Peano Axioms

The Peano axioms or assumptions Peano are a set of axioms for natural numbers entered by Giuseppe Peano in the nineteenth century .

Summary

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• 1 Description
• 2 Peano’s five axioms
• 3 Natural number operations
• 4 Bibliography
• 5 Works consulted
• 6 External links

Description

The axioms have been used practically unchanged for a variety of meta-mathematical investigations, including questions about consistency and completeness in Number Theory .

Peano’s axioms do not deal with the meaning of “natural number”, but they suppose it and try to find a simple system of axioms that characterize natural numbers and allow us to deduce from them all the properties of natural numbers, using the rules of logic.

The five axioms of Peano

1. 1 is a natural number.
2. If n is a natural number, then the successor of n is also a natural number.
3. 1 is not the successor of any natural number.
4. If there are two natural numbers n and m with the same successor, then n and m are the same natural number.
5. If 1 belongs to a set, and given any natural number, the successor of that number also belongs to that set, then all natural numbers belong to that set. This is the axiom of induction, and it captures the idea of mathematical induction.

There is a debate about whether to consider 0 as a natural number or not. It is generally decided in each case, depending on whether it is needed or not. When resolving to include 0, then some minor adjustments need to be made:

1. 0 is a natural number.
2. If n is a natural number, then the successor of n is also a natural number.
3. 0 is not the successor of any natural number.
4. If there are two natural numbers n and m with the same successor, then n and m are the same natural number.
5. If 0 belongs to a set, and given any natural number, the successor of that number also belongs to that set, then all natural numbers belong to that set. This is the axiom of induction, and it captures the idea of ​​mathematical induction.

Natural number operations

First, we define the addition of natural numbers by the formulas

1. m + 1 = S (m) where m is a natural number and S denotes the successor function of N in N- {0}.
2. p + S (m) = S (p + m).

Next, we define the multiplication of natural numbers by the following conditions

1. mx 1 = m .
2. px S (m) = p x m + p.