The Axiom Scheme of Replacement
by Thomas Forster
Abstract: The axiom scheme of replacement is - after the axiom of choice - the most controversial of the axioms of Zermelo-Fraenkel set theory, and is less well understood. It is also the most recent. Some people object to it on the grounds that it implies the existence of sets much bigger than any normal mathematician would take an interest in. This is true, but it misses the point. The axiom scheme also has things to tell us about the real line (there are facts abut sets of reals that cannot be proved without it) and it is an essential prop in any project of showing that facts about general mathematical (i.e., not sets) objects can be proved in set theory in an implementation-insensitive way.