E-infinity communication 12
The Menger-Urysohn transfinite theory of dimension as used in E-infinity theory
Before continuing our discussion we must give you some literature on the subject as well as give the various people who contributed to E-infinity their due by mentioning their achievements. The easiest way to start studying the transfinite theory of dimension is to look at the work of Nada, Crnjac, Iovane, He and Zhong. Prof. Shokry Nada is a professor of topology who got his Ph.D. from Southampton, UK and although he is originally Egyptian he is since many years part of the full time staff of the University of Qatar, Dept. of Mathematics. I strongly recommend contacting him personally but only on scientific questions. He has no patience for gossip or triviality and will not answer to things like those using the internet for entertainment on an unacceptable level. I give here without any particular order seven papers from these various authors that will help in studying the Menger-Urysohn theory tailored to E-infinity and physics. All papers are from Chaos, Solitons & Fractals. On the mathematical theory of transfinite dimensions and its application in physics, 42, 2009, p.530. The mathematical theory of finite and infinite dimensional topological spaces and its relevance to quantum gravity, 42, 2009, pp. 1974. Partially ordered sets, transfinite topology and the dimension of Cantorian-fractal spacetime, 42, 2009, pp. 1796. On the Menger-Urysohn theory of Cantorian manifolds and transfinite dimensions in physics, 42, 2009, p. 781. Density manifolds, geometric measures and high-energy physics in transfinite dimensions, 42, 2009, p. 1539. From Menger-Urysohn to Hausdorff dimensions in high energy physics, 42, 2009, p. 2338. From the numerics of dynamics to the dynamics of numerics and vice versa in high energy particle physics, 42, 2009, p. 1780. It is however recommendable to go back to the original contribution of Menger and Urysohn. The papers of Urysohn are most in French and German. Menger published mainly in German and English. There is an important classical book by Hurwitz, a Polish mathematician writing in German. You will find all of that referred to in the papers of Nada, Iovane, He and Crnjac. Regarding the golden mean in physics, we forgot to mention a few papers which were initially considered outlandish but in the meantime, that is no longer the case. They are important papers by Leonard J. Malinowski, Electronic golden structure of the periodic chart, CS&F, 42, 2009, p. 1396. The other papers you will find on Elsevier's Science Direct.
To understand E-infinity fast we recommend that you take no short cuts. You must read at least one review paper by Mohamed El Naschie or Marek-Crnjac from the beginning to the end. At a minimum you should read El Naschie's paper in David Finkelstein's journal, Int. Journal of Theoretical Physics, Vol. 37, No. 12, 1998 from the beginning to the end. It is fair to say that after flirting with nonlinear sciences due to his background in the theory of stability and bifurcation as well as Rene Thom's catastrophe theory, El Naschie started to seriously enter into nonlinear science and chaos due to the encouragement of his friend, the chaos pioneer, Otto Rössler. Much of what he knows about chaos was taught to him by his Israeli friend, the well known scientist Itamar Procaccia from the Wiseman Inst. as well as the legendary figure of chaos, Mitchell Feigenbaum. El Naschie was highly impressed by the overall personality of Mitchell although he did not share his passion for red wine and smoking. El Naschie is almost a vegetarian, unlike Mitchell who lives from red meat.
Maybe it is time here to correct some old mistakes and omissions. El Naschie in retrospect was always at pain to acknowledge that Indian meteorologist Marie Selvam may have been the first to notice the E-infinity theoretical value of the inverse electromagnetic constant. He mentioned that on many occasions before but he asked me to mention that again on this occasion. In addition we must acknowledge that Carlos Castro did some important work on E-infinity and it is with sadness that we note that he moved towards blogs and internet publications, leaving serious discussion with his old friends on science. However these few words are said because we cannot rewrite history as we like. The truth remains always the truth, no matter how painful it is. A person who is outside our group but works with incredible dedication on fractal time is a German American Susie Vrobel. We should mention her work like Fractal Time and the Gift of Natural Constraints, Tempos in Sci. & Nature, Structures, Relations and Complexity, Vol. 879, June 1999. Another person outside our group who works with dedication on fractals in cosmology is the South African Jonathan J. Dickau and we recommend his paper Fractal cosmology, CS&F, 41, 2009, 2102. Now we should return to our main subject.
