Infinite decreasing chains in the Mitchell order

It is known that the behavior of the Mitchell order substantially changes at the level of rank-to-rank extenders, as it ceases to be well-founded. While the possible partial order structure of the Mitchell order below rank-to-rank extenders is considered to be well understood, little is known about the structure in the ill-founded case. The purpose of the paper is to make a first step in understanding this case, by studying the extent to which the Mitchell order can be ill-founded. Our main results are (i) in the presence of a rank-to-rank extender there is a transitive Mitchell order decreasing sequence of extenders of any countable length, and (ii) there is no such sequence of length $\omega_1$.


Introduction
Definition 1.1. Let E, E be two extenders. We write E E if E is represented in the (well-founded) ultrapower of V by E .
The relation , known as the Mitchell order, was introduced by Mitchell in [Mit74] to construct canonical inner models with many measurable cardinals. The Mitchell order, which was initially introduced as an ordering on normal measures, has been extended to extenders and plays a significant role in inner model theory. As a prominent notion in the theory of large cardinals, the study of the Mitchell order and its structure has expanded in recent decades. The behaviour of the Mitchell order on extenders depends on the type of extenders in consideration and naturally becomes more complicated when restricted to stronger types of extenders. A fundamental dividing line in the behaviour of the Mitchell order is its well-foundedness: Mitchell ([Mit83]) has shown that is well-founded when restricted to normal measures. The question of the well-foundedness of was further studied by Steel [St93], and Neeman [Ne04], who showed that it fails exactly at the level of rank-to-rank extenders.
Definition 1.2. Let E be an extender with associated ultrapower embedding i E : V → Ult(V, E). We say E is a rank-to-rank extender iff whenever λ > crit(E) is least such that i E (λ) = λ, then V λ ⊆ Ult(V, E).
Due to their similarity with embeddings j : V → M with V λ+2 ⊆ M , which have been shown by Kunen to be inconsistent with ZFC, the large cardinal strength witnessed by rank-to-rank extenders is considered to be located near the top of the large cardinal hierarchy. More precisely, rank-to-rank extenders naturally arise from the large cardinal axiom I2. The known dividing line of well-foundedness naturally breaks the question of the general possible behaviour into two (i) which well-founded partial orderings can be isomorphic to the Mitchell order on measures/extenders below the rank-to-rank level? and (ii) which ill-founded partial orderings can be isomorphic to the Mitchell order on a set of rank-to-rank extenders? 1 Concerning question (i), the possible structure of the Mitchell order on normal measures has been extensively studied in [Mit83], [Bal85], [Cum93], [Cum94], [Wit96], [BN16], [BN15]. It has been shown in [BN15] that it is consistent for every well-founded partial ordering to be isomorphic to the restriction of to the set of normal measures on some measurable cardinal κ (the exact consistency strength of this property has not been discovered). In this work, we make a first step towards expanding the study of the Mitchell order in the ill-founded case, and address question (ii). Specifically, we focus on the extent to which the well-foundedness of the Mitchell order fails on rank-to-rank extenders, by considering possible ordertypes of infinite decreasing chains in . The main results of this paper are the following two theorems. Theorem 1.3. Assume there exists a rank-to-rank extender E. Then for every countable ordinal γ there is a sequence of rank-to-rank extenders of length γ, (E α | α < γ), on which the Mitchell order is transitive and strictly decreasing.
Theorem 1.4. There is no ω 1 -sequence of extenders which is strictly decreasing and transitive in the Mitchell order.
Our presentation of the proof of Theorem 1.4 goes through a proof of a weak version of Steel's conjecture, which addresses transitive ω-sequences of extenders. 2 This presentation replaces a previous ad-hoc proof. The authors would like to thank Grigor Sargysyan for pointing out the connection with Steel's conjecture, which led to the current concise proof of Theorem 1.4. The (full) conjecture was recently proved by Goldberg ([Go]) building on his remarkable study of the internal relation. We believe that our proof of the weaker statement is of interest due to the fact that it only employs elementary concepts of the theory of extenders.
