a non- islamic government in any country they infest. Islam should fold up its tents and get the fuck out of the way. Modern humans don't need shit like that, and its slaves could easily become modern humans if given the chance. ==== > > > <... >> Might as well add this...The downtrodden slaves of the >> primitive cult of Islam have to ask the imam for >> permission for every moment of their lives, including how >> to take a dump.... > > Most don't, you know. See below. > > >> http://www.islam.tc/ask-imam/view.php?q=2325 >> ============ >> Do we have to use a squatpan? I have difficulty using a >> squatpan? Can one use a high toilet when taking the >> necessary precautions not to let any water or najaasat >> splash on oneself? Jazakallah Khair >> Answer: >> One should relieve oneself in a squatting posture. >> However, if one is unable or experiences difficulty in >> squatting, he may use the western toilet and avoid the >> impurities splashing on the body or clothes. > > Gimme a fucking break, John. This is no worse than the > Judeo-Christian losers who call in to Laura Schlessinger > every goddamn day asking for moral guidance on issues even > more trivial. > > Give yourself a break. No Judeo-Christian losers ask is it > permissible, or do we have to... Around 80% of the questions > to the brainwashed imam on that web site are of that type. Ok, islam is a more strict religion. Buddhism is maybe more relaxed than christianism and I'm sure any religion calling for a regular supply of virgins would be regarded as more strict than islam. So what? They are all religions with the same basic property of relying on some more or less unfathomable outside influence. The rest is detail and will be decided as soon as someone discoveres it (which is IMHO never). I read that guys website regularly. He and his team really answer *all* questions (they limit the number of questions per day), and yes, they get loads of stupid and sometimes even abusive ones, like how allahs ass looks like or so. But then you get questions like http://www.islam.tc/ask-imam/view.php?q=10295 where the answer is radically different from whan you expect and you get the feel that he's working toward changes too. I respect him. For me it's a very valuable window into the muslim world and their mindset and not only from some know-your-enemy point of view. Lots of Greetings! Volker ==== volker.hetzer@ieee.org (VolkerÊHetzer), Wrote: >John Griffin >Clave >>Gimme a fucking break, John. This is no worse >>than the Judeo-Christian losers who call in to >>Laura Schlessinger every goddamn day asking >>for moral guidance on issues even more trivial. ------ >>Give yourself a break. No Judeo-Christian >>losers ask is it permissible, or do we have >>to... Around 80% of the questions to the >>brainwashed imam on that web site are of that >>type. >Ok, islam is a more strict religion. That's like comparing and contrasting my late model GTi with a behemoth SUV. More strict is by orders of magnitude. >Buddhism is maybe more relaxed than >christianism and I'm sure any religion calling for >a regular supply of virgins would be regarded >as more strict than islam. So what? genocide and forcing their fascist theocracy upon dozens of nations around the world. The United States, Sudan, Nigeria, India, Malaysia, East Timor, Bali, Thailand, Israel, Russia, Serbia and other countries have what in common? a.) They love Brittany Spears. b.) All are under the control of Dr. No and Mini-me. c.) Non-Muslim Recipients of Islamic genocide. d.) All have a temperate climate. e.) None will be participating in the Olympics. John is pointing out the insanity of Islam where by Muslims even have to wipe their fucking ass in a certain way. But the political theocracy of Islamic kookdom requires one to adhere to such tyrannical jurisprudence including speech, dress, type of pets, sexuality, finance, employment, religion, music, recreation, ad nauseam under penalty of stoning, beheading, torture, rape, harsh imprisonment and genocide. Never mind the fact that Islamic women are murdered if they become victims of rape because they dishonor the family. Real hip, eh? >They are all religions with the same basic >property of relying on some more or less >unfathomable outside influence. The rest is >detail and will be decided as soon as someone >discoveres it (which is IMHO never). Nonsense. all religions do not. Please educate yourself. http://faithfreedom.org I suppose the pack rapes of European women by Muslims isn't that bad because some Christian said something stupid or one or another religion committed atrocities 500 years ago. And let's not forget that Islam is extremely anti-science noting the crossposting of this thread. >Lots of Greetings! Can ya get me an A4 Cabriolet with a diesel so I can run it on Bio-Diesel, a renewable fuel source? -- Keith >Volker Cosmic upheaval is not so moving as a little child pondering the death of a sparrow in the corner of a barn. -Anouk Aimee, French Actor Death is better, a milder fate than tyranny, Aeschylus (525BC-456BC), Agamemnon I wear no Burka. - Mother Nature ---------- ---------- ==== > <... >> Might as well add this...The downtrodden slaves of the >> primitive cult of Islam have to ask the imam for >> permission for every moment of their lives, including how >> to take a dump.... Most don't, you know. See below. >> http://www.islam.tc/ask-imam/view.php?q=2325 >> ============ >> Do we have to use a squatpan? I have difficulty using a >> squatpan? Can one use a high toilet when taking the >> necessary precautions not to let any water or najaasat >> splash on oneself? Jazakallah Khair >> Answer: >> One should relieve oneself in a squatting posture. >> However, if one is unable or experiences difficulty in >> squatting, he may use the western toilet and avoid the >> impurities splashing on the body or clothes. Gimme a fucking break, John. This is no worse than the > Judeo-Christian losers who call in to Laura Schlessinger > every goddamn day asking for moral guidance on issues even > more trivial. Give yourself a break. No Judeo-Christian losers ask is it > permissible, or do we have to... Around 80% of the questions > to the brainwashed imam on that web site are of that type. Listen to her show before you say nonsense like that. Jim ==== >> message >> <...