Axiom schema of replacement

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The substitution axiom is an axiom that Abraham Fraenkel in 1921 suggested as a supplement to Zermelo set theory of 1907 and later became an integral part of the Zermelo -Fraenkel set theory ZF. It states informally that the images of sets are also quantities. In the predicate logic language, the substitution axiom is specified as an axiom schema that includes infinitely many axioms. Therefore, it is now often referred to as replacement pattern.

Clarification

Substitution axiom for each binary predicate:

The premise of the axiom states that the binary predicate is quite clear ( functional), that is, to each there is at most one with. The amount formed in the replacement axiom is uniquely determined due to the Extensionalitätsaxioms and as quoted.

Importance

The substitution axiom fills a gap of Zermelo set theory. Fraenkel discovered in 1921 namely, that certain countable sets of set theory by Georg Cantor one can not construct with Zermelo's axioms, and therefore supplemented his axiom, with the amount shown each and every particular enumeration of the set of natural numbers is a lot. The same gap named a year later Thoralf Skolem and gave a more precise formalization of this. This axiom integrated Zermelo 1930 in his IF system and remarked that the axiom of separation, and the pair axiom can be derived from the substitution axiom, so that these older axioms of Zermelo set theory in the IF system are dispensable.