1 Beyond the possible (partial) ordering structure of the Mitchell order, the investigation can be further extended to non-transitive relations, as the Mitchell order need not be transitive in general (see [St93]). This direction is not developed in this paper. 2 See Section 4 for a formulation of Steel's conjecture.

Basic definitions and observations
We start with fixing some notation for extenders. A (κ, λ)-extender is a sequence E = (E a | a ∈ [λ] <ω ) with crit(E) = κ and lh(E) = λ. For a ∈ [λ] <ω , µ a denotes the least µ such that a ⊆ j(µ), if we are deriving an extender from j. i E : V → Ult(V, E) (we identify Ult(V, E) with its transitive Mostowski collapse) denotes the ultrapower embedding by E. We also write M E for Ult(V, E). For a rank-to-rank extender E we write λ E for the least λ > crit(E) such that i E (λ) = λ. Moreover, we write κ E 0 = crit(E) and κ E n+1 = i E (κ E n ) for n < ω, and call (κ E n | n < ω) the critical sequence of E. Note that for any rank-to-rank extender E, λ E = sup n<ω κ E n = lh(E). For every n < ω let E κ E n be the cutback of E to the measures E a , a ∈ [κ E n ] <ω . We have that Ult(V, E) = n N n where N n = Ult(V, E κ E n ). Moreover, for each n < ω, N n+1 contains V κ E n+1 and in particular i E "κ E n ∈ N n+1 . It follows from a standard argument that κ E n N n+1 ⊆ N n+1 . The following observations will be useful in comparing extenders in different ultrapowers.
Lemma 2.1. Assume that M = n<ω N n is an increasing union of transitive classes N n , and there exists a sequence of cardinals κ n | n < ω , such that κn N n ⊆ N n for all n < ω. Let E ∈ M be a rank-to-rank extender of height lh(E) = λ = n κ n .
(1) M E = n N n is an increasing union, and E a is σ-complete, there exists some n such that the set Lemma 2.2. Suppose E 2 , E 1 , E 0 are three rank-to-rank extenders of the same length λ = λ E i , i = 0, 1, 2, such that E 2 is Mitchell order below E 1 , and both E 2 , E 1 are Mitchell order below E 0 . Then Ult(V, E 0 ) also witnesses that E 2 is Mitchell order below E 1 .
Proof. The fact E 2 E 1 means that E 2 is represented in the V -ultrapower by E 1 , by a function f and a generator a of E 1 . Take k < ω such that . The reason it is not immediate that the relation E 2 E 1 also holds in M E 0 is that the function f need not belong to M E 0 . However, we can argue that M E 0 does see some witnessing function f * by using approximations. Assume without loss of generality that f (ν) is an extender for every ν ∈ dom(f ). Indeed, notice that for every n < ω, the function f n with dom(f n ) = dom(f ) such that for every ν, f n (ν) is the restriction of the extender f (ν) to length Let E(η) for some ordinal η denote the set of all (κ, η)-extenders. Now, working in M E 0 and utilizing the fact that both E 1 , E 2 belong to the model, we consider the tree T of all pairs (τ, n) such that τ : represents the restriction of E 2 to length κ E 1 n+1 . The tree order < T is given by (τ, n) < T (τ , n ) if n < n and τ (ν) extends τ (ν) for all ν ∈ dom(τ ). It is clear that a cofinal branch in T translates to a function F for which [a, F ] E 1 represents E 2 , and vice versa. Therefore f ∈ V witnesses that T has a cofinal branch in V , and thus, by absoluteness of well-foundedness, it must also have one in M E 0 .
Steel gives in [St93] a folklore example that for rank-to-rank extenders the Mitchell order need not be well-founded. We recall it here because some of the ideas will be used later.
Proposition 2.3 (Folklore). Let E be a rank-to-rank extender. Then there is a strictly decreasing sequence of length ω in the Mitchell order on which is transitive.
Proof. Consider the following sequence of rank-to-rank extenders (E n : n < ω).
is the canonical embedding associated to E n . Then it is straightforward to check that every E n is a V -extender and E n+1 E n for all n < ω.
