DAISUKEIKEGAMI
Abstrat. Foralargenaturallass offoringnotions,weprove
generalequivalene theorems betweenforing absoluteness
state-ments,regularityproperties, andtransendeneproperties overL
andtheoremodelK. Weuseourresultstoansweropenquestions
fromsettheoryofthereals.
1. Introdution & Bakground
Foring absoluteness statements have been investigated by Judah,
Brendle, Halbeisen, Amir, Bagariaand others [18, 6, 12, 1, 3℄. These
statements of the form \Every -statement is absolute between the
groundmodeland itsforing extensions withP" aretypially
indepen-dentofthe axiomsof ZFC ,and anoftenbeproved tobeequivalentto
statements about regularity properties. Typial equivalene theorems
are:
Theorem 1.1 (Bagaria, Woodin, [2, 29℄). Every 1
3
-statement is
ab-solute between the ground model and its Cohen foring extensions if
and only if every 1
2
set has the Baire property.
Theorem1.2(Ikegami,[14℄). Every 1
3
-statementisabsolutebetween
the ground modeland itsSaks foring extensions if and onlyif every
1
2
-set eitherontains aperfet subsetorisdisjointfromaperfet set.
The mentioned regularity properties are in turn equivalent to
tran-sendene properties over L. For instane, Judah and Shelah proved
that the Baire property of all 1
2
-sets is equivalent to the
transen-dene statement \for all reals x, there is a Cohen real over L[x℄" [19℄;
similarly,Brendle and Lowe showed that the statement \every 1
2 set
either ontains a perfet subset or is disjoint from a perfet set" is
equivalentto \forall reals x, there isa real not in L[x℄" [8℄.
2000MathematisSubjet Classiation. 03E15,28A05,54H05.
In this paper, we shall prove a general abstrat result underlying
both Theorems 1.1 and 1.2, by onneting (for a large lass of
for-ingsP) 1
3
-P-absoluteness, aregularityproperty atthe 1
2
-level,anda
transendene property relatedtoP. The ase of Cohen foring might
suggest that the right transendene property is the existene of
P-generis, but this already fails in the ase of Saks foring. 1
In order
to deal with this situation, Brendle, Halbeisen and Lowe introdued
the notion of quasi-generi reals [7℄. In many ases of ... forings
(suh asCohen foring),the notionsof quasi-generiity and generiity
oinide; in general, the existene of quasi-generis gives us the right
transendene property for our general theorem. We prove:
Theorem 1.3. For any foring P in a large lass of foring notions 2
,
the followingare equivalent:
(1) 1
3
-P-absoluteness holds,
(2)every 1
2
-set of reals isP-measurable, and
(3) for any real a and T 2 P, there isa quasi-P-generi real x2[T℄
over L[a℄.
We shall start by dening and investigating the basi onepts in
x2 and x3. We then state and prove the main result of the paper
(thepreiseversionofTheorem1.3)anditsimmediateonsequenesin
x4. AmongtheonsequenesisageneralSolovay-styleharaterization
theorem(inthetraditionof[26℄). Inx5,wemoveonto 1
4
-absoluteness
and prove the analogues of the results from x4 under the assumption
ofappropriatelargeardinalaxioms. Theseproofsuse somebasifats
of inner modeltheory. In x6,we give appliations ofour main results,
answeringanopen questionfrom[7℄; nally,inx7,welista numberof
interesting open questions.
2. Basi onepts
From now on, we will work in ZFC. We assume that readers are
familiar with the elementary theories of foring and desriptive set
theory. (For basi denitions not given in this paper, see [15, 22℄.)
When we are talking about \reals", we mean elements of the Baire
spae orof the Cantor spae.
In this setion, we introdue the notions we will need for the rest.
Westart with introduing the foring absoluteness we willfous on:
1
Inthemodelafteradding!
1
manyCohenrealstoL,everyprojetiveseteither
ontainsorisdisjointfromaperfetset,but thereisnoSaksreal overL.
2
Denition 2.1 (
n
-P-absoluteness). Let P be a foring notion and
n be a natural number with n 1. Then 1
n
-P-absoluteness is the
followingstatement:
\forany 1
n
-formula', real r inV, and P-generi lter
Gover V, V '(r)i V[G℄'(r)".
Denition 2.2 (Projetive forings). Letn beanaturalnumberwith
n 1. A partial order P is 1
n
(resp. 1
n ,
1
n
) if the sets P,
P ,
and ?
P
are 1
n
(resp. 1
n ,
1
n
), where P = (P;
P
) and ?
P
is the
inompatibility relation in P. We say P is projetive if it is 1
n for
some n1.
