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Andrey Shvets
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Andrey Shvets

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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.
Proof
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»)
Link:    http://andreyshvets100.blogspot.com.by/2016/03/the-theorem-of-parallel-sets-worlds.html
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Теорема о параллельных множествах (мирах)
Теорема
о параллельных множествах (мирах) Пусть существуют два бесконечных множества A и B с
общей единицей дискретности (например, состоящие из целых чисел). На обоих
множествах изначально определена функция следования, которая является базовой
функцией. П...
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The proof (?) of Fermat's theorem
Mathematics considers natural numbers as points on a number line. I think it is a bit lopsided. Natural numbers are not just a series of points on a number line with an interval of 1. Let's try to build another series of intervals 3 ^ 0.5. All points of thi...
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Andrey Shvets

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THE THEOREM OF PARALLEL SETS (WORLDS)
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 ex...
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Другой подход к решению теоремы Ферма
Математика рассматривает натуральные числа как точки
на числовой прямой. Мне кажется, что это немного однобоко.  Натуральные числа это не только ряд точек на
числовой прямой с интервалом 1. Попробуем построить другой ряд с интервалом
3^0.5 . Все точки этого...
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Andrey Shvets

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Help prove or disprove my hypothesis:
Theorem (Axiom)
There is an ordered set of natural numbers, which is given by the function f (x) (x - a natural number). The operation of addition is defined on this set only if the successor function S (x) associated with the operation of addition in the form of:
f (n) + f (n + 1) = f (k) (1)
Or
f (k) + f (n) = f (n + 1) (2)
k, n - integers

If the set has a pair of consecutive numbers that satisfy the condition (1) or (2), the set has an infinite number of triples that are solution of the equation
a + b = c (3)

If the set does not have a pair of consecutive numbers that satisfy the condition (1) or (2), the set has no members who are solution of the equation (1). On this set the operation of addition is not defined, because it is not connected with the following function.

Corollary.
It is easy to show that the sets that are defined function f (x) = x ^ n (n> 2) does not satisfy the condition (1) or (2). Therefore, the equation a ^ n + b ^ n = c ^ n (n> 2) has no solution in nonzero whole numbers.
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We're talking about the three worlds. The first world - ours. Second World sets: 1,4,9,16,25,36 ... working on this set "normal" addition: 9 + 16 = 25
Third World - many arguments: 1,2,3,4,5 But here another addition. Imagine that you - a resident of the third world. Rule (1) - the god who created an infinite number of examples of adding 3 + 4 = 5, etc. You will create a theory which will link all of these examples in the common law. This is knowledge. And it will be based on the already well-known laws, and given only a function of repetition in the third world. :)

Can you give an example of a function that corresponds to the condition (1), which has at least one solution, but that does not have parental triples?
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