>> Might as well add this...The downtrodden slaves of the > primitive cult of Islam have to ask the imam for > permission for every moment of their lives, including > how to take a dump.... >> Most don't, you know. See below. > http://www.islam.tc/ask-imam/view.php?q=2325 >> ============ > Do we have to use a squatpan? I have difficulty using a > squatpan? Can one use a high toilet when taking the > necessary precautions not to let any water or najaasat > splash on oneself? Jazakallah Khair >> Answer: > One should relieve oneself in a squatting posture. > However, if one is unable or experiences difficulty in > squatting, he may use the western toilet and avoid the > impurities splashing on the body or clothes. >> Gimme a fucking break, John. This is no worse than the >> Judeo-Christian losers who call in to Laura Schlessinger >> every goddamn day asking for moral guidance on issues >> even more trivial. >> Give yourself a break. No Judeo-Christian losers ask >> is it permissible, or do we have to... Around 80% of >> the questions to the brainwashed imam on that web site are >> of that type. > > Listen to her show before you say nonsense like that. I've listened to people who listen to her show. Most of them need to loosen up and suck a dick, eat some pussy, or just get laid. They're misfits. Only the dozen most fucked up of all Christians regularly ask their church for permission for anything. The ordinary Christian might ask for advice, but since they won't be stoned, beaten with clubs, have a stone wall toppled on them or thrown off the top of a tall building for the crime of thinking for themselves, as those downtrodden slaves of the primitive islam cult would, they don't think of it as permission. ==== > message >> <...>> Might as well add this...The downtrodden slaves of the > primitive cult of Islam have to ask the imam for > permission for every moment of their lives, including > how to take a dump.... >> Most don't, you know. See below. > http://www.islam.tc/ask-imam/view.php?q=2325 >> ============ > Do we have to use a squatpan? I have difficulty using a > squatpan? Can one use a high toilet when taking the > necessary precautions not to let any water or najaasat > splash on oneself? Jazakallah Khair >> Answer: > One should relieve oneself in a squatting posture. > However, if one is unable or experiences difficulty in > squatting, he may use the western toilet and avoid the > impurities splashing on the body or clothes. >> Gimme a fucking break, John. This is no worse than the >> Judeo-Christian losers who call in to Laura Schlessinger >> every goddamn day asking for moral guidance on issues >> even more trivial. >> Give yourself a break. No Judeo-Christian losers ask >> is it permissible, or do we have to... Around 80% of >> the questions to the brainwashed imam on that web site are >> of that type. Listen to her show before you say nonsense like that. I've listened to people who listen to her show. Most of them > need to loosen up and suck a dick, eat some pussy, or just get > laid. They're misfits. Only the dozen most fucked up of all Christians regularly ask > their church for permission for anything. The ordinary Christian > might ask for advice, but since they won't be stoned, beaten > with clubs, have a stone wall toppled on them or thrown off the > top of a tall building for the crime of thinking for > themselves, as those downtrodden slaves of the primitive islam > cult would, they don't think of it as permission. > Islam is no more a primitive cult than christianity would be, had we been the unlucky ones and ended up as the third world. Then they would be the progressive ones, and we would be declaring the occasional crusades against them. They would be the enlightened ones, while our masses would be firmly under the heels of despotic kings and dukes - and even they would be under the heels of cardinals and the pope. It isn't very long ago that things were actually like that. As things turned out, we beat the islamic world to a dominant posistion, largely through widespread colonialism and conquest, amidst much raping and pillaging. Our wealthy status secured, we developed away from a feudal culture. Science made great headway (something it was actually doing for a time in the islamic world when the christian world was in the dark ages). My point is that it is through equal parts of luck and aggression that we can now call ourselves enlightened and them primitive. If the western economy were to collapse and bring us down to their standards of living, our civilized facades would crumble in a matter of decades. Incidentally, it is an economy which is maintained substantially through the continuing exploitation of poorer countries around the world. Is it really any wonder that so many hate us? All that aside, I'm no friend of cults or religious dogma. I think it would be best for both islam and christianity to simply go away. They have done/are doing more harm than good. Of course, it's not going to happen anytime soon. Perhaps one day we will have advanced to a point where such archaic nonsense has become useless to us... ==== [snip] > If the western economy were to collapse and bring us down to their standards > of living, our civilized facades would crumble in a matter of decades. > Incidentally, it is an economy which is maintained substantially through the > continuing exploitation of poorer countries around the world. Is it really > any wonder that so many hate us? Nobody hates you people. You are so loving and kind people. Searching life on Mars and ignoring it in Africa. -Abhi. ==== > [snip] > If the western economy were to collapse and bring us down to their standards > of living, our civilized facades would crumble in a matter of decades. > Incidentally, it is an economy which is maintained substantially through the > continuing exploitation of poorer countries around the world. Is it really > any wonder that so many hate us? Nobody hates you people. You are so loving and kind people. Searching > life on Mars and ignoring it in Africa. Eh? You realize, of course, that that was my whole