Letn beanaturalnumberwithn 1. Apartial orderPisprovably
1
n
if there are 1
n
-formula and 1
n
-formula suh that the
state-ment \ and dene the same partial order with the inompatibility
relation"is provable in ZFC.
All typialforings related to the regularity properties are provably
1
2
. In this paper, we are onlyinterested inprojetive forings.
In some of our main results, we shall need a strengthening of the
standard notion of properness for projetive forings:
Denition 2.3. A projetive foring P is strongly proper if for any
ountable transitive model M of a nite fragment of ZFC ontaining
the real parameter in the formuladening P, if P M
; M
P ;?
M
P
are
sub-sets of P;
P ;?
P
respetively, then for any ondition p in P M
, there
is an (M;P)-generi ondition q below p,i.e., if M \A is a maximal
antihain in P", then A\M is predense below q. 3
Here (M;P)-generi onditionsarethe sameas(X;P)-generi
ondi-tionsforountableelementarysubstruture X ofH
: ifPisprojetive,
X isaountableelementarysubstruture of H
forsome enoughlarge
regular and M is the transitive ollapse of X, then a ondition p is
(M;P)-generi i it is (X;P)-generi in the usual sense. In partiular,
if P is projetive and strongly proper, then P isproper.
All the typialexamples of proper, 1
2
-forings are strongly proper.
Butthereisaproper, provably 1
3
-foringwhihisnotstronglyproper
(for the details,see the papers [5, 4℄by Bagariaand Bosh).
3
AlthoughwewillnotexpliitlymentionthenitefragmentofZFCwewilluse
forthedenitionofstrongproperness,itwillbeenoughlargesothatweanproeed
alltheargumentsinthispaperasusual. Fromnowon,wesay\ountabletransitive
We use strong properness instead of properness, as it allows us to
leaveoutthequantiation\2H
"whihwouldinrease the
omplex-ity of our statements in the relevant results (Proposition 2.17,
Theo-rem 5.3, Theorem 5.6) beyond projetive.
Next, we introdue a lass of forings ontaining all the tree-type
forings. Apartial orderP isarborealif itsonditions are perfet trees
on!(resp. 2)ordered byinlusion. Butthis lassof foringsontains
sometrivialforingssuhasP=f <!
!g. Weneedthefollowingstronger
notion:
Denition 2.4. A partial order P is strongly arboreal if it is arboreal
and the following holds:
(8T 2P) (8t2T) T
t 2P;
where T
t
=fs2T jeither st ors tg.
Withstronglyarborealforings,weanode generiobjetsbyreals
inthestandardway: letPbestronglyarborealandGbeP-generiover
V. Letx
G =
S
fstem(T)jT 2Gg, wherestem(T)is the longestt2T
suhthatT
t
=T. Thenx
G
isarealandG=fT 2Pjx
G
2[T℄g,where
[T℄isthe set of allinnitepathsthrough T. Hene V[x
G
℄=V[G℄. We
allsuh real x
G
a P-generi real over V.
Almostalltypialforingsrelatedtoregularitypropertiesarestrongly
arboreal:
Example 2.5. (1) Cohen foring (C): let T
0 be
<!
!. Consider the
partial order f(T
0 )
s j s 2
<!
!g;
. Then this is strongly arboreal
and equivalent toCohen foring.
(2) random foring (B): onsider the set of all perfet trees T on 2
suhthat forany t2T,[T
t
℄has apositiveLebesgue measure, ordered
by inlusion. Then this foring is strongly arboreal and equivalent to
randomforing.
(3)Hehler foring (D): for (n;f)2D, let
T
(n;f) =
n
t2 <!
! j eithert f n or
t f n and 8m 2dom(t)
t(m)f(m) o
:
Then the partial order (fT
(n;f)
j (n;f) 2 Dg;) is strongly arboreal
and equivalent toHehler foring.
(4)Mathias foring: for aondition (s;A) of Mathias foring, let
T
(s;A)
=ft2 <!
! jt is stritly inreasing and sran(t)s[Ag:
Then fT
(s;A)
(5) Saks foring, Silver foring, Miller foring, Laver foring (S,V ,
M , L, respetively): these forings an be naturally seen as strongly
arboreal forings.
We nowintroduea generaldenition of aregularity property
asso-iatedwithanarbitraryarborealforing. Setsofrealswitharegularity
propertyshouldbeapproximatedby somesimplesets (e.g.,Borelsets)
modulo some\smallness" asBaireproperty and Lebesgue
measurabil-ity. Therefore we rst introdue \smallness" for eah arboreal foring
by deiding a-ideal asfollows:
Denition2.6. LetPbeanarborealforing. Aset ofrealsAisP-null
if for any T inP there is a T 0
T suh that [T 0
℄\A=;. N
P
denotes
the setof allP-null setsand I
P
denotes the-idealgenerated byP-null
sets.
