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3.2.1. Axioms of a Count

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.39

3.2.2. Axioms of Actions

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

3.2.3. Basic Properties of Numbers

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

37

The content of Peano’s axioms is as follows: (A1) 1 is a natural number; (A2) For any natural number n there is a natural number denoted by n' and called the number following n; (A3) If m'=n' for any positive integers m, n then m = n; (A4) The number 1 does not follow any natural number i.e. n' is never equal to 1; (A5) If the number 1 has some property P and for any number n with the property P the next number n' also has the property P then any natural number has the property P.

38

In the Euclid's "Elements" there is something similar to this axiom: "1. An unit is that by virtue of each of the things that exist is called one. 2. A number is a multitude composed of units” (Book VII, Definitions).

39

So, count is the nominate starting numbers in a finished (counted) form so that on their basis it becomes possible using a similar method to name any other numbers. All this of course, is not at all difficult, but why is it not taught at school and simply forced to memorize everything without explanation? The answer is very simple – because science simply does not know what a number is, but in any way cannot acknowledge this.

40

The axioms of actions were not separately singled out and are a direct consequence of determining the essence a notion of number. They contribute both to learning and establish a certain responsibility for the validity of any scientific research in the field of numbers. In this sense, the last 6th axiom looks even too categorical. But without this kind of restriction any gibberish can be dragged into the knowledge system and then called it a “breakthrough in science”.