MandrakeRoot

So what is it made off this particle ? And how theoretically (supposing this particle exists) could it travel faster than lightspeed ? (wouldnt it need mass zero in order to be able to do so ?, but then again what would it be made off) Are you a physicist Dave ? Mandrake

Thank you for all these replies. So if i understood correctly the answer would be that is a current debate whether or not it is possible to travel faster than lightspeed ? For what i understood of the relativity theory, was that things should be looked at locally in comoving frames, and in such a frame it was easily deduced that something couldn't travel faster than light. But in the book i was reading it was equally written that such frames where just a local representation. Much like you can do locally euclidian stuff on a sphere if it is large enough or in some distored space with some regularity. So in fact i never quite understood why something would make it impossible to travel faster than light speed globally, all in respecting the fact that in each comoving frame you do not cross this speed limit. I hope that this time my question is more clear ? By the way what is a tachyon ? (Isn't a particle that appears in star trek stuff ? It really exists ?, and what is it ?) Mandrake

Hi, When you formalyl define numbers, with the Peano axioms at the basis. The irrationals are defined as the limits of a sequence of rational numbers. So the decimal representation of an irrational number itself is also defined and in this definition 0.999999999etc is equal to one. So the equality holds by definition. Just like 1 + 1 = 2 holds by definition. Mandrake

If i remember correctly the decimal representation is defined to have this property. Thus making 0.4999999999999999etc = 0.5 and 0.9999999999etc = 1. The geometric mean argument is i think the most easily understood. A pseudo-intituive argument could be something like : Let x = 0.99999999999999999etc. It is easily seen that x <= 1 and also that for any n > 1, 9*sum_m=1^n (1/10)^m < x, So for any eps > 0, 1 - eps < x <= 1. It follows that x could only be equal to 1. Which basicaly comes down to the geometric mean argument. Hey Radical edward your argument is not finished yet...you have to show why it is impossible for a rational to be in between x and 1, not using the fact they are equal; Mandrake,

Yeah that is the most easy way to find roots if you suspect that one of them is "easily" found. I think i probably misunderstood the nature of the problem. Mandrake

Hi, It is easily verified that x = 5,is a solution to f(x) = 0, right ? So since it is a polnomial the only thing you will need to proof is that you can write every polynonomial as the product of (x - r), where r is a root of the polynomial and you are finished. By the way there are explicit formulas for finding the roots of third degree polynomials (The formulaes of Cardano). So try proving that if p(x) is a polynomial of degree n and x_1,...x_n are its roots (counting multiplicities) that p(x) = prod_i=1^n (x - x_i) Then show a = 5 is a root of your polynomial and you are done Mandrake

Yeah i think i would agree with that. It might be very usefull in the future. The first physicist to occupy them with elementary particles and radioactivity, had no idea that would ever be used and in the time that was considered to be abstract physics without any use. Mandrake

I have to disagree about set theory not having applications. It answers questions like the place of the axiom of choice for instance, a vital theorem for Hahn-Banach and the existence of functionals, which is again very important for many questions in operator theory and thus for quantum mechanics. Also the axiom of choice is very often implicitely used in many proofs. Even though direct applications are maybe rare, the questions posed do have important implications in other fields of mathematics. A cardinal is a set, but also the size of a set. Two sets are said to have teh same cardinality if you can create a bijection in between them. Mandrake,

You are right. Though i believed we were talking of physical systems beyond the influence of human nature ? Because you can then also include the possibility of stopping the choatic pendulum,making it perfectly predictable. Mandrake

I dont think that there will be less randomness in the future then there is now. I dont see why flipping a coin would influence future flippings of coins for instance. If you considere many coins then already now you can say something about the ratio. With the same number of coins in the future, your "prediction" will nto be more accurate then it is now. Weather predictions have become more accurate in the last years, but it is still very hard to predict very accurately. Ten years ago the grid used to predict global weather patterns was about 100 km * 100 km, and now it is approx 10 km * 10 km, so tehre is some progress. Predictions will never be always correct you know, that would be contrary to the concept "random". Since if you can always predict accurately then it is not random. Mandrake,

I have a question about faster than light speed. For what i understood of relativity, it is impossible for anything to have a speed faster than light in a local frame, but the thing i dont understand is how does that make faster than light speed impossible globally ? I hope i posed the question clearly ? Mandrake

Each set of numbers serves pretty much a certain goal. Naturals would allow you to count. Fractions to solve stuff like ax = b, when a and b are natural numbers. Irrationals are limits of fractions and complex numbers allow you to express roots of polynomials. So each expansion would sort out of the before ordinary, but it depends a little what you call ordinary i guess ? Mandrake,

