Suslin set theoretical foundation of E-infinity theory

12th December, 2010.
E-infinity communication No. 54

Suslin set theoretical foundation of E-infinity theory

Descriptive set theory which forms an important basis and deep theoretical underpinning of E-infinity theory was the joint discovery of Nikolai Luzin, the real founder of the famous Moscow School of Mathematics and a young brilliant student of his, Mikhail Suslin in the summer of 1916. In fact there is a birth certificate for this theory with both Luzin and Suslin as parents and no one less than the famous Polish mathematician Waclaw Sierpinski (the father of the Sierpinski fractal triangle or gasket) was a witness. The certificate is dated the afternoon of October 1916 but no day is given. There is a serious reason for this joke. It seems that P.S. Alexandrov, the great Russian topologist whose work on wild topology combining topology with Cantor sets (which is crucial to El Naschie’s E-infinity) wanted to claim Luzin and Suslin’s discovery for himself. Alexandrov was a great mathematician and was actually a very close friend of P. Urysohn whose dimensional theory is about the most important thing in E-infinity theory. Never the less it seems Alexandrov had a character deficiency. Dishonesty among the greatest of scientists seems to be as old as the history of science and it is still with us today. There are far worse plagiarists than this and far more famous scientific disputes engulfing as famous people as Newton versus Leipnitz, Einstein versus Poincaré as well as Lorenz and even D. Hilbert and so on and so on. This is extremely sad but it is fact. Great minds do not always necessarily imply great characters. Scientists are only human and sometimes very human.
A good place to start studying descriptive set theory is to start with trees and trees on products. After that we have to concentrate on polish spaces. The most important examples of polish spaces for E-infinity are the n-dimensional cube studied by M. El Naschie and the Hilbert cube studied by Ji-Huan He. After that comes the Cantor space used in several papers by El Naschie on E-infinity and descriptive set theory, all published in CS&F. Finally we should mention the Bair space. In E-infinity ordinary differentiation and integration are replaced by Weyl scaling while Suslin scaling is used as a fundamental operation on sets. These things sound more difficult than they are and in praxis the situation is much simpler when we are dealing with a concrete problem. Finally we should mention that E-infinity may be regarded as a space made of a Borel set. El Naschie repeatedly mentioned that this idea was given without any mathematics at all by the very great theoretical physicist J.A. Wheeler. El Naschie did not do more than adding the mathematics using random Cantor sets with golden mean Hausdorff dimension.
E-infinity group.