§ Permutation models
- These are used to show create models of
ZF + not(Choice).
- Key idea: if we just have ZF without atoms, then a set has no non-trivial
∈ preserving permutations.
- Key idea: if we have atoms, then we can permute the atoms to find non-trivial automorphisms of our model.
- Key idea: in ZF + atoms, the
ordinals come from the ZF fragment, where they live in the kernel [ie the universe formed by repeated application of powerset to the emptyset ]. Thus, the "order theory" of ZF + atoms is controlled by the ZF fragment.
- Crucially, this means that the notion of "well ordered" [ie, in bijection with ordinal ] is determined by the ZF fragment.
- Now suppose (for CONTRADICTION) that
A is well ordered. This means that we Now suppose we have an injection
f: ordinal -> A where
A is our set of atoms.
A possesses non-trivial structure preserving automorphisms, so too must
ordinal is a subset of
A. But this violates the fact that
ordinal cannot posses a non-trivial automorphism.
- Thus, we have contradiction. Ths means that
A cannot be well-ordered, ie, there cannot be an injection
f: ordinal -> A.