The well-foundedness of the Mitchell order

Abstract

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Suitable Extender Models I

“…This extends earlier results of Steel [16] which deal with generalized iteration trees of length ω and for ultimately a different purpose (wellfoundedness of the Mitchell order), not for the purpose of identifying a notion of iteration that might be relevant to the inner model theory of coherent sequences of (long) extenders.…”

“…(2) The fundamental nature of the generalization of iteration tree in the sense of [7], that is provided by Definition 60, is not that long extenders are allowed, it lies in the cancellation rule; (3b). Then sp(E) coincides with Steel's notion of the space of an extender, [16]. Now define…”

“…We generalize the definition to the case of long extenders. For iteration trees of length ω this is actually also due to Steel, [16].…”

“…this is the definition of [16] for iteration trees of length ω. We have chosen the definition based on the parameters (SPT(E), ρ(E)) simply in order to conform to the definitions of [7].…”

“…We prove that the generalization of the Martin-Steel Theorem on the existence of maximal realizable branches does hold for this notion of iteration tree (and in particular this eliminates the family of counterexamples identified by Neeman). The special cases of both this definition and the theorem, for iteration trees of length ω, are due to Steel, [16]; but for a different purpose namely analyzing the Mitchell order for arbitrary extenders. All current theorems on the iterability of V or of structures derived from extenders in V , trace back either directly, or indirectly, to the Martin-Steel Theorem.…”