point? No need to get sarcastic with me. Save it for the self-righteous christians :-) -Abhi. ==== > >> [snip] >> If the western economy were to collapse and bring us >> down to their > standards >> of living, our civilized facades would crumble in a >> matter of decades. Incidentally, it is an economy which >> is maintained substantially through > the >> continuing exploitation of poorer countries around the >> world. Is it > really >> any wonder that so many hate us? >> Nobody hates you people. You are so loving and kind >> people. Searching life on Mars and ignoring it in Africa. > > Eh? You realize, of course, that that was my whole point? > No need to get sarcastic with me. Save it for the > self-righteous christians :-) Give him a break. He's an idiot. One day, slugs like that fool are making that kind of infantile attempts to deride the U.S. for not living up to some imaginary obligation to save the primitives from themselves, and the next day the same fool is yammering about how they should all hate us for arrogantly thinking we can save the world. Meanwhile, he and his 29 brothers and sisters are eating the seeds we give them for next year's crops. By the way, the self-righteous christian missionaries are the people the snivelling idiot was parroting. ==== >> If the western economy were to collapse and bring us down >> to their standards of living, our civilized facades would >> crumble in a matter of decades. Incidentally, it is an >> economy which is maintained substantially through the >> continuing exploitation of poorer countries around the >> world. Is it really any wonder that so many hate us? > > Nobody hates you people. You are so loving and kind people. > Searching life on Mars and ignoring it in Africa. We do more for the Africans than the Africans do. Here...use these free condoms. Fuck you. That ain't manly. Now let's talk about what Africans do for us, and why we're obliged to solve each others' problems. Africans are cool. One of my favorite writers, Wilbur Smith, is an African. ==== >> If the western economy were to collapse and bring us down > to their standards of living, our civilized facades would > crumble in a matter of decades. Incidentally, it is an > economy which is maintained substantially through the > continuing exploitation of poorer countries around the > world. Is it really any wonder that so many hate us? >> >> Nobody hates you people. You are so loving and kind people. >> Searching life on Mars and ignoring it in Africa. We do more for the Africans than the Africans do. Here...use these free condoms. >Fuck you. That ain't manly. Now let's talk about what Africans do for us, and why we're >obliged to solve each others' problems. Africans are cool. One of my favorite writers, Wilbur Smith, is >an African. I went to a college that had a bunch of Gambians on their soccer team. Nice guys, but really jumpy. As I recall, none of them were in any big hurry to get back there, either. It's pretty bad when you consider Oklahoma an improvement over your native country. -- V.G. People are more violently opposed to fur than leather, because it is easier to harrass rich women than it is motorcycle gangs. - Bumper Sticker (This sig file contains not less than 80% recycled SPAM) Sarcasm is my sword, Apathy is my shield. ==== >> message >> message > .. >> <...>> Might as well add this...The downtrodden slaves of >> the primitive cult of Islam have to ask the imam for >> permission for every moment of their lives, >> including how to take a dump.... >> Most don't, you know. See below. >> http://www.islam.tc/ask-imam/view.php?q=2325 >> ============ >> Do we have to use a squatpan? I have difficulty >> using a squatpan? Can one use a high toilet when >> taking the necessary precautions not to let any >> water or najaasat splash on oneself? Jazakallah >> Khair >> Answer: >> One should relieve oneself in a squatting posture. >> However, if one is unable or experiences difficulty >> in squatting, he may use the western toilet and >> avoid the impurities splashing on the body or >> clothes. >> Gimme a fucking break, John. This is no worse than > the Judeo-Christian losers who call in to Laura > Schlessinger every goddamn day asking for moral > guidance on issues even more trivial. >> Give yourself a break. No Judeo-Christian losers ask > is it permissible, or do we have to... Around 80% > of the questions to the brainwashed imam on that web > site are of that type. >> Listen to her show before you say nonsense like that. >> I've listened to people who listen to her show. Most of >> them need to loosen up and suck a dick, eat some pussy, or >> just get laid. They're misfits. >> Only the dozen most fucked up of all Christians regularly >> ask their church for permission for anything. The ordinary >> Christian might ask for advice, but since they won't be >> stoned, beaten with clubs, have a stone wall toppled on >> them or thrown off the top of a tall building for the >> crime of thinking for themselves, as those downtrodden >> slaves of the primitive islam cult would, they don't think >> of it as permission. > > Islam is no more a primitive cult than christianity would > be, had we been the unlucky ones and ended up as the third > world. Then they would be the progressive ones, Not a chance. One half of their slaves have virtually no rights, as promulgated by the pervert mohammed and some lackeys. There are some women in Pakistan (I think) who say they're going to build a mosque for females, with a female imam. The slavemasters (imams, mullahs) are going berserk. They're embarrassed to even talk about such a thing. The women who're doing that will soon be dead. The islam cult is designed for the 7th century and just about every form of progress is strictly forbidden by at least one of its laws. How would you like to know that you can be killed for going outdoors alone? How would you like to be told that it's understandable and that you need to repent if some asshole sees the shape of your butt through your islam-dictated coverings and rapes you? That's the way it is for islam-enslaved women, and that's only one example of why it's accurate to refer to moslems as primitive. >and we > would be declaring