Axiom of the infinite one

In axiomatic Theory of the units and in the branches of the Logical , the Mathematical , and data processing, the axiom of infinite the is one of the axioms of the set theory of Zermelo-Fraenkel. It states that there exists an infinite unit.

In the formal Language of axiomatic of Zermelo-Fraenkel, the axiom is written:

$\ exists \ Omega (\ empty \ in \ Omega \ wedge \ forall X (X \ in \ Omega \ Rightarrow X \ cup \ {X \} \ in \ Omega))$

or in other words: there exists a unit ω ; such that the Empty set $\ empty$ belongs to ω and such as all the times where X is an element of ω , the unit formed by taking the union of X with its singleton { X } is also an element of ω .

To include/understand this axiom, let us invite first of all $x \ cup \ {X \}$ the successor of X . Let us note that the Axiome of the pair and the Axiome of extensionnality enable us to build the singleton { X }, and the Axiome of the meeting is used to us to form the union. The successors are used to define and code the entireties in the theory of the natural whole numbers. In the coding of the entireties, zero are the empty set $(0= \ empty)$ , and 1 is the successor of 0:

$1=0 \ cup \ {0 \} = \ empty \ cup \ {\ empty \} = \ {\ empty \} = \ {0 \}$

In the same way, 2 is the successor of 1:

$2=1 \ cup \ {1 \} = \ {0 \} \ cup \ {1 \} = \ {\ empty, \ \ {\ empty \} \} = \ {0, \ 1 \}$

and so on. A consequence of this definition is that each integer is equal to the whole of all the integers which precede it. We could plan to form, by using this process, the whole of all the natural integers; but it proves that by using only these axioms construction is impossible. The axiom of infinite ensures the existence of this unit ω and it defines it by a method similar to that of the reasoning by recurrence, by initially supposing that ω contains zero, then while imposing that the successor of any element of ω is also in ω .

This unit can contain other elements that the natural integers (which form a subset of this first), but we can apply the Schéma of axioms of comprehension to withdraw the undesirable elements, releasing the unit ω of all the natural integers. This unit is single according to the Axiome of extensionnality. Thus the axiom affirms primarily that:

There exists a unit containing all the natural integers.

The axiom of infinite is also one of the axioms of von Neumann-Bernays-Gödel.

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