Inductive set

In mathematics, inductive sets are sets that contain the empty set and where each set also contains its subsequent set . The infinity axiom says that there is an inductive set. ${\displaystyle M}$ ${\displaystyle \emptyset }$ ${\displaystyle x}$ ${\displaystyle x'=x\cup \{x\}}$

definition

A set is an inductive set if and only if it has the following two properties ${\displaystyle M}$

• ${\displaystyle \emptyset \in M}$
• ${\displaystyle \forall x\in M:x'\in M}$

met, where denotes the successor of . ${\displaystyle x':=x\cup \{x\}}$ ${\displaystyle x}$

Importance in mathematics

Natural numbers

With the help of inductive sets, the set of natural numbers is defined in set theory according to an idea by Richard Dedekind :

${\displaystyle {\begin{aligned}\mathbb {N} :=\bigcap \{x\mid x\;{\text{induktiv}}\}.\end{aligned}}}$

Since the intersection of inductive sets is inductive again, the set of natural numbers is the smallest inductive set. consists of the iterated descendants of the empty set: ${\displaystyle \mathbb {N} }$

${\displaystyle \mathbb {N} =\{\emptyset ,\emptyset ',\emptyset '',\emptyset ''',\ldots \}=\{0,1,2,3,\ldots \}}$

In order to be able to define the natural numbers in this way, one needs two axioms: The axiom of infinity and the axiom of exclusion: The axiom of infinity ensures that there is at least one inductive set. However, if one forms the intersection over all inductive sets, one obtains the class of natural numbers. The axiom of disposal ensures that the cut over sets is also a set and that the class of natural numbers is really a set.

Within the Zermelo-Fraenkel set theory it can be shown that the set thus constructed fulfills the Peano axioms . thus catches the intuitive concept of the natural number exactly in set theory. Instead of and , as in arithmetic, one usually writes or . ${\displaystyle \mathbb {N} }$ ${\displaystyle \mathbb {N} }$ ${\displaystyle x'}$ ${\displaystyle \emptyset }$ ${\displaystyle x+1}$ ${\displaystyle 0}$

With the help of the definition of inductive sets, the proof method of complete induction can be justified (hence the name inductive ): If it is to be shown that all natural numbers have a certain property , then consider the set . If one now shows that it is true and that it also follows, then is inductive. Since the smallest inductive quantity is, and therefore applies . So every natural number has a property . ${\displaystyle e}$ ${\displaystyle E:=\{n\in \mathbb {N} \mid e(n)\}\subseteq \mathbb {N} }$ ${\displaystyle e(0)}$ ${\displaystyle e(n)}$ ${\displaystyle e(n+1)}$ ${\displaystyle E}$ ${\displaystyle \mathbb {N} }$ ${\displaystyle \mathbb {N} \subseteq E}$ ${\displaystyle \mathbb {N} =E}$ ${\displaystyle e}$

Transfinite ordinal numbers

Other inductive sets are the transfinite ordinals , for example . Here the natural numbers are included as a subset, but an infinite ordinal number, i.e. H. greater than any natural number. ${\displaystyle \omega +\omega =\{0,1,2,\ldots ,n,n+1,\ldots ,\omega ,\omega +1,\omega +2,\ldots \}}$ ${\displaystyle \omega }$

Individual evidence

1. ↑ Richard Dedekind : What are and what are the numbers? Vieweg, Braunschweig 1888, § 6, 71.β, reduced by definition 44, 37 and 17 in somewhat unusual terminology with an implicitly defined number of successors. Adopted into the Zermelo set theory in free verbalization with reference to Dedekind .