the occasional crusades against them. > They would be the enlightened ones, while our masses > would be firmly under the heels of despotic kings and dukes > - and even they would be under the heels of cardinals and > the pope. It isn't very long ago that things were actually > like that. As things turned out, we beat the islamic world > to a dominant posistion, largely through widespread > colonialism and conquest, amidst much raping and pillaging. Holy shit. Perpetuating the It's all our fault nonsense is ridiculous. I don't know exactly what you mean by not very long ago, but modern humans did in fact move past the days when raping an pillaging were okay. If the Catholic Church, for example, had laid down 24-hour-per-day rules and violently enforced them, we might be in the sad state the mohammedans are in today, enslaved and in mortal fear of even mentioning ways to improve our lives. Advocating any change is Speaking against Islam, a crime punishable by death. If the civilized world had tried to evolve under that sort of rule, with bosses watching over everyone to make sure they did their rites on schedule from dawn to dusk, there would have been no progress at all. Speaking of colonialism, moslems are required to kill all the indigenous people whenever they take over a geographical area. You can see that sort of mass murder today in at least one country in Africa. (Okay, they're allowed to keep a few alive as slaves. You can see that today in the same places.) A moslem living in any civilized country is required to try to overthrow its government. There will never be peace on earth as long as we allow those parasites to infest the modern world with that attitude. > Our wealthy status secured, we developed away from a feudal > culture. Science made great headway (something it was > actually doing for a time in the islamic world when the > christian world was in the dark ages). My point is that it > is through equal parts of luck and aggression that we can > now call ourselves enlightened and them primitive. If the > western economy were to collapse and bring us down to their > standards of living, our civilized facades would crumble in > a matter of decades. Incidentally, it is an economy which > is maintained substantially through the continuing > exploitation of poorer countries around the world. Is it > really any wonder that so many hate us? Resentment of our freedoms and resentment of rules against trying to emulate us, while being threatened with death for trying, is one of the reasons they hate modern humans. However, the big reason they hate modern humans is that they're taught hatred from day one of their lives. > All that aside, I'm no friend of cults or religious dogma. > I think it would be best for both islam and christianity to > simply go away. Could be. It can't happen because they're too entrenched as moral compasses (although moral prison better describes islam) and there's nothing to replace them in that regard. No modern government could take over that duty and teach anything resembling morality because there could never be agreement on what that means. For example, we might think murder is a violation of fundamental morality, but someone else might say its an absolutely understandable and justifiable reaction to an insult, and the new ACLU would fight any condemnation of murder on that basis. >They have done/are doing more harm than > good. Of course, it's not going to happen anytime soon. > Perhaps one day we will have advanced to a point where such > archaic nonsense has become useless to us... That would be cool! However, I'm afraid that very soon after that ideal state of affairs attains, new religions will necessarily spring up. The skinny little intellectuals of the world will invent gods and ascribe superpowers to them, and then sell the idea to stop the bullies, thieves etc. from behaving like savages, beating them up, screwing their women and eating their lunch. That's a likely depiction of how and why the current gods were created and the attendant religions were contrived. JMHO... ==== [...piggybacked cuz my server didn't get John's reply...] <...Dr. Laura... I've listened to people who listen to her show. Most of them > need to loosen up and suck a dick, eat some pussy, or just get > laid. They're misfits. Right. You know what goes on on her show because you've read the writings of her listeners. Poor. > Only the dozen most fucked up of all Christians regularly ask > their church for permission for anything. Which means *nothing* in this context. You don't know that the people who write in to the Imam do so regularly. > The ordinary Christian > might ask for advice, but since they won't be stoned, beaten > with clubs, have a stone wall toppled on them or thrown off the > top of a tall building for the crime of thinking for > themselves I won't deny that some do, but you don't know that the ordinary Muslim lives in fear of such. There are a *lot* of them, you know, and many live here in the West, or in what even you would describe as modern cities and circumstances elsewhere. You might ridicule those questions based on your own cultural biases and ignorances, but I have no reason to believe that they're not being asked in all sincerity, and out of a desire to do the Right Thing. Again, listen to all the fundy Laurettes before you jump to what'll ultimately wind up being more embarrassing and false conclusions. > as those downtrodden slaves of the primitive islam > cult would, they don't think of it as permission. You're really making yourself look silly in front of those of us who *have* listened to her show. Hardly a caller passes who doesn't ask her what their moral obigations are. Not what's permitted, but Am I morally obilgated... In THOSE. EXACT. FUCKING. WORDS. It's like a mantra with them. Sheesh even. There are a *huge* number of fundy Christians who look to Dr. Laura in *exactly* the same way as some Muslims look to that Imam. For you to argue against that without any direct knowledge of what her show's like is PRETTY FUCKING LAME, and in this forum, pretty shameful. If you want to ridicule Islam, I suggest you find another way. Perhaps one that doesn't have such an obvious