Mathematical Proof and the Principles of Mathematics/Sets/Replacement


The Axiom Schema of Replacement says that if one replaces each of the elements of a set according to some formula, then the result is also a set.

Axiom Schema (Replacement)

Let   be a property such that for each   there is a unique   such that   holds. There exists a set   consisting of all the   for which there exists some   such that   holds.

Technically the formula is allowed to have finitely many free variables, and is often written  .

As for the Axiom Schema of Comprehension, there is an axiom in the schema for every possible property  .

As for the Axiom of Foundation, most of mathematics can be accomplished without the Axiom Schema of Replacement. However, the axiom allows for the construction of certain infinite sets that are important in set theory itself.