Preface
This boo k i s a n expositor y introductio n t o stationar y towe r forcing , a
construction isolate d b y W . Hug h Woodi n i n th e wak e o f wor k o f Fore -
man, Magido r an d Shela h [12]. Th e firs t chapte r present s som e relevan t
background material , an d i s also intended t o serv e a s a warmup fo r reader s
unfamiliar wit h thi s are a o f se t theory . I n th e secon d chapte r w e presen t
the stationar y towe r an d develo p it s basi c properties , an d i n th e thir d w e
present some of the major application s of this method to the study of forcing-
absoluteness an d set s o f reals i n inne r models . Th e majo r result s presente d
here includ e th e following :
(1) I f ther e i s a prope r clas s o f Woodi n cardinals , the n th e theor y o f
L(M) i s invariant unde r al l set forcin g extensions ;
(2) I f ther e i s a prope r clas s o f measurabl e Woodi n cardinals , the n
all forcin g extension s satisfyin g th e Continuu m Hypothesi s agre e
about al l Y!\ sentences;
(3) I f there exist infinitel y man y Woodin cardinals below a measurabl e
cardinal, the n th e Axio m o f Determinacy hold s i n L(M);
(4) I f ther e i s a prope r clas s o f Woodin cardinals , the n th e collection s
of universall y Bair e set s o f reals , oo-homogeneousl y Susli n set s o f
reals an d oo-weakl y homogeneousl y Susli n set s o f reals coincide ;
(5) I f ther e exis t a supercompac t cardina l 5 an d a prope r clas s o f
Woodin cardinals , the n afte r an y forcin g makin g V^+ i countable ,
the theor y o f th e leas t inne r mode l o f Z F containin g th e ordinal s
and th e universall y Bair e sets of reals is invariant unde r al l furthe r
set forcin g extensions .
Items (3) , (4 ) an d (5 ) her e ar e prove d usin g th e mai n theore m o f [30] .
The boo k i s mostl y base d o n lectur e note s fro m a graduat e se t theor y
course give n b y Woodi n a t U.C . Berkele y i n th e Sprin g o f 1996. Sectio n
2.6 i s adapte d fro m a n emai l fro m Woodi n t o Jame s Cummings . Sectio n
3.2 i s a n update d proo f o f Woodin's S^-absolutenes s result , base d o n a se t
of hi s handwritte n note s fro m 1985. Sectio n 3. 3 include s a proo f fro m a
paper b y Neema n an d Zapleta l [33 ] an d materia l fro m a preprin t o f Stee l
[46]. Sectio n 3. 4 is based o n a set o f notes written b y Woodin especiall y fo r
inclusion i n this book . Excep t wher e noted , al l of the result s i n Chapter s 2
and 3 are du e t o Woodin . Th e histor y o f muc h o f th e materia l i n Chapte r
1 i s documented i n [30] . Th e book s [19, 21] ar e als o good sources .
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