**2. Transfinite sets and quantum mechanics**

Let us examine the basic concept of a line or more generally a curve. Classical geometry used in classical mechanics and general relativity the fact that a line is a one-dimensional object, while a point is zero-dimensional. Furthermore, it would seem at first sight that a line consists of infinite number of points and that it is simply the path drawn by a zero-

Quantum Gravity in Cantorian Space-Time 89

Fig. 1. A vision of T. Right's cosmos as a form of sphere packing, on all scales [1].

Fig. 2. A vision of a fractal-like universe, with clusters of clusters ad infinitum as envisaged by the Swedish astronomer C. Charlier who lived between 1862 and 1934. This work was clearly influenced by the work of the Swedish astrophysicist A. Swedenborg (1688–1772) [1].

dimensional point moving in the two or three-dimensional space. Classical geometry similar to classical mechanics has made various tacit simplifications and ignored several subtle topological facts [2].

If a line is one-dimensional and if it is made of infinite number of points then the sum of infinitely many zeros should be equal to one. That is of course not true. On the other hand we know that there is a curve called the Peano-Hilbert curve which is area filling and twodimensional [1, 7-20]. By contrast, we can construct a three dimensional cube known as the Menger sponge which has a fractal dimension more than two and less than three, namely log <sup>20</sup> *D* as explained for instance in the classical book of Mandelbrot [2].

log 3

The existence of all these non-conventional forms described in modern parlance, following Mandelbrot, as fractals, may be traced back to the archetypal transfinite set known as Cantor triadic set [6].

A Cantor set is a set of disjoint points which possesses the same cardinality as the continuum. It may be this coincidence that makes it an ideal compromise between the discrete and the continuum. It is transfinite discrete. Our Cantorian space-time which we will use to "topologize" physics is based on these transfinite sets. The main idea behind the Cantorian space-time approach is to replace the formal analysis of quantum mechanics and the Riemannian space-time geometry of general relativity by a transfinite fractal Cantorian space-time manifold [1, 8, 11, 13].
