Contemporary Mathematics

Volume 425, 2007

A stationary-tower-free proof of the derived model theorem

J. R. Steel

0

In this note, we give a proof of one direction of a version of Woodin's derived

model theorem:

THEOREM 0.1 (Woodin). Let). be a limit of Woodin cardinals, let G be V-

generic overCol(w, .),let JR.*= U{JR.nV[G

fa]

I

a.},

and let Hom*= {p[T]n

JR.* I

::la

.(T E V[G

fa]/\

V[G

fa]

f=

T is

).

absolutely complemented)}; then

(1) L(JR.*,Hom*)

f=

AD+,

(2) Hom* = {A

~JR.*

I

A is Suslin and co-Suslin in L(JR.*, Hom*)}.

Woodin proved the theorem in perhaps 1986 or 1987, using stationary tower

forcing and (through the work of Martin and the author) iteration trees. The

proof we give here uses only iteration trees. Stationary tower forcing is replaced by

"genericity iterations", as it can be in certain related contexts as well. We believe

that the unity of method in the resulting proof gives it some interest.

The first stationary-tower-free proof of the special case L(JR.*)

f=

AD+ was

discovered by the author in the early 90's. That proof uses fine-structural mice

and Woodin-style genericity iterations to replace the stationary tower forcing. The

fine-structural mice were needed because Woodin-style genericity iterations require

w1

+

1-iterable structures, and we do not know how to prove that kind of iterability

for countable

M -- V.

In the mid 90's, Neeman found a new kind of genericity

iteration that requires only w

+

1 iterabilty, which we do know how to prove for

countable

M -- V.

This gave the proof that L(JR.*)

f=

AD+

with the greatest

conceptual economy; only the tools of

[3]

were used. Neeman's work can be found

in

[4]

and

[5].

Our proof here uses Neeman's genericity iterations to prove the full

0.1.

The basic structure of our proof is that of Woodin's original proof. That proof,

along with related material, is exposited in

[6].

Larson's monograph

[2]

is an

excellent source on stationary tower forcing, including some of the material in

[6],

although it does not prove Theorem 0.1 itself.

Woodin actually proved a stronger version of Theorem 0.1. (See

[6,

p. 28] for

a statement of this version.) We do not know whether our proof here gives that

The author would like to thank the Logic Institute at the University of Muenster for its

generous hospitality during the preparation of this paper, and Ralf Schindler for a conversation

on the topic of the paper.

@2007 American Mathematical Society

1

http://dx.doi.org/10.1090/conm/425/08113