Inductive set (axiom of infinity)

As inductive amounts amounts are in mathematics called containing the empty set and where for any amount of their succession amount is included. The axiom of infinity states that there is an inductive set.

Definition

A lot is just then an inductive set if it the two following properties

Satisfied ( with the successor of designated ).

Importance in mathematics

Natural Numbers

Using the inductive sets the set of natural numbers is defined in set theory:

Since the average amount of induction is inductive again, the set of natural numbers is the smallest amount of inductive. thus consists of the iterated successors of the empty set:

In order to define the natural numbers so you need two axioms: the axiom of infinity and the axiom: The axiom of infinity ensures that there is at least one inductive set. If now, however, is the average of all inductive quantities is obtained so that the class of the natural numbers. The axiom ensures that the cut above amounts also is a lot and that the class of natural numbers so that really is a lot.

Within the Zermelo -Fraenkel set theory can be shown that the amount thus constructed satisfy the Peano axioms. thus captures the intuitive notion of the natural number a quantitative theo -driven precision. Instead, and therefore to write as in arithmetic or mostly.

From the definition by inductive quantities can be the method of proof of complete induction to justify ( hence the name inductive): Should be shown that all natural numbers have a certain property, then consider the set. If you point now that applies and also follows so is inductive. Since the smallest inductive set is valid and thus. So has every natural number property.

Transfinite ordinals

More inductive sets are the transfinite ordinal numbers, for example. Here are the natural numbers as a subset include, but is an infinite ordinal number, that greater than any natural number.

• Set theory