We presented a list of statements, definable relations, functions, and constants, that are absolute for transitive models of enough set theory. We showed that absolute functions are closed under composition, although functions are not. We also verified that being a well-ordering is absolute. The same argument actually shows:
Theorem. The statement “ is a well-founded relation on ” is absolute for transitive models of .
This is a key result very useful in a variety of situations. Notice that we are not claiming that being well-orderable is absolute; in fact, it is not. The difference is that in the first case we are given a witness to the well-orderability, and claim that no matter in which transitive model the witness is observed, in all of them it has the property of being a well-ordering. The second case only states that there is a witness, and a given model may very well fail to produce such a witness unless it is a model of the axiom of choice.