Judeo-Christian parallel. Jim ==== > [...piggybacked cuz my server didn't get John's reply...] > > >> message > > <...Dr. Laura... >> I've listened to people who listen to her show. Most of >> them need to loosen up and suck a dick, eat some pussy, >> or just get laid. They're misfits. > > Right. You know what goes on on her show because you've > read the writings of her listeners. > > Poor. I have no idea what goes on on her show, and I don't care. I knew it appeals to idiots, and now I know it appeals to you. Is this a coincidence? >> Only the dozen most fucked up of all Christians >> regularly ask their church for permission for anything. > > Which means *nothing* in this context. You don't know that > the people who write in to the Imam do so regularly. I do know that the imam's answers will eventually cover 1440 minutes of the day of the life of every mohammedan. It might occur to you, or maybe I'll have to explain it, that when the imam lays down the law to one slave for the minute he's taking a leak or something, it applies to all of them, and when he says that a woman must drop whatever slave labor she's up to and open her legs on demand, he's telling all of them. >> The ordinary Christian >> might ask for advice, but since they won't be stoned, >> beaten with clubs, have a stone wall toppled on them or >> thrown off the top of a tall building for the crime of >> thinking for themselves > > I won't deny that some do, but you don't know that the > ordinary Muslim lives in fear of such. There are a *lot* > of them, you know, and many live here in the West, or in > what even you would describe as modern cities and > circumstances elsewhere. Moslems in this country are required to try to overthrow our government. (They say encouraged, rather than required, but encouragement from someone who stood over you with a whip while you learned absolute obedience in your childhood doesn't exactly mean what it means to a free person.) > You might ridicule those questions based on your own > cultural biases and ignorances, but I have no reason to > believe that they're not being asked in all sincerity, and > out of a desire to do the Right Thing. > > Again, listen to all the fundy Laurettes before you jump to > what'll ultimately wind up being more embarrassing and > false conclusions. I already said those people are a bunch of fucking jerkoffs. >> as those downtrodden slaves of the primitive islam >> cult would, they don't think of it as permission. > > You're really making yourself look silly in front of those > of us who *have* listened to her show. I already said those people are a bunch of fucking jerkoffs. > Hardly a caller passes who doesn't ask her what their moral > obigations are. Not what's permitted, but Am I morally > obilgated... > > In THOSE. EXACT. FUCKING. WORDS. It's like a mantra > with them. Does the Laura thing tell them how they'll be punished by whatever church they're in, in the event that they stray? No. > Sheesh even. There are a *huge* number of fundy Christians > who look to Dr. Laura in *exactly* the same way as some > Muslims look to that Imam. For you to argue against that > without any direct knowledge of what her show's like is > PRETTY FUCKING LAME, and in this forum, pretty shameful. Christians are mostly nuts. They follow a shitload of rules because of internal conflicts. They don't worry about the rev coming along with a truckload of river rocks and smashing their skulls with them. > If you want to ridicule Islam, I suggest you find another > way. Perhaps one that doesn't have such an obvious > Judeo-Christian parallel. I'm only comparing islam to individuals' rights and freedom. Oops, I mean contrasting. I'm about as far from a fan of any religion as it gets. Well, maybe the Rastafarians are okay, except for their weird choice about the downfall of islam. ==== [...piggybacked cuz my server didn't get John's reply...] >> message <...Dr. Laura... >> I've listened to people who listen to her show. Most of >> them need to loosen up and suck a dick, eat some pussy, >> or just get laid. They're misfits. Right. You know what goes on on her show because you've > read the writings of her listeners. Poor. I have no idea what goes on on her show, and I don't care. I > knew it appeals to idiots, and now I know it appeals to you. Is > this a coincidence? > Only the dozen most fucked up of all Christians >> regularly ask their church for permission for anything. Which means *nothing* in this context. You don't know that > the people who write in to the Imam do so regularly. I do know that the imam's answers will eventually cover 1440 > minutes of the day of the life of every mohammedan. It might > occur to you, or maybe I'll have to explain it, that when the > imam lays down the law to one slave for the minute he's taking a > leak or something, it applies to all of them, and when he says > that a woman must drop whatever slave labor she's up to and open > her legs on demand, he's telling all of them. > The ordinary Christian >> might ask for advice, but since they won't be stoned, >> beaten with clubs, have a stone wall toppled on them or >> thrown off the top of a tall building for the crime of >> thinking for themselves I won't deny that some do, but you don't know that the > ordinary Muslim lives in fear of such. There are a *lot* > of them, you know, and many live here in the West, or in > what even you would describe as modern cities and > circumstances elsewhere. Moslems in this country are required to try to overthrow our > government. (They say encouraged, rather than required, but > encouragement from someone who stood over you with a whip while > you learned absolute obedience in your childhood doesn't exactly > mean what it means to a free person.) You might ridicule those questions based on your own > cultural biases and ignorances, but I have no reason to > believe that they're not being asked in all sincerity, and > out of a desire to do the Right Thing. Again, listen to all the fundy Laurettes before you jump to > what'll ultimately wind up being more embarrassing and > false conclusions. I already said those people are a bunch of fucking jerkoffs. > as those downtrodden slaves of the