V l l l
PREFACE
This book doe s not attemp t t o give an encyclopedic accoun t o f stationar y
tower forcing . I n particular , a grea t dea l o f wor k ha s bee n don e o n th e
stationary towe r whic h w e d o no t discus s here ; see , fo r instance , [4 , 5 , 10,
11, 28 , 3 1 , 33 , 35 , 38 , 53 , 56 , 57] .
The presentatio n i s aime d a t a graduat e studen t wh o ha s take n intro -
ductory graduat e course s i n se t theor y an d logi c (a s wa s th e cours e th e
book i s base d on) . Th e primar y prerequisite s ar e th e basic s o f logi c (mod -
els, elementar y submodels , elementar y embeddings , theories , satisfaction ,
quantifiers), forcin g an d constructibilit y (throug h Chapte r VIII , § 3 o f [26] ,
say). Infinit e forcin g iteration s ar e no t required , thoug h th e reade r shoul d
be familia r wit h forcin g wit h partia l order s an d wit h Boolea n algebras , an d
the relationshi p betwee n th e tw o methods . Som e standar d forcin g fact s ar e
collected i n th e appendix . Th e reade r shoul d b e familia r wit h ultrapowe r
constructions, whic h ar e quickl y reviewe d i n th e first section . Sharp s (a s
presented i n [19, 21], fo r instance ) appea r a t severa l point s i n th e book .
None o f th e result s presente d her e i s du e t o th e autho r (asid e fro m th e
very mino r Exampl e 2.7.10, whic h probabl y woul d hav e bee n obviou s t o an y
expert i n determinac y wh o though t abou t it) . I have , however , modifie d
many o f th e origina l proof s fro m m y note s an d i n som e case s replace d the m
entirely (th e proo f o f Lemm a 3.1.14 , fo r instance , use s a constructio n fro m
[27]).
I a m gratefu l t o th e man y peopl e wh o answere d m y question s o n th e
material i n thi s book , an d wh o pointe d ou t error s i n earlie r drafts . I woul d
especially lik e to than k Joh n Stee l fo r lettin g m e cri b fro m hi s preprin t cite d
above, an d fo r enlightenin g m e o n a numbe r o f issues . Aaro n Siege l an d
David Asper o eac h mad e a larg e numbe r o f helpfu l comments . Mos t o f all ,
I than k Hug h Woodin , fo r al l th e obviou s reasons , an d especiall y fo r hi s
support o f thi s project .
0.1. Notatio n
Under th e usua l notation , whic h w e us e here , V b = 0 , V^+ i i s th e se t
of subset s o f V
a
an d V\ = LL A V* r limi t A . I f M i s a mode l o f ZFC ,
we sometime s writ e M
a
fo r V^f. However , w e sometime s us e subscript s t o
index a famil y o f models , suc h a s M
a
(a ft), an d w e hop e tha t ther e wil l
be n o confusion . I f G C P i s a ^/-generi c filter, the n Vs[G] i s th e se t o f
realizations b y G o f th e P-name s i n V
s
, wherea s V[G]
S
i s Vp G].
If / : X Y i s a functio n an d A C X, the n
f[A] = {y€Y\3xeAf(x)=y}.
If ft is a cardina l an d X i s a set, [X] K i s the se t o f unordered subset s o f [X] o f
size ft, an d [X] K an d V
K
(X) bot h denot e th e se t
U
7
A J ^ ] 7 ,
Likewise , i f ft
is a n ordina l an d an d X i s a set , the n X
K
i s the se t o f ordere d (i n ordertyp e
ft) subset s o f X o f siz e ft (also , th e se t o f function s fro m ft t o X) , an d X
K
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