Yeah i think i pretty much agree with dave, that it just depends on what you call a concept. Since the complex right can be interpreted as the real plane i am sure there is some interpretation possible. For instance you can see multiplication with "i" as rotation over 90 degrees counterclockwise or something like so. Mandrake

I still would say all numbers are concepts, because you cant say something as "there are three apples" without having a concept of what "three" exactly is. They way i see it there are two separate worlds, one wherein you create you numbers mathematically and so in fact you define some sets with some properties. And the other world is the one in which we live wherein we will apply these concepts in order to facilitate our lives. You can easily apply N to the real worlds once you have a concept of more or less. If you have some collection of appels and you can say that some collection contains more then another. You can basically order them from nothing to a lot of appels. Then you call no (entire) appels "zero appels", the next best thing being the least amount of (entire) appels more then no appels, you call it one etc;... Like so you can also introduce fractions. Since pi would be the surface of a circle with unit radius i would say that the concept "Pi" can be found in the real world. Since basically pi is historically defined as being the surface of the circle with unit radius. By the way an irrational number is just that and its decimal representation is a representation and not the number ! 0.49999999999(and so on and so on) represents the same number as 0.50000000000, but they are different representations of the same thing. The point i am trying to make is that numbers are all concepts, but they are here since we have found them a use in average day live. So they are all concepts that can be applied in every day life. Mandrake

I would even say all numbers are concepts, since none of them "exist" in the real world. But they have each been constructed (as concepts) to help us do something we couldn't do without. The natural numbers help us to count things, the fractions to solve equations of the type nx = m, where n and m are natural numbers, which are equations naturally arising in ratio type of questions. The irrationals help us to number even more things in the real world, such as the example above. And then finally the complex numbers are also a very usefull tool in solving polynomials equations, but also in differential equations and many other fields. All numbers can be constructed mathematically from the hypothesis that there exists an infinite set. So all numbers are concepts and a world without is perfectible imaginable. In set theory it is possible to join an element to an ordered set and to extend the ordering such that all elements in the original set are smaller then this "new" element. This procedure applied to N would give a set with a maximal element, which is denoted often with infinity. Zero is the algebraic "unit" element for the addition, i.e., the unique element such that a + 0 = a for all a in some set with an addition defined on. Mandrake,

How about x^(ln y + 1) = exp( (lny + 1)lnx ) = exp (ln x) exp(ln x ln y) = x y^(ln x) Remember that by definition x^alpha is defined as exp (alpha ln x) Mandrake

Let me include the definitions of the different things a set B is kappa-closed if for every xi < kappa and every non empty A subseteq B intersection xi , we have sup A = union A element of B. since a cardinal is in fact an ordinal, and for ordinals alpha, beta, alpha < beta if alpha is an element of beta. xi < kappa means xi element of kappa and union A is defined as the set of all elements x such that there is an y in A for which x in y. Since each element of an ordinal is again an ordinal, it can be shown that sup A will be also an ordinal and thus the requirement sup A in B makes sense. a set B is said to be cofinal in kappa if for every xi < kappa we can find an eta in B such that xi <= eta (<= meaning < or =) The cofinality of a set is defined as the cardinality of the least set cofinal in it. A regular cardinal kappa is a cardinal for which cofinality(kappa) = kappa, and thus the least subset of kappa being cofinal in it must have cofinality kappa. (e.g. in N the set of even numbers is cofinal and omega = cardinality(N) is a regular cardinal). By the way cf(kappa) <= kappa for all cardinals kappa. Transfinite recursion is just some theorem saying that if you arrive to assign a set to each function f by some operation G, then there is an operation F such that F(alpha) = G(F|alpha). So in fact if you can define something for ordinals/cardinals smaller then alpha, then you can somehow extend it to include all ordinals. Like recursively defining a sequence x_n+1 = x_n + 2 or something like so, you define x_0 = 0 and then you recipe and then you have a sequence. I hope i could clarify the question ?

I would reason as follows : Since there are infinitely many prime numbers and since they are a subset of N, the set of prime numbers is clearly a countably infinite set and has thus the same cardinality as N. Mandrake

I would like to know the following: Let kappa be a regular cardinal and B a kappa-club (kappa-closed and cofinal in kappa). And let lambda be another regular cardinal strictly smaller than kappa. Why is it possible to choose a strictly increasing sequence {alpha_nu : nu < lambda} in B, such that the suppremum of {alpha_nu : nu < lambda} has cofinality lambda ? My first guess would be to use the fact that for every nu < lambda and every non empty A subseteq B intersection nu, sup A an element of B is. And then do something with transfinite recursion, but why would B even contain any ordinals smaller then lambda ? I hope someone could shed some light on this question. Thanks in advance, Mandrake