primitive islam >> cult would, they don't think of it as permission. You're really making yourself look silly in front of those > of us who *have* listened to her show. I already said those people are a bunch of fucking jerkoffs. Hardly a caller passes who doesn't ask her what their moral > obigations are. Not what's permitted, but Am I morally > obilgated... In THOSE. EXACT. FUCKING. WORDS. It's like a mantra > with them. Does the Laura thing tell them how they'll be punished by > whatever church they're in, in the event that they stray? No. Sheesh even. There are a *huge* number of fundy Christians > who look to Dr. Laura in *exactly* the same way as some > Muslims look to that Imam. For you to argue against that > without any direct knowledge of what her show's like is > PRETTY FUCKING LAME, and in this forum, pretty shameful. Christians are mostly nuts. They follow a shitload of rules > because of internal conflicts. They don't worry about the rev > coming along with a truckload of river rocks and smashing their > skulls with them. If you want to ridicule Islam, I suggest you find another > way. Perhaps one that doesn't have such an obvious > Judeo-Christian parallel. I'm only comparing islam to individuals' rights and freedom. > Oops, I mean contrasting. I'm about as far from a fan of any religion as it gets. Well, > maybe the Rastafarians are okay, except for their weird choice > about the downfall of islam. > You'd make a great virtual Jehova's Witness. Just a knockin' on ng doors annoying the shit out of people. Try alt.trailerliving. You can practice on your neighbors. ==== > >> [...piggybacked cuz my server didn't get John's >> reply...] > message >> <...Dr. Laura...>> I've listened to people who listen to her show. Most > of them need to loosen up and suck a dick, eat some > pussy, or just get laid. They're misfits. >> Right. You know what goes on on her show because you've >> read the writings of her listeners. >> Poor. >> I have no idea what goes on on her show, and I don't care. >> I knew it appeals to idiots, and now I know it appeals to >> you. Is this a coincidence? > Only the dozen most fucked up of all Christians > regularly ask their church for permission for > anything. >> Which means *nothing* in this context. You don't know >> that the people who write in to the Imam do so >> regularly. >> I do know that the imam's answers will eventually cover >> 1440 minutes of the day of the life of every mohammedan. >> It might occur to you, or maybe I'll have to explain it, >> that when the imam lays down the law to one slave for the >> minute he's taking a leak or something, it applies to all >> of them, and when he says that a woman must drop whatever >> slave labor she's up to and open her legs on demand, he's >> telling all of them. > The ordinary Christian > might ask for advice, but since they won't be stoned, > beaten with clubs, have a stone wall toppled on them > or thrown off the top of a tall building for the > crime of thinking for themselves >> I won't deny that some do, but you don't know that the >> ordinary Muslim lives in fear of such. There are a >> *lot* of them, you know, and many live here in the West, >> or in what even you would describe as modern cities and >> circumstances elsewhere. >> Moslems in this country are required to try to overthrow >> our government. (They say encouraged, rather than >> required, but encouragement from someone who stood over >> you with a whip while you learned absolute obedience in >> your childhood doesn't exactly mean what it means to a >> free person.) >> You might ridicule those questions based on your own >> cultural biases and ignorances, but I have no reason to >> believe that they're not being asked in all sincerity, >> and out of a desire to do the Right Thing. >> Again, listen to all the fundy Laurettes before you jump >> to what'll ultimately wind up being more embarrassing >> and false conclusions. >> I already said those people are a bunch of fucking >> jerkoffs. > as those downtrodden slaves of the primitive islam > cult would, they don't think of it as permission. >> You're really making yourself look silly in front of >> those of us who *have* listened to her show. >> I already said those people are a bunch of fucking >> jerkoffs. >> Hardly a caller passes who doesn't ask her what their >> moral obigations are. Not what's permitted, but Am >> I morally obilgated... >> In THOSE. EXACT. FUCKING. WORDS. It's like a mantra >> with them. >> Does the Laura thing tell them how they'll be punished by >> whatever church they're in, in the event that they stray? >> No. >> Sheesh even. There are a *huge* number of fundy >> Christians who look to Dr. Laura in *exactly* the same >> way as some Muslims look to that Imam. For you to argue >> against that without any direct knowledge of what her >> show's like is PRETTY FUCKING LAME, and in this forum, >> pretty shameful. >> Christians are mostly nuts. They follow a shitload of >> rules because of internal conflicts. They don't worry >> about the rev coming along with a truckload of river rocks >> and smashing their skulls with them. >> If you want to ridicule Islam, I suggest you find >> another way. Perhaps one that doesn't have such an >> obvious Judeo-Christian parallel. >> I'm only comparing islam to individuals' rights and >> freedom. Oops, I mean contrasting. >> I'm about as far from a fan of any religion as it gets. >> Well, maybe the Rastafarians are okay, except for their >> as soon as I bring about the downfall of islam. > > You'd make a great virtual Jehova's Witness. Just a > knockin' on ng doors annoying the shit out of people. Try > alt.trailerliving. You can practice on your neighbors. You need to shun serious discussions like the plague. You have no fucking brains and therefore you couldn't possibly have anything to say. You can only remind us that you're a damned fool and that there's nothing you like more than my pecker, pervert. By the way, I did make a JW once, in high school. Not bad. You could have tried her brother. ==== [...piggybacked cuz my server didn't get John's reply...] >> message <...Dr. Laura... >> I've listened to people who listen to her show. Most of >> them need to loosen up and suck a dick, eat some pussy, >> or just get laid. They're misfits. Right. You know what goes on on her show because you've > read the writings of her listeners. Poor. I have no idea what goes on on her show, and I don't care. I > knew it appeals to idiots, and now I know it appeals to you. Is > this a coincidence? No, it's just one of those embarrassingly wrong assumptions on your part. <...snip spin and froth...