Claim 1. The Mitchell order is transitive on (E n : n < ω).
Proof. Let n < ω. We show that E n+2 E n , the rest follows analogously.

Countable decreasing sequences in the Mitchell order
We now prove Theorem 1.3 and show that there can be strictly decreasing transitive sequences in the Mitchell order of any countable length.
Proof of Theorem 1.3. Let E be a rank-to-rank extender with critical sequence (κ n | n < ω) and λ E = λ. In what follows, all extenders will have the same length λ. We start by introducing some notation for sequences of extenders as constructed in the proof of Proposition 2.3. For a rank-to-rank extender F , write S 1 (F ) = i F (F ) and S n+1 (F ) = i S n (F ) (S n (F )). Now, the decreasing sequences in the Mitchell order we construct will be of the following form.
Definition 3.1. Let E = (E α | α < γ) be a sequence of rank-to-rank extenders. Then we say that E is guided by an internal iteration iff there are well-founded models ( for some n ≥ 1 and all α + 1 < γ, (2) M ν = M * ν for all limit ordinals ν ≤ γ is given as the direct limit of for all limit ordinals ν < γ, E ν = j * 0,ν (E 0 ) and Ult(V, E ν ) is wellfounded, and (5) the following diagram commutes and all maps in the diagram are given by internal ultrapowers.
Before we prove the existence of sequences of extenders guided by internal iterations, we show an abstract claim which will allow us to extend any such sequence by one further element. This is shown as in the proof of Theorem 2.2 in [St93] (where this particular argument is attributed to Martin). We sketch the proof here for the reader's convenience. Next, we argue that sequences as in Definition 3.1 are in fact as desired.
Claim 3. If a sequence E = (E α | α < γ) of extenders is guided by an internal iteration, the embeddings in the diagram witness that E β E α for all α < β < γ. In particular, E is a decreasing transitive sequence in .
Proof. We start by showing the following subclaim.
Proof. The claim is immediate when α is a limit ordinal or α = 0. Let α = β + 1 be a successor ordinal and assume inductively that For n > 1, the result is similarly obtained by applying the Lemma n many times.
Using this subclaim and the fact that all maps in the diagram witnessing that E is guided by an internal iteration are internal ultrapowers, it is straightforward to verify that E α E n whenever n < α, using the fact there are internal iterations from M n+1 (the ultrapower of V by E n ), and up to M * n+1 (i.e., vertical maps), and from M * n+1 to M * α . More generally, to see that E α E β for all β < α, suppose β = η + n for some limit ordinal η and The proof of Claim 2 shows that the following diagram commutes and moreover that the top row is an internal iteration of Ult(V, E η+n ). This, in combination with the subclaim and the fact that all maps in the diagram witnessing that E is guided by an internal iteration are internal ultrapowers as well, suffices to argue that E α E η+n .
We now turn to prove following claim, which immediately yields the theorem.
Let (E α | α < γ) be the concatenation of the sequences E n , n < ω. The choice of E n+1 to be minimal in M * αn with respect to the well ordering j * 0,αn (< w ) guarantees that the sequence E m | m > n belongs to M * αn , and thus also the tail of the iteration M * α , j * α,β | α n < α ≤ β < γ . Note that crit(j * 0,γ ) > κ since crit(j * αn,α n+1 ) > κ for all n < ω. Let j * 0,γ : V → M * γ = M γ be the direct limit embedding of the system. The reflecting a minimal counterexample argument used to show that internal iterations by normal ultrafilters are well-founded (see e.g., [Je03,Theorem 19.7] for normal ultrafilters or [Di18, Proposition 5.8] for rank-to-rank extenders), can also be used to show that M γ is well-founded.
Moreover, it is clear from the construction that, letting E γ = j * 0,γ (E), the resulting sequence (E α | α ≤ γ) of rank-to-rank extenders is guided by an internal iteration. The only condition that needs a small argument is the following.
