U_Mathematics

The whole of the current mathematics can be treated in only one theory: the set theory. Currently, the theory of reference is the axiomatic set theory of Zermelo-Fraenkel. This theory was developed at the beginning of the 20th century to solve the paradoxes discovered in the set theory of Georg Cantor (the father of the set theory). Among the concepts that were problematic, there were: the Set of all sets, the Set of all ordinals, etc., and more generally what is called the "great sets".

But in reality all these problems come from the fact that the set theory does not take place in the most natural framework where it should take place : the Set of all things. In other words, the TOTAL Universe !

The U_Mathematics is the Universal Set Theory,
that is to say the set theory which takes place in the greatest set: the TOTAL Universe.

When we work in the TOTAL Universe (the Set of all things), all the branches of mathematics (and also of the the other sciences) are unified into one theory: the Universal Set Theory (Universal Theory of sets).

But to be able to make the Mathematics of the TOTAL Universe without encountering the paradoxes mentioned, we should not any more reason in the classical logic. This one is too narrow to manage great sets like the TOTAL Universe. That Universe has fractal structure, so it (he) needs a fractal (or cyclical) logic. That alos means we must change our conception of the equality. The current conception of the equality is the identity, and we must now work with the notion of equivalence, which is more general and more powerful.

Illustration of fractal logic with the Fractal (or Triangle) of Sierpinski

In the classical logic, to claim that "A IS B" and in the same time "A IS NOT B", is what is called a contradiction or a paradox, as the paradoxes found in the set theory of Cantor. But the problem comes from the fact that we were up to now reasoning in the paradigm of identity, instead of the paradigm of equivalence

In the example of the fractal of Sierpinski above, A and B are not the same thing in the meaning of the identity, since these are two different objects. And yet, A and B are the same thing in the meaning of equivalence, the two objects are equivalent. Indeed, both are the same triangle of Sierpinski at two different scales. All the properties of A are also those of B and vice versa. The only difference between them is that A is a sub-fractal of B. So, in the paradigm of the equivalence, two different things can yet be the same thing!

The notion of equivalence is very closely associated to the fractal structure, but also to the very important notion of cycle.

The equalities of the kind "0 = 1", "0 = 2", "0 = 3", "0 = 4" or again "0 = 2π", which are typical equalities of the cycle or of the circle, are wrong in the meaning of the identity (the classical paradigm). But these equalities are eminently true when we reason in cyclic logic. They all are equivalences, and each equivalence corresponds to a precise cycle. For example, "0 = 4" is what we call the "Cycle 4" (for more details, see Cyclic Numbers, Fractal Algebra, Universal Algebra).