Axiom schema of replacement

In mathematics, the Axiom schema of replacement is an axiom scheme in set theory that allows for the creation of new sets by replacing items in an existing set according to a specific rule, called a function. If you have a set and you apply a well-defined rule to each item in that set, the new items you create will also form a set. The rules must always be clear and give a unique result for each input.

For example, imagine you have a set ${\displaystyle S=\{1,2,3\}}$ and you apply the rule "double each number". The output will be a new set, ${\displaystyle T=\{2,4,6\}}$. The Axiom schema of replacement tells us that this new collection of items is also a valid set. We might write this as ${\displaystyle T=2S}$, which is the function of replacement.

The Axiom schema of replacement is important in set theory because it allows us to build new sets, including very large and infinite sets, while keeping everything logically and mathematically consistent.

The Axiom schema or replacement doesn't require the Axiom of choice, but together, they form the foundation for Zermelo-Fraenkel set theory (ZF or ZFC), which is one of the most widely used foundations for modern mathematics.