THE THEOREM OF PARALLEL SETS (WORLDS)
Suppose there are two infinite sets A and B, with a common unit of discreteness (e.g. .consisting of integers). A successor function is initially defined on both sets, which is the basic function. Suppose that there exists a reflection A to B along an arbitrary, but always on the entire set, rule. Mapping does not change the sequence. Such sets are said to be parallel.
Let an elementary rule 1 be set on set A, which establishes a relationship between some elements of set A, and there is an infinite number of examples of application of this rule. The rule is consistent on the whole set A, and is due to the laws and properties of the set, i.e., the examples are not random.
Reflections of elements A (bound by Rule 1), on B will be the elements of set B, and will be bound with each other by an elementary rule 2, which can differ from Rule 1. Rule 2 will be self-sufficient for set B, i.e. it is formulated through a previously defined functions and rules on set B. And one of the examples of rule 2 will consist of consecutive members of the set.
If B is a mapping of A, then A is a mapping of B too. Examples of rule 1 on set A are not accidental. Hence, they may not be reflections of random examples 2 on set B. Therefore, two examples are not random and are bound by a certain rule 2.
If A is a reflection of B, then rule 2 should not be explained by rules and functions of A. Therefore, Rule 2 will be self-sufficient for set B, that is, it can be formulated only in terms of functions and rules of B.
The base sequence function is initially defined on set B. Therefore, Rule 2 should be formulated through the sequence function, i.e., on the example of successive members.
Corollary 1 (Gödel's incompleteness theorem)
Some functions and rules are initially defined on set B . However, any rules that depend on set A and the reflection function A on B can be formulated through them. But any such rule is to be self-sufficient for set B. Thus, the same rules and axioms can give rise to any rules on set B . Consequently, these new rules are not based only on the existing axioms. And at the same time, on the basis of these axioms one cannot prove that the new rules are not based only on them, because it would violate the condition of self-sufficiency of new rules on set B.
Corollary 2 (Fermat's Last Theorem)
Let B be an infinite set of integers x and A - an infinite set, each element of which is equal to xn. A and B are parallel sets. On set A rule 1 is valid : an + bn = cn. If there is one example of this rule, there exists an infinite number of examples. Therefore there is an infinite number of reflections on set B. For example, if n = 2 on set A: 9 + 16 = 25, and on set B: 3 + 4 = 5 there is rule 2 by the theorem of parallel sets, which binds all mappings on set B, i.e. binds all roots of the equation an + bn = cn in nonzero whole numbers. And one example is to consist of a sequence of elements.
And, therefore, on the contrary, if there are no solutions in successive elements, then there is no any solutions at all. For n> 2, one can easily show that there are no roots of the equation in the whole serial numbers. Consequently, there are no solutions in non-zero integer numbers in general.
Corollary 3 (Physics)
Let worlds of all observers be parallel sets that are mutually reflected. Each world is self-sufficient.
Reflections can be different and produce different rules. Since the same phenomenon can be the manifestation of an electrostatic force for one observer, it can be the manifestation of a magnetic force for the other.
But the basic function of parallel worlds is a successor function, and no reflection changes the sequence of events. Therefore, mapping can change everything (space, time, etc.), but the order of events (theory of relativity) will always remain the same for all observers. And any law can be illustrated by the example of the sequence of events between which there are no other events, there is nothing (quantum mechanics).
Corollary 4 (Philosophy)
Parallel sets can be combined into subsets. Our world is one of these subsets. It is a reflection of other sets and worlds, but at the same time is self-sufficient. This leads to contradictory Сorollary 1.
Philosophers found contradictory essence of all elementary propositions of our world long ago:
«Then let us say that, and we may add, as it appears, that whether the one is or is not, the one and the others in relation to themselves and to each other all in every way are and are not and appear and do not appear.» (Plato, «Parmenides»)