Proof. As M γ is well-founded and E γ is an extender in M γ , Ult(M γ , E γ ) is well-founded. We prove the subclaim by defining an elementary embedding π : Eγ . π is well-defined since for a ∈ [λ] <ω , µ a < λ. Therefore, This finishes the proof of Theorem 1.3.

A bound on the length of decreasing sequences in the Mitchell order
Steel proved in [St93] that in a Mitchell order decreasing sequence of rankto-rank extenders, the extenders cannot all have the same critical point. He conjectured the following stronger statement.
Conjecture 4.1 (Steel). Suppose that (E m | m < ω) is a sequence of rankto-rank extenders which is strictly decreasing in . Let λ be the unique ordinal such that λ = λ Em for all sufficiently large m. Then sup m<ω crit(E m ) = λ.
Theorem 4.2 below establishes Steel's Conjecture for the special case that the Mitchell order is transitive on the sequence (E m | m < ω).
Theorem 4.2. Suppose that (E m | m < ω) is a sequence of rank-torank extenders, which is strictly decreasing and transitive in . Let λ be the unique ordinal such that λ = λ Em for all sufficiently large m. Then sup m<ω crit(E m ) = λ.
Proof. Suppose otherwise, and let γ 0 be the minimal ordinal for which there exists a -decreasing and transitive sequence E = (E m | m < ω) such that γ 0 = sup m<ω κ Em 0 < λ E = sup m<ω λ Em . We assume without loss of generality that λ E = λ Em for all m < ω. Let n < ω be the integer for which κ E 0 n ≤ γ < κ E 0 n+1 . We move to the ultrapower M E 0 . By our assumption, E m ∈ M E 0 for every m > 0, and by Lemma 2.2, M E 0 sees that E m is Mitchell order below E k for every 0 < k < m < ω. Since M E 0 is not closed under ω-sequences of its elements (in V ) the sequence (E m | 1 ≤ m < ω) need not belong to M E 0 . Nevertheless, we may define in M E 0 the tree T * of all finite sequences of rank-to-rank extenders (E * m | m < N ), which are strictly Mitchell order decreasing, transitive, have length λ E , and satisfy γ 0 = sup m<N κ E * m 0 . The sequence (E m | 1 ≤ m < ω) witnesses that T * has a cofinal branch in V . So by absoluteness of well-foundedness there is a cofinal branch in M E 0 as well. We can now reflect this from M E 0 back to V . Using the fact that κ E 0 n ≤ γ 0 < κ E 0 n+1 = i E 0 (κ E 0 n ), we conclude that in V , there exists some γ −1 < κ E 0 n ≤ γ 0 , and a sequence E * = (E * m | m < ω) of rank-to-rank extenders which is strictly -decreasing and transitive such that sup m<ω κ E * m 0 = γ −1 < λ E * . This is a contradiction to the minimality of γ 0 . Now we can obtain Theorem 1.4 as a corollary.
Proof of Theorem 1.4. Suppose otherwise. Let E = (E α | α < ω 1 ) be a sequence of extenders which is strictly decreasing and transitive in the Mitchell order. We may assume that all E α are rank-to-rank extenders and that there exists some λ E such that λ Eα = λ E for all α < ω 1 . In particular, cf(λ E ) = ω and we may choose a cofinal sequence (ρ n | n < ω) in λ E . By a straightforward pressing down argument, we can find an uncountable set I ⊆ ω 1 and some n * < ω such that κ Eα 0 < ρ n * for all α ∈ I. Taking (α n | n < ω) to be the first ω many ordinals of I, it follows that (E αn | n < ω) is strictly decreasing and transitive in the Mitchell order, with sup n<ω κ Eα n 0 ≤ ρ n * < λ E . This contradicts Theorem 4.2.

Questions
After studying the length of the Mitchell order for rank-to-rank extenders, a natural question that arises is about the structure this order can have.
Question 5.1. Suppose there is a rank-to-rank extender. Can the tree order on the infinite binary tree 2 <ω be realized by a Mitchell order?
We can even ask the following more general question.
Question 5.2. Suppose there is a rank-to-rank extender. Can any tree order on ω be realized by a Mitchell order?