It is known that in differentially closed fields of characteristic zero, the ranks of stability RU, RM and the topological rank RH need not to be equal. Pillay and Pong have just shown however that the ranks RU and RM coincide in a group definable in this theory. At the opposite, we will show in this paper that the ranks RM and RH of a definable group can also be different, and even lead to non-equivalent notions of generic type.

Morley rank and Lascar rank are equal on generic types of definable groups in differentially closed fields with finitely many commuting derivations.

In this paper, we shall prove two theorems involving the construction of core models with infinitely many Woodin cardinals. We assume familiarity with [12], which develops core model theory the one Woodin level, and with [10] and [6], which extend the fine structure theory of [5] to mice having many Woodin cardinals. The most important new problem of a general nature which we must face here concerns the iterability of Kc with respect to uncountable iteration trees.

Our first result is the following theorem, a slightly stronger version of which was proved independently and earlier by Woodin. The theorem settles positively a conjecture of Feng, Magidor, and Woodin [2].

Theorem. Let Ω be measurable. Then the following are equivalent:

(a) for all posets,

(b) for every poset,

(c) for every poset ℙ ∈ VΩ, Vℙ ⊨ there is no uncountable sequence of distinct reals in L(ℝ)

(d) there is an Ω-iterable premouse of height Ω which satisfies “there are infinitely many Woodin cardinals”.

It is an immediate corollary that if every set of reals in L(ℝ) is weakly homogeneous, then ADL(ℝ) holds. We shall also indicate some extensions of the theorem to pointclasses beyond L(ℝ), and mice with more than ω Woodin cardinals.

We consider only computable languages, and countable structures, with universe a subset of ω, which we think of as a set of constants. We identify sentences with their Gödel numbers. Thus, for a structure , the complete (elementary) diagram, Dc(), and the atomic diagram, D(), are subsets of ω. We classify formulas as usual. A formula is both Σ0 and Π0 if it is open. For n > 0, a formula, in prenex normal form, is Σn, or Πn, if it has n blocks of like quantifiers, beginning with ∃, or ∀. For a formula θ, in prenex normal form, we let neg(θ) denote the dual formula that is logically equivalent to ¬θ—if θ is Σn, then neg(θ) is Πn, and vice versa.