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Peano Axioms | Number System | Discrete Mathematics

The Peano axioms can also be referred to as Peono's postulates. An Italian mathematician was introduced the five types of axioms in the number system in 1889. These axioms are used to provide the rigorous foundation of natural numbers, i.e., 0, 1, 2, 3, and so on, which can be used in the set theory, number theory, and arithmetic. A finite set of rules and symbols can be generated by Peano axioms, which enable the infinite set.

There are five Peano axioms, which are described as follows:

  • Zero is a natural number.
  • In the natural number, there is a successor for every natural number.
  • The successor of any natural number cannot be zero.
  • The two original numbers will be the same if two natural numbers contain the same successor.
  • The set will be contained the natural numbers if there is 0 in a set, and the successor of every number is also in the set.

To understand this concept, we should also have to understand equality, which is described as follows:

In the case of mathematical axiomatically, we should not assume anything at all. We should also not assume something as rudimentary as how equality behaves. In the number system, "=" is used just as a symbol as long as we use it to declare some important properties. In this section, we will specify natural numbers N axiomatically. Before going deep, we will establish the properties that "=" should have. First property is that every natural number must be equal to itself.

An assumption, axioms, or postulate is a starting point or premises for further argumentation and reasoning. It is a type of statement that is assumed as true. The G.Peano develops the axioms, which are described as follows:

1. P1. 0 ∈ N ; 0 is a natural number:

In the different versions of Peano axioms, there are basically 5 axioms that replace 0 with 1. This yields a newly identical set of natural numbers, which is called "positive whole numbers". With the help of this context, we will find out whether a mathematician will be included or not number 0 in the natural number. For this, the standard practice will be followed, which includes 0 as a natural number.

At this point, there is only one natural number that is guaranteed to exist that is 0. The next axioms will use the successor so that it can construct other natural numbers. As the name implies, the successor function is a type of function S that contains domain name N. So, according to the next axioms, the co-domain S is also N.

2. ∀x ∈ N ⇒ x=x ;

It is a type of reflexive equality. The closure of equality axiom is the fourth axiom, which states that if natural number and "anything" are equal to each other, then that "anything" will also be the natural number.

3. ∀ x, y ∈ N ; and if x=y ⇒ y=x ;

It is a type of Symmetric equality. If one natural number and "another number" are equal to each other, in this case, the second number will also be equal to the first one. It can be called the axioms of symmetry.

4. ∀ x, y, z ∈ N ; and if x=y & y = z ⇒ x=z;

It is a type of Transitive equality. According to this property, suppose we have three natural numbers and the first number and second number are equal to each other, and that second number and third number are equal to each other. In this case, the first number will also be equal to the third number. It can be called the transitivity axioms.

5. ∀ a, b ; if a ∈ N and a = b ⇒ b

It is a type of natural number.

6. P2. If x ∈ N ⇒ S(x) ∈ N.

In the initial of Peano, 1 is the first natural number, which was employed in place of 0. In the Peano axioms, the most recent formulations start with the help of number 0 because in arithmetic, 0 is treated as the additive identity.

If x is a type of natural number, the successor of x will also be a natural number.

In the axioms, the S(x) is used to indicate the successor of x.

Intuitively, we can interpret S(x) in the form of x+1.

As we know above about natural numbers, but it is not enough. We are still very far from having the natural numbers.

The unary representation of natural numbers is described by the axioms 1 and 6 like this:

The next two axioms will be used to define the attribute of this representation.

7. If n ∈ N ; S(n) ≠ 0 .

If there is a case when n ∈ N, then the successor of n can't be 0.

8. ∀ a, b ∈ N; if S(a) = S(b) ⇒ a = b.

Here S is used as an injection, which means there is a unique successor of every number. The injection is also known as one-to-one mapping.

There are some significant implications for the preceding axioms. Axioms 1 ruled out that N can be defined as simply 1 and 0. To prove this, we will assume that S(0) = 1 already exists, and because of the injection mapping, S(1) = 1 will not be possible.

With the help of axioms 6, it ruled out the possibility of S(1) = 0. As a result, S(1) will be another natural number, and we will call this number as 2.

Thus, 2 = S(1).

Because of the same reasoning, S(2) can't be 0, 1, or 2.

As a result, S(2) will be other natural numbers, and we will call these numbers as 3. If we follow the same pattern, we can conclude that N must contain all the natural numbers with which we are aware. At this point, we know that the number 0 must be contained by N, and we will also know its successor 1 = S(0), successor 2 = S(1), successor 3 = S(2), and many more. Thus, it has proved that there is a unique successor for every number.

So if there is any case where two successors are the same, it will show that both the successors indicate the same number because there is always a unique successor for every number.

9. If V is an inductive set; i.e., 0 ∈ V and every natural number n ∈ V, then S(n) ∈ V ⇒ N ⊂ V

According to the above 8 axioms, there is a guarantee that {0, 1, 2, 3, ....} ∈ N. There is also an induction set which contains {0, 1, 2, 3,....}. The axiom 9 will generate the result which says N ⊂ {0, 1, 2, 3,...} must be true. In the result, we will get the set equality which we want, i.e., N = {0, 1, 2, 3,....}.

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