The vital so called bijection formula of E-infinity theory basically says the following. If you want to know the Hausdorff dimension of a Cantorian set in n dimensions then all that you need is the inverse of the zero set raised to n minus 1. As we said earlier and we will prove it again, the Hausdorff dimension of the zero set is the golden mean. So if we want to know what the Hausdorff dimension is in two dimensions, we take the inverse of the golden mean and raise it to two minus one. That means we would have 1 divided by the golden mean which because of the nice property of Feigenbaum's golden mean renormalization is exactly equal 1 plus the golden mean. For three dimensions you can easily work out the result to be 2 plus the golden mean. For four dimensions you obtain the famous formula which is 1 over the golden mean to the power of 4 minus 1. This is 1 over the golden mean to the power of 3 which is our famous number 4 plus the golden mean to the power of 3 equals 4.23606799.... What is interesting now is to look into the zero case and the empty set case. Let us see what the dimension for n equals 1 is. This would be the inverse of the golden mean to the power of 1 minus 1. This would be equal to the inverse of the golden mean to the power of 0. This means it is unity. In E-infinity this is called the normality condition. The one dimension is a special case. Here the Menger-Urysohn extension of the topological dimension and the Hausdorff dimension coincide and are equal to unity. Let us go one step further and ask what the Hausdorff dimension is in dimension 0. I mean now the Menger-Urysohn dimension 0. Then we have 1 divided by the golden mean to the power of 0 minus 1 equals minus 1. That means it is the golden mean to the power of 1 which is equal to the golden mean. Thus we have proven the assumption. Now we go to the second most vital step and ask what the Hausdorff dimension is when the Menger-Urysohn topological dimension is equal to minus 1. Remember this is the empty set as defined classically. The result is we have 1 over the golden mean to the power of minus 1 minus 1 which means to the power of minus 2. This means we have the golden mean proper to the power of plus 2. Now we have resolved indirectly the famous two-slit experiment. The most conservative explanation is due to the work of Mohamed El Naschie together with his late teacher, fatherly friend and mentor, Prof. Dr. Dr. habil Werner Martienssen who sadly died a few weeks ago. Martienssen and El Naschie decided to give the wave the Menger-Urysohn dimension of the empty set. This is a Hausdorff dimension equal to the golden mean to the power of two. The particles on the other hand get a Menger-Urysohn zero and a Hausdorff dimension equal to the golden mean. All these are of course interpreted as probabilities but I am running ahead of my theory. I will repeat all of that later on. El Naschie asked a trivial question but in doing so, he solved a major problem. He said why stop at the empty set? Why not ask if there are emptier sets? Being a naive engineer he said why should I integrate from minus 1 to plus infinity. From many engineering problems of struts on an elastic foundation extending relatively from minus infinity to plus infinity, some time referred to as full or half space in theory of elasticity, he is used to integrate from minus infinity to plus infinity. So he found emptier and emptier sets with Hausdorff dimensions equal to golden mean to the power of 3, then golden mean to the power of 4, then to the power of 5 and so on. At minus infinity you will have the golden mean to the power of infinity. Since the golden mean is smaller than one, to be raised to infinity you have an absolute 0. That way El Naschie discovered as it would turn out later, a unidirectional system giving a hint at the unidirectionality of time starting at a singularity. When he reached this result he wrote a paper in some Pergamon mathematical journal and dedicated it to his teacher and mentor, Ilya Prigogine who was rather excited about it and said so in correspondence. Related results were also communicated to K.F. von Weizsäcker who was very excited about it and wrote as much. Weizsäcker advised El Naschie to take the work of David Finkelstein seriously and told him that his own work was quite near to that of Finkelstein. My recounting of all these anecdotes may not be very accurate because I know them all second hand from someone who was told by someone that El Naschie told him. However the big paper dedicated to Ilya Prigogine is there for everyone to read and so is the complimentary letter of Weizsäcker to El Naschie. I am saying all these relatively unimportant facts because there is a whole industry now on the internet whose main reason d'être is to shed doubt and spread doubt and rumors and lies about E-infinity and its founders. El Naschie's theory was not invented in 2008 when all this noise started with an article published in Scientific American. El Naschie has been working on his theory since the late 80's. Only Garnet Ord and maybe Laurent Nottale were first chronologically speaking.
I think we have covered substantial ground on Menger-Urysohn theory and the extension of the classical empty set to the absolute empty set by El Naschie. The relevant papers which are cited in the pure mathematical literature quite a bit will be given to you in the next communication. It is important to understand now that we have subdued infinity. In Cantorian spacetime and fractal spacetime there is no ultraviolet or infrared catastrophe. Everything is regularized automatically because we are working in a naturally renormalized geometry. Everything has at least two major dimensions to describe it. Therefore the old fashioned uniqueness of dimension theorem does not apply and do not restrict us anymore. That is why Cantor's theory is a paradise from which we should not be evicted as David Hilbert asserted. E-infinity is based on Cantor's paradise. There is no room here for cheap jokes except from those uncorrectable philistines whose jokes just pollute every site in the blogosphere. I am sorry for using these harsh words but E-infinity or not, we are luckily all human. Until next time, all the best.
