Folklore

There exist several traditional methods of construction of the natural entireties but that of the entireties of von Neumann is often looked like simplest.

Method of von Neuman

On the basis of the Set theory, one identifies 0 with the Empty set, then one builds the successor of a Entier naturalness like the whole of the natural entireties which precede it. More precisely, the natural entireties are built starting from the following rules:

1. the empty set $\backslash \; emptyset$ is a natural entirety noted 0.

2. If NR is a natural entirety, the unit NR U {NR} is also a natural entirety, called the immediate successor of NR.
3. Entire naturalness is built starting from rules 1 and 2.

For example, the immediate successor of 0 is: 0 U {0} = {0} = 1

That of 1:1 U {1} = {0} U {1} = {0,1} = 2

That of 2:2 U {2} = {0,1} U {2} = {0,1,2} = 3

The axiom of infinite is necessary to ensure the existence of a unit containing all the natural entireties. The intersection of all the whole of this type (containing 0 and closed for the operation successor) is then the whole of the natural entireties. One can check that this last satisfies the Axiomes of Peano.

See too

• Addition of the natural entireties.
• Whole naturalness
• ordinal Number
• Cardinal number

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