## § 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
`ordinal`

s 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. - Since
`A`

possesses non-trivial structure preserving automorphisms, so too must `ordinal`

, since `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`

.