All arithmetic actions are components of the definition the essence of the number. In a compact form they are presented as follows:

1. Addition: n=(1+1…)+(1+1+1…)=(1+1+1+1+1…)

2. Multiplication: a+a+a+…+a=a×b=c

3. Exponentiation: a×a×a×…×a=a^>b=c

4. Subtraction: a+b=c → b=c−a

5. Division: a×b=c → b=c:a

6. Logarithm: a^>b=c → b=log_>ac

Hence, necessary definitions can be formulated in the form of axioms.

Axiom 1. The action of adding several numbers (summands) is their

association into one number (sum).

Axiom 2. All arithmetic actions are either addition or derived from

addition.

Axiom 3. There are direct and inverse arithmetic actions.

Axiom 4. Direct actions are varieties of addition. Besides the addition

itself, to them also relate multiplication and exponentiation.

Axiom 5. Inverse actions are the calculation of function arguments.

These include subtraction, division and logarithm.

Axiom 6. There aren’t any other actions with numbers except for

combinations of six arithmetic actions.40

The consequence to the axioms of actions are the following basic properties of numbers due to the need for practical calculations:

1. Filling: a+1>a

2. The neutrality of the unit: a×1=a:1=a

3. Commutativity: a+b=b+a; ab=ba

4. Associativity: (a+b)+c=a+(b+c); (ab)c=a(bc)

5. Distributivity: (a+b)c=ac+bc

6. Conjugation: a=c → a±b=b±c; ab=bc; a:b=c:b; ab=cb; log_>ba=log_>bc

These properties have long been known as the basics of primary school and so far, they have been perceived as elementary and obvious. The lack of a proper understanding of the origin of these properties from the essence the notion of number has led to the destruction of science as a holistic system of knowledge, which must now be rebuilt beginning from the basics and preserving herewith everything valuable that remains from real science.

The presented above axiomatics proceeds from the definition the essence the notion of number and therefore represents a single whole. However, this is not enough to protect science from another misfortune i.e. so that in the process of development it does not drown in the ocean of its own researches or does not get entangled in the complex interweaving of a great plurality of different ideas.

In this sense, it must be very clearly understood that axioms are not statements accepted without proof. Unlike theorems, they are only statements and limitations synthesized from the experience of computing, without of which they simply cannot be dispensed. Another meaning is in the basic theorems, which are close to axioms, but provable. One of them is the Basic or Fundamental theorem of arithmetic. This is such an important theorem that its proof must be as reliable as possible, otherwise the consequences may be unpredictable.

Pic. 33. Initial Numbers Pyramids

The earliest known version of the theorem is given in the Euclid's "Elements" Book IX, Proposition 14.

If a number be the least that is measured by prime numbers, it will not be measured by any other prime number except those originally measuring it.

The explain is following: “Let the number A be the least measured by the prime numbers B, C, D. I say that A will not be measured by any other prime number except B, C, D”. The proof of this theorem looks convincing only at first glance and this visibility of solidity is strengthened by a chain of references: IX-14 → VII-30 → VII-20 → VII-4 → VII-2.