Euratom Association 85748 Garching bei München

A mathematical framework is proposed for the " big bang ". It starts with some Cantor set and assumes the existence of a transformation from that set to the continuum set used in conventional physical theories like general relativity and quantum mechanics. Let us assume that the big bang started from a Cantor set (see e.g. [1]) with the power of continuum and an arbitrary cardinality higher than the one of countable sets (see e.g. [2]). Since a Cantor set has zero measure, it can be considered as physically empty. However, if the inverse construction out of a Cantor set to some sets of the reals exists, for example by appropriate " rearrangement " of its elements, then a nonzero measure can be produced and, possibly, the continuum of the physical world. This is not unlikely because the simplest Cantor set [1] has the same power as the interval [0,1], though, originally, it has been constructed by annihilating the measure of that interval. Moreover it is known that a continuous surjection from the Cantor set to the interval [0,1] exists [1]. However, in general , such transformations can imply very sophisticated tools for an infinite set (see e.g. [3]). A nonvanishing measure is essential to introduce spacetime and Hilbert space in order to formulate general relativity and quantum mechanics. Also 1 the theory of " superstrings " could be the result of such a procedure, but alternative theories could be described in the same frame as well. The physical questions arising within the first millisecond [4] of the big bang are then related to the nature of the nontrivial " rearrangement " mentioned above and to the choice of cardinality of the Cantor set. This would be a framework permitting somehow to describe " everything out of almost nothing " to compare with ideas suggested in Ref. [5]. On the other hand, the mere existence of " rearrangements " does not explain , for example, the inflation in size of the early universe needed in order to match the backward calculations (see e.g. [4]) from the present universe to its size at about a millisecond after the big bang. Such an inflation corresponds in our scheme to an " explosion " of measure, which is similar to a " phase change " or an instability triggered by a specific parameter like " temperature " in statistical mechanics or thermodynamics. A …