> You have no idea what you're talking about, you know. Jim ==== >> [...piggybacked cuz my server didn't get John's >> reply...] > message >> <...Dr. Laura...>> I've listened to people who listen to her show. Most > of them need to loosen up and suck a dick, eat some > pussy, or just get laid. They're misfits. >> Right. You know what goes on on her show because you've >> read the writings of her listeners. >> Poor. >> I have no idea what goes on on her show, and I don't care. >> I knew it appeals to idiots, and now I know it appeals to >> you. Is this a coincidence? > > No, I knew that. > it's just one of those embarrassingly wrong assumptions > on your part. I haven't made any assumptions. Just to humor your attempt to deflect what I was saying originally, I stated that Dr. Laura doesn't give orders and that the islamic slavemasters do. That's not an assumption. It's a fact. > <...snip spin and froth... > You have no idea what you're talking about, you know. What you mean is that I have no idea what you're talking about. I do, of course, but it's merely a discursion from what I was talking about and it's not interesting or relevant. Dr. Laura and her fans are fools, but islam is still an abomination. ==== >> Gimme a fucking break, John. This is no worse than the >> Judeo-Christian losers who call in to Laura Schlessinger >> every goddamn day asking for moral guidance on issues even >> more trivial. >> Give yourself a break. No Judeo-Christian losers ask is it >> permissible, or do we have to... Around 80% of the questions >> to the brainwashed imam on that web site are of that type. Listen to her show before you say nonsense like that. Jim > Most of the people who call the Dr. Laura show are an embarrassment to the entire species. -- V.G. People are more violently opposed to fur than leather, because it is easier to harrass rich women than it is motorcycle gangs. - Bumper Sticker (This sig file contains not less than 80% recycled SPAM) Sarcasm is my sword, Apathy is my shield. ==== >> Gimme a fucking break, John. This is no worse than the > Judeo-Christian losers who call in to Laura Schlessinger > every goddamn day asking for moral guidance on issues even > more trivial. >> Give yourself a break. No Judeo-Christian losers ask is it > permissible, or do we have to... Around 80% of the questions > to the brainwashed imam on that web site are of that type. >>Listen to her show before you say nonsense like that. >>Jim Most of the people who call the Dr. Laura show are an embarrassment >to the entire species. Which species would that be, VG? -- Jimmy Snibbler ==== in alt.fan.art-bell: > Gimme a fucking break, John. This is no worse than the >> Judeo-Christian losers who call in to Laura Schlessinger >> every goddamn day asking for moral guidance on issues even >> more trivial. >> Give yourself a break. No Judeo-Christian losers ask is it >> permissible, or do we have to... Around 80% of the questions >> to the brainwashed imam on that web site are of that type. >>Listen to her show before you say nonsense like that. >>Jim >Most of the people who call the Dr. Laura show are an embarrassment >>to the entire species. Which species would that be, VG? Whichever one they belong to. -- V.G. People are more violently opposed to fur than leather, because it is easier to harrass rich women than it is motorcycle gangs. - Bumper Sticker (This sig file contains not less than 80% recycled SPAM) Sarcasm is my sword, Apathy is my shield. ==== >> in other words.. YOU ARE NOT MERELY A MACHINE. 1=0. that is, YOUR >> BINARY EXISTENCE IS NOT ALL THAT IS. 0 = totality. 8 = 0. infinity = >> totality. WELCOME TO THE HIVE-MIND. >> >> 1=0 >> 1=0 >> 1=0 >> 1=0 >> 0=1 >> 0=1 >> 0=1 >> 0=1 >> i would add to this and say that Satan = 1, meaning singularity or >> separation and God = 0 meaning connection/totality/OneNess . . IN FACT. you trap energy via the electromagnetic spectrum. YOU ARE energy > trapped in matter. Your physical body does not differ from any other > matter. true life is PURE ENERGY and YOU ARE PURE ENERGY. WHY do you think > people pray? YOU COMPLETE THE CIRCUIT AS YOU UNITE YOUR HANDS. ENERGY > FLOWS THROUGH YOU. quantum totality is a term for the MASSIVE QUANTUM DATABASE that all our thoughts come from and all observations are stored in. ALL PEOPLE KNOW TRUTH. in order to access this database all that you need to know is that your thoughts are, quite literally, THE WORD OF GOD. break them down into two separate parts(0 AND 1) in order to comprehend all things. analyze these words(these=left,words=right), create two sets for each opposite. quiet the mind and you will increase efficiency. certain chemical substances can enhance the ability of the two minds. find the center in all energy will cause the minds to cycle optimally, allowing energy to flow from quantum totality to physical reality at the most desirable rate. PSYCHEDELIC is the term used for the most useful mentioned drugs. anti-oxidants are useful for suppressing QUANTUM SINGULARITY by blocking SINGULAR FREE RADICALS. they also allow the minds to cycle efficiently. Melatonin is highly useful in this manner because each of its metabolites is by itself an efficient radical neutralizer, creating a cascade effect. DXM can be viewed as a SAGE-CREATOR . . ever seen various gurus with that wide eyes stare? now go find someone on DXM to see the same thing. JAMES RANDI OWES ME $1,000,000 - - - - NORMAL implies static linearity. there IS ONLY PARANORMAL. ==== That it -- the check was voided. Give up! The Psychedelick Pope Saint Isidore of Laytonville ^.85^ Patron Saint of the Internet ^.85^ ¡¡^.85^ ¡¡ http://apple2.org.za/gswv/me AOXOMOXOA and ENESSA QUA ONNICA ==== > JAMES RANDI OWES ME $1,000,000 You owe everyone on this planet a million dollars for being such a fucking stupid embarrassment to the human race, unless you're just another poorly designed and ineptly implemented automaton. ==== - ==== > in other words.. YOU ARE NOT MERELY A MACHINE. 