The well-foundedness of the Mitchell order

Extract

Let E ⊲ F iff E and F are extenders and E ∈ Ult(V, F). Intuitively, E ⊲ F implies that E is weaker—embodies less reflection—than F. The relation ⊲ was first considered by W. Mitchell in [M74], where it arises naturally in connection with inner models and coherent sequences. Mitchell showed in [M74] that the restriction of ⊲ to normal ultrafilters is well-founded.

The relation ⊲ is now known as the Mitchell order, although it is not actually an order. It is irreflexive, and its restriction to normal ultrafilters is transitive, but under mild large cardinal hypotheses, it is not transitive on all extenders. Here is a counterexample. Let κ be (λ + 2)-strong, where λ > κ and λ is measurable. Let E be an extender with critical point κ and let U be a normal ultrafilter with critical point λ such that U ∈ Ult(V, E). Let i: V → Ult(V, U) be the canonical embedding. Then i(E) ⊲ U and U ⊲ E, but by 3.11 of [MS2], it is not the case that i(E) ⊲ E. (The referee pointed out the following elementary proof of this fact. Notice that i ↾ V_λ+2 ∈ Ult(V, E) and X ∈ E_a ⇔ X ∈ i(E)_i(a). Moreover, we may assume without loss of generality that = support(E). Thus, if i(E) ∈ Ult(V, E), then E ∈ Ult(V, E), a contradiction.)

By going to much stronger extenders, one can show the Mitchell order is not well-founded. The following example is well known. Let j: V → M be elementary, with V_λ ⊆ M for λ = j_oω(crit(j)). (By Kunen, V_λ+1 ∉ M.) Let E₀ be the (crit(j), λ) extender derived from j, and let E_n+1 = i(E_n), where i: V → Ult(V, E_n) is the canonical embedding. One can show inductively that E_n is an extender over V, and thereby, that E_n+1 ⊲ E_n for all n < ω. (There is a little work in showing that Ult(V, E_n+1) is well-founded.)