axiom schema of replacement

The axiom schema of replacement is a fundamental principle in Zermelo-Fraenkel set theory (ZFC). It addresses the formation of new sets by stating that if you have a set and a function that maps each element of that set to another set, then the collection of all those mapped-to sets (the range of the function) is also a set.12

This schema is crucial for developing the theory of ordinal numbers, which describe positions in ordered sequences. It allows for the construction of increasingly larger ordinal numbers beyond the natural numbers, extending the concept of counting infinitely. The axiom schema of replacement is powerful enough that the axiom schema of separation can be derived from it.2