Disposal axiom

The axiom of disposal comes from the Zermelo set theory of 1907 and is therefore also part of the expanded Zermelo-Fraenkel set theory ZF , which is relevant today . It informally states that all subclasses of sets are also sets. In the language of predicate logic, the axiom of separation is specified as a schema of axioms that includes an infinite number of axioms; hence it is often referred to as the disposal scheme today.

Clarification

Axiom of exclusion for each predicate in which the variable does not appear: ${\displaystyle P(x)}$ ${\displaystyle M}$

${\displaystyle \forall A\colon \exists M\colon \forall x\colon (x\in M\iff x\in A\land P(x))}$

Verbalization and notation:

For every set there is a set that contains exactly the elements from for which applies. Due to the axiom of extensionality , this set is uniquely determined and is noted as. ${\displaystyle A}$ ${\displaystyle x}$ ${\displaystyle A}$ ${\displaystyle P(x)}$ ${\displaystyle \{x\in A\mid P(x)\}}$

meaning

Zermelo introduced the axiom of separation into set theory because the axiom common in naive set theory at the end of the 19th century that every class is a set generates Russell's antinomy . Russell, however, adopted this naive comprehension axiom in his logic and was therefore forced to severely restrict the syntax of the admissible predicates in his type theory to avoid contradictions. In contrast to Russell, Zermelo did not restrict the syntax, but showed with his axiom of exclusion, which strongly weakened comprehension, that the class contradicting Russell's antinomy is no longer a set. In this way he achieved a much simpler and more powerful set theory. ${\displaystyle \{x\mid P(x)\}}$

However, Abraham Fraenkel showed in 1921 that the Zermelo set theory with separation axiom was too weak to derive the set theory of Georg Cantor , and for this reason added a stronger replacement axiom that filled the gap. Zermelo integrated this axiom into his ZF system in 1930 and noted that the axiom of disposal can be derived from it, so that it can be dispensed with in the ZF system. Obviously , the set of rejections is obtained as by substitution axiom. ${\displaystyle \{x\in A\mid P(x)\}}$ ${\displaystyle \{y\mid \exists x\in A\colon P(x)\land x=y\}}$

Individual evidence

1. ^ ^{A b} Ernst Zermelo: Investigations on the basics of set theory , 1907, in: Mathematische Annalen 65 (1908), 261-281 , there Axiom III p. 263f.
2. ^ Bertrand Russell: Mathematical logic as based on the theory of types , in: American Journal of Mathematics 30 (1908), p. 250.
3. ↑ Abraham Fraenkel: On the foundations of the Cantor-Zermeloschen set theory , 1921, in: Mathematische Annalen 86 (1922), 230-237.
4. ↑ Ernst Zermelo: Limits and Quantities , Fundamenta Mathematicae 16 (1930), p. 31 Comment on redundancy.