1=0. that is, YOUR > BINARY EXISTENCE IS NOT ALL THAT IS. 0 = totality. 8 = 0. infinity = > totality. WELCOME TO THE HIVE-MIND. > > 1=0 > 1=0 > 1=0 > 1=0 >> 0=1 > 0=1 > 0=1 > 0=1 >> i would add to this and say that Satan = 1, meaning singularity or > separation and God = 0 meaning connection/totality/OneNess . . >> IN FACT. you trap energy via the electromagnetic spectrum. YOU ARE energy >> trapped in matter. Your physical body does not differ from any other >> matter. true life is PURE ENERGY and YOU ARE PURE ENERGY. WHY do you think >> people pray? YOU COMPLETE THE CIRCUIT AS YOU UNITE YOUR HANDS. ENERGY >> FLOWS THROUGH YOU. > > quantum totality is a term for the MASSIVE QUANTUM DATABASE that all our > thoughts come from and all observations are stored in. ALL PEOPLE KNOW > TRUTH. in order to access this database all that you need to know is > that your thoughts are, quite literally, THE WORD OF GOD. break them down > into two separate parts(0 AND 1) in order to comprehend all things. > analyze these words(these=left,words=right), create two sets for each > opposite. quiet the mind and you will increase efficiency. certain > chemical substances can enhance the ability of the two minds. > > find the center in all energy will cause the minds to cycle optimally, > allowing energy to flow from quantum totality to physical reality at > the most desirable rate. PSYCHEDELIC is the term used for the most useful > mentioned drugs. anti-oxidants are useful for suppressing QUANTUM > SINGULARITY by blocking SINGULAR FREE RADICALS. they also allow the minds > to cycle efficiently. Melatonin is highly useful in this manner because > each of its metabolites is by itself an efficient radical neutralizer, > creating a cascade effect. > > DXM can be viewed as a SAGE-CREATOR . . ever seen various gurus with that > wide eyes stare? now go find someone on DXM to see the same thing. > > JAMES RANDI OWES ME $1,000,000 > > - - - - > > NORMAL implies static linearity. there IS ONLY PARANORMAL. . . . . NOW that you know that FREE-RADICALS/QUANTUM-SINGULARITIES/RADIATION is the TRUE-EVIL of the physical world you can comprehend Sin. Unite your hands at the finger tips, allowing energy to flow . . Quiet your Mind and ask, like you mean it, for GOD to forgive you of ALL YOUR SINS SINS I N G U L A R I T I E S - - - - AGAIN, God is telling you that ALL-IS-ONE . . SEPARATION/SINGULARITY is FALSE. if you are still skeptical, for God's sake go read Super String Theory . . http://www.superstringtheory.com . . http://www.superstringtheory.com/basics/basic7.html the very last part states: . . string theory is, but judging from all of these relationships, it must be a very interesting and rich theory, one where distance scales, coupling strengths and even the number of dimensions in spacetime are not fixed concepts but fluid entities that shift with our point of view. - - - - YOUR THOUGHTS CREATE REALITY. ==== Turn your repeater OFF NOW! The Psychedelick Pope Saint Isidore of Laytonville ^.85^ Patron Saint of the Internet ^.85^ ¡¡^.85^ ¡¡ http://apple2.org.za/gswv/me AOXOMOXOA and ENESSA QUA ONNICA ==== > YOUR THOUGHTS CREATE REALITY. Jump off a tall building, and on the way down think hard about the nice soft, large, foam rubber cushion at the end of your 10-sec journey that will gently reintegrate you with the street below. Mind the air-ground interface. -=-=-=-=- ==== I am working through (solo) a couple of differential equations texts. One is from '73, the other is a dover republication of one from '63. I do not have the books with me, and I cannot remember all of the author's names, but I think Bauer was one for the '73, and Tenenbaum for the '63. I find it useful to play them off of each other when I get lost. (not an unusual occurance) At least in terms of reasonably simple first order or second order equations solvable with first order means, the '73 book uses a definite integral approach vs. the 63's solve for C method. The reason I find it interesting is because the '73 book is the only text I've seen that uses this method for differential equations that have variable starting conditions. I only have the two diff eq specific books, but my other calculus books all devote a chapter or two to them. To clarify what I mean, take a simple equation like: y' + 2y = e^x (solution valid through some point (x_0, y_0)) Find the integrating factor as e^2x and rewrite: [e^(2x)*y' + 2*e^(2x)*y] = d/dx[e^(2x)*y] = e^3x At this point, the 73 book would write that as (using t instead of x and noting some solution phi(t) and rebranding the point as (t_0, y_0)): d/dt[e^(2t)*phi(t)] = e^3t Int(t_0, t, e^(2s)*phi(s)) = 1/3*Int(t_0, t, 3*e^3t) = 1/3*(e^3t - e^(3*t_0)) e^(2t)*phi(t) - e^(2*t_0)*phi(t_0) = 1/3*(e^3t - e^(3*t_0)) e^(2t)*phi(t) - e^(2*t_0)*y_0 = 1/3*(e^3t - e^(3*t_0)) phi(t) = e^(2*(t_0-t))*y_0 + 1/3*(e^t - e^(3*t_0 - 2t) etc, while other texts would change this to be: d/dx[e^(2x)*y] = e^3x ... integrating ... e^(2x)*y = 1/3*e^3x + C, C = e^(2x_0)*y_0 - 1/3*e^3x_0 ... and rewrite ... y = 1/3*e^x + e^(2(x_0-x))*y_0 - 1/3*e^(3*x_0-2x) y = e^(2(x_0-x))*y_0 + 1/3*(e^x - e^(3*x_0-2x)) which is the same solution as the 'phi' method. Now, it is pretty apparent as to why the former works - my question is, why the favor for the later method ? Many of the problems I have done to date I have done with both methods in an attempt to come up with my own theory. I, personally, have noticed that I am more prone to simple errors going through the definate integral route. One other reason I've noted is that when you have variables seperable, and could, in the second method, write something like dy/y = x dx, in the first method you have something like phi'(t)/phi(t) = x, and must integrate, with a change of variables (u = phi'(t), du/u, limits from t_0 -> t to phi(t_0) -> phi(t)). This is less straightforward, at least to me. As I know many of you are mathematicians and teachers, I was wondering if anyone could tell me what the reasons are for the preference of the second method.