Example 2.7. (1) Cohen foring C: C-null sets are the same as
nowhere dense sets and I
C
is the meager ideal.
(2)randomforingB: B-nu ll sets are the sameasLebesguenullsets
and I
B
is the Lebesgue null ideal.
(3) Hehler foring D: D-null sets are the same as nowhere dense
setsinthedominatingtopology,i.e.,thetopologygeneratedby f[s;f℄j
(s;f)2Dg where
[s;f℄=fx2 !
!js xand (8ndom(s)) x(n)f(n)g:
Hene I
D
is the meager idealin the dominating topology.
(4)Mathiasforing: aset of reals AisMathias-nullifran(x)jx2
A\A
0
gisRamsey null ormeagerinthe Ellentuk topology,where A
0
is the set of stritly inreasing innite sequenes of natural numbers.
Also, Mathias-nullsets form a-ideal by a standard fusionargument.
(5) Saks foring S: in this ase, I
S = N
S
by a standard fusion
argument. TheidealI
S
isalledtheMarzewskiidealandoftendenoted
by s
0 .
As with Saks foring, all the typial non- tree-type forings
ad-mitting a fusion argument satisfy the equation I
P = N
P
. Sine I
P is
Borelgeneratedforany arborealforing,theondition()in
The-orem 4.4(whihwe willstate in x4)holds for allthe typialtree-type
strongly arboreal forings.
Now we introdue the regularity property for eah arboreal foring:
Denition 2.8. Let P be arboreal. A set of reals A is P-measurable
if for any T in P there is a T 0
T suh that either [T 0
℄\A 2 I
P or
[T 0
As we expet, P-measurability oinides with the known regularity
property for Pwhen P is :
Proposition 2.9. Let Pbe astrongly arboreal, foring and letP
beasetofreals. ThenP isP-measurableithereisaBorelsetB suh
that P4B 2I
P .
Proof. Thediretionfromright toleftfollowsfrom thefat thatevery
Borel set of reals is P-measurablewhihwill be proved inLemma 3.5.
Fortheotherdiretion,suppose P isP-measurableandwewillnda
Borelset approximatingP moduloI
P
. SineP isP-measurable,theset
D=fT 2Pj either [T℄\P 2I
P
or[T℄nP 2I
P
g is dense. Wetakea
maximalantihainAinDanddeneB = S
f[T℄jT 2A and [T℄nP 2
I
P
g. Then sine A is ountable, B is Borel and P4B 2 I
P
beause D
is dense.
This argument does not work for non- forings suh as Saks
foring. 4
But P-measurability is almost the same as the regularity
properties for non- forings P, e.g., for Mathias foring, a set of
reals A is Mathias-measurable i fran(x)j x 2A\A
0
g is ompletely
Ramsey(orhastheBairepropertyintheEllentuktopology),whereA
0
isthesetofallstritlyinreasinginnitesequenesofnaturalnumbers.
Also, forSaks foring, the followingholds:
Proposition 2.10 (Brendle-Lowe). Let be a topologially
reason-able pointlass, i.e., it is losed under ontinuous preimages and any
intersetion between a set in and a losed set. Then every set in
is S-measurablei every set in has the Bernstein property. 5
Proof. See [8,Lemma 2.1℄.
Next we introdue atehnial ideal I
P
whih we need later:
Denition 2.11. Let P be an arboreal foring. A set of reals A is in
I
P
if for any T in P there isa T 0
T suh that [T 0
℄\A isin I
P .
Question 2.12. Let Pbeastrongly arboreal,properforing. Can we
proveI
P =I
P
?
4
Forexample,assumingevery 1
1
-set hastheperfetsetproperty,every 1
1 -set
of reals hasthe Bernstein property(i.e., either it ontainsaperfet orthere is a
perfetsetdisjointfromtheset)butfora 1
1
-setofrealsA,Aisapproximatedby
aBorelset modulo I
S
i Ais Borel. Thisis beauseI
S
restritedtoanalyti sets
(oro-analytisets)isthesetofallountable setsofreals.
5
In general, the Bernstein property does not imply S-measurability while the
We givesome easy observations onerning toQuestion 2.12:
Lemma 2.13. Let P be strongly arboreal foring.
(1)The ideal I
P
is asubset of I
P
.
(2) A set of reals A is P-measurable i for any T in P there is a
T 0
T suh that either [T 0
℄\A 2 I
P
or [T 0
℄nA 2 I
P
holds. Hene
weget the same notionofmeasurabilityeven ifwereplaeI
P by I
P
in
the denition of P-measurability.
