Pic. 32. Giuseppe Peano

However, now when it becomes clearer and without any special difficulties, all science can receive a new and very powerful impetus for its development. And then namely on such a solid basis, science acquires the ability to overcome with an incredible ease such complex obstacles, which in the old days, when there was no understanding the essence of numbers, they seemed to science as completely impregnable fortresses36.

This path was first paved at the end of the 19th century by Peano axioms.37 We will make changes to them based on our understanding the essence of the number.

Axiom 1. A number is natural if it is added of units.38

Axiom 2. The unit is the initial natural number.

Axiom 3. All natural numbers form an infinite row, in which each

following number is formed by adding unit to the previous number.

Axiom 4. The unit does not follow any natural number.

Axiom 5. If some proposition is proven for unit (the beginning of

induction) and if from the assumption that it is true for a natural

number N, it follows that it is also true for a natural number

following N (induction hypothesis), then this sentence will be true

for all natural numbers.

Axiom 6. In addition to natural numbers, there can exist another

numbers derived from them, but only in the case if they possess all

without exception the basic properties of natural numbers.

The first axiom is a direct consequence from definition the essence of number, so Peano simply could not have it. Now this first axiom conveys the meaning of defining the notion of number to all another axiom. The second, fourth, and fifth axioms are preserved as in Peano version almost unchanged, but the fourth axiom of Peano is completely removed from this new system as redundant.

The second axiom has the same meaning as the first one in the Peano list, but is being specified in order to become a consequence of the new first axiom.

The third axiom is the new wording of Peano's second axiom. The notion of the natural row is given here more simply than by Peano where you need to guess about it through the notion of the “next” number. The fourth axiom is exactly the same as the third axiom of Peano.

The fifth axiom is the same as by Peano, which is considered the main result of the entire system. In fact, this axiom is the formulation the method of induction, which is very valuable for science and in this case allows to justify and build a count system. However, a count is present in one or another form not only in natural numbers, but also in any other numbers, therefore one more final axiom is needed.

The sixth axiom extends the basic properties of natural numbers to any numbers derived from them because if it turns out that any quantities obtained by calculations from natural numbers, contradict their basic properties, then these quantities cannot belong to the category of numbers.

Now arithmetic gets all the prerequisites in order to have the status the most fundamental of all scientific disciplines. From the point of view the essence of a count everything becomes much simpler and more understandable than until now. On the basis of this updated system of axioms there is no need to “create” natural numbers one after another and then “prove” the action of addition and multiplication for the initial numbers. Now it’s enough just to give names to these initial numbers within the framework of the generally accepted number system.

If this system is decimal, then the symbols from 0 to 9 should receive the status of the initial numbers composed of units in particular: the number “one” is denoted as 1=1, the number “two” is denoted as 2=1+1, the number “ three ” as 3=1+1+1 etc. up to the number nine. Numbers after 9 and up to 99 adding up from tens and ones for example, 23=(10+10)+(1+1+1) and get the corresponding names: ten, eleven, twelve … ninety-nine. Numbers after 99 are made up of hundreds, tens and units, etc. Thus, the names of only the initial numbers must be preliminarily counted from units. All other numbers are named so that their quantity can be counted using only the initial numbers.