*Theorem 1:*

Let *X* be polish and *Y X* be a Borel set. Then either *Y* is countable or else it contains a Cantor set. In particular every uncountable standard Borel space has cardinality 2.

A Cantor space is homeomorphic to a triadic Cantor set and also to the random Cantor set [17]. The relation between the triadic Cantor set and the Cantor space establishes the relationship between the Cantor space and the Cantorian space-time, since the Cantorian space-time is a hierarchical infinite dimensional Cantor set with the expectation Hausdorff-Besicovitch dimension <sup>3</sup> 4 4.236067. 

In particular, it has been shown [17] that when interpreting (0) 1 *Cd* in the bijection formula as

the average (0) <sup>1</sup> <sup>2</sup> *Cd* of the fundamental Wisse-Abbot theorem and taking N = 128, 96 Quantum Gravity

where [dx] means that we are summing over all possible paths of the concerned particle. What is important here is to realize that from all of infinitely many paths which a quantum particle can take some are more probable than others. The probability of the actual path, that is to say the amplitude of an event is the sum over the amplitude corresponding to all paths. Thus we have a weight assigned to each path in the Feynman formulation of quantum mechanics.

In the Cantorian space-time theory we proceed in an analogous way. However, instead of summing over all paths, we sum over all dimensions of infinite dimensional hierarchical Cantorian space-time. El Naschie has recently demonstrated that E-Infinity is a Suslin operation and the so-called Suslin A operation [9]. In this theory Suslin scaling replace the

classical Lagrangian and the classical calculus using descriptive set theory [16, 17].

[16]. In descriptive set theory and theory of polish spaces it is shown that [16, 17]:

When a space is polish and when *A = 2 = [0,1],* then we call *C= 2N* the Cantor space.

When a space is polish and when *A = N* then we call *B* = *NN* the Baire space.

*Definition 1:* 

*Definition 2:* 

*Definition 3:* 

*Theorem 1:* 

the average (0)

Two cases are of considerable importance.

homeomorphic to a Cantor space [16, 17].

Besicovitch dimension <sup>3</sup> 4 4.236067.

<sup>1</sup> <sup>2</sup>

In particular, it has been shown [17] that when interpreting (0)

**7. Cantorian space-time and Newton's non-dimensional gravity constant** 

Quantum non-dimensional gravity constants can be derived from descriptive set theory

When a space *AN* is viewed as the product of infinitely many copies of *A* with discrete topology and is completely metrizable and if *A* is countable then the space is said to be polish.

Now we can proceed to explain the relationship between the Cantor space and Cantorian space-time. The relationship comes from the solution of the cardinality problem of a Borel set in polish spaces. Thus, we call a subset of a topological space a Cantor set if it is

Let *X* be polish and *Y X* be a Borel set. Then either *Y* is countable or else it contains a

A Cantor space is homeomorphic to a triadic Cantor set and also to the random Cantor set [17]. The relation between the triadic Cantor set and the Cantor space establishes the relationship between the Cantor space and the Cantorian space-time, since the Cantorian space-time is a hierarchical infinite dimensional Cantor set with the expectation Hausdorff-

1

*Cd* of the fundamental Wisse-Abbot theorem and taking N = 128,

*Cd* in the bijection formula as

Cantor set. In particular every uncountable standard Borel space has cardinality 2.

( *ew* 128 is the inverse coupling constant measured at the electroweak scale) then the bijection formula

$$d\_{\mathbb{C}}^{(n)} = \left(\frac{1}{d\_{\mathbb{C}}^{(0)}}\right)^{n-1}$$

gives for *N* = *n* =*ew* 128 the following

$$d\_{\mathbb{C}}^{\text{(128)}} = \left( \left\langle \frac{1}{d\_{\mathbb{C}}^{(0)}} \right\rangle \right)^{\text{128-1}} = 2^{\text{127}} \equiv (1.70141)(10)^{38}$$

where (128) <sup>2</sup>*<sup>N</sup> C dC* . The value <sup>38</sup> (1.70141)(10) is the non-dimensional gravity constant *<sup>G</sup>* which is defined as

$$\overline{\alpha\_{\rm G}} = \frac{\hbar c}{G m\_p^2} \equiv (1.7)(10)^{38}$$

It is of interest to mention that a similar result was found empirically by F. Parker Rhodes which was the subject of extensive discussions by Noyes [18]

$$\overline{\alpha\_{\rm G}} = 2^{127} + 137 = (1.7)(10)^{38}$$

Here *ħ* is the Planck quantum, *c* the speed of light, *G* the Newton's gravity constant, *mp* the Planck mass.