(3)If P is, then I
P =I
P
.
(4) If I
P = N
P
, then I
P = I
P
. Hene I
P = I
P
for any typial
tree-type stronglyarboreal foring admittinga fusionargument.
(5) (Brendle) Suppose P satises the following ondition: for any
maximal antihain A in P, there is a maximal antihain A 0
suh that
foranytwoelementsT;T 0
ofA 0
,[T℄and[T 0
℄are disjointand A 0
renes
A, i.e., for anyT 0
inA 0
there isa T inA with T 0
T. Then I
P =I
P
.
Saksforing isatypialexampleof theondition in(5). Butwedo
not knowofanystronglyarboreal Psatisfyingthe onditionbut whih
are neither nor satisfy I
P =N
P .
Proof. We willprove only (5). The rest are straightforward. Suppose
P satises the above ondition and let A be in I
P
. We prove A is in
I
P
. Sine A is in I
P
, the set of all Ts in P suh that [T℄\ A 2 I
P
is dense in P. Hene we an take a maximal antihain A ontained
in this set. By the ondition, we may assume for any two distint
elements T
1 , T
2
of A, [T
1 ℄, [T
2
℄ are pairwisedisjoint. For eahT in A,
[T℄\A 2 I
P
. So we an pik fN
n;T
j n 2 !g suh that eah N
n;T is
P-null and S
n2! N
n;T
=[T℄\A. LetN
n =
S
T2A N
n;T
for eah n2 !.
Sine A= S
n2! N
n
, the proof is omplete if we prove the following
Claim 2.14. For eah n2!, N
n
is P-null.
Proof of Claim 2.14. TakeanyT 0
inP. SineAisamaximalantihain,
we an take a T 2 A suh that T and T 0
are ompatible. Take a
ommon extension T 00
. Then [T 00
℄\N
n = [T
00
℄\N
n;T
beause of the
property of A. But we know that N
n;T
isP-null. Hene we an take a
further extension of T 00
disjointfrom N
n
.
Next, we introdue quasi-P-generiity for arboreal forings P and
ompare it with P-generiity. Quasi-generireals are obvious
general-ization ofCohen reals and random reals:
with B
2 I
P
, x is not inB
, where B
is the deoded Borel set from
.
Example 2.16. (1) Cohen foring (C): quasi-C-generi reals are the
same as Cohen reals by denition. Hene quasi-C-generiity oinides
with C-generiity.
(2) random foring (B): quasi-B-generi reals are the same as
ran-dom reals by denition. Hene quasi-B-generiity oinides with B
-generiity.
(3)Hehlerforing(D): quasi-D-generirealsarethesameasHehler
reals. Hene quasi-D-generiity oinides with D-generiity.
(4)Saksforing(S): ifM isaninnermodelofZFC,quasi-S-generi
reals over M are the reals whih are not in M beause any Borel set
in I
S
= N
S
is ountable and this is also true in M if the ode is in
M by Shoeneld absoluteness. Therefore, quasi-S-generiitydoes not
oinidewith S-generiity.
The last example explains the dierene between generiity and
quasi-generiity and shows that the equivalene for Saks foring we
mentioned in the introdutionis a speial ase of Theorem 4.3 whih
we willprove later. 6
As is expeted, generiity impliesquasi-generiity for allthe typial
stronglyarborealforingsandtheonverseistrueformostforings:
Proposition2.17. LetPbeastronglyarboreal,stronglyproper,
prov-ably 1
2
foring. Then
(1) The set f j B
2 I
P
g is 1
2
. Hene the statement \ odes a
Borel set inI
P
" is absolutebetween inner models of ZFC.
(2)If M is a transitive modelof ZFC and a real x is P-generi over
M, then xis quasi-P-generi overM.
(3)Suppose P isalsoprovably ,i.e., there isa formuladening
P and the statement \ is " is provable in ZFC. Then if M is an
inner modelof ZFC and a real x is quasi-P-generi over M, then x is
P-generi over M.
Proof. See x3.
In [31℄, Zapletal starts from a -ideal I on a Polish spae X and
onsiders the quotient of the set of all Borel sets in X modulo I and
develops the general theory of this foring as a Boolean algebra. Let
us ompare hissetting with our setting:
6
Itiseasytohektheondition()inTheorem4.3forSaksforingbynoting
thattheidealI
S
Proposition 2.18. Suppose P is a strongly arboreal, proper foring.
Then the map i: P! B=I
P
nf0g dened by
i(T)= the equivalenelass represented by [T℄;
is a dense embedding,where B denotes the set of allBorel sets of the
reals and B=I
P