116c- Lecture 15

May 21, 2008

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 \Delta_0 functions are not. We also verified that being a well-ordering is absolute. The same argument actually shows:

Theorem. The statement \text{``} R is a well-founded relation on A\text{''} is absolute for transitive models of \mathsf{ZF}-\mathsf{Power\,set}.

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 (at least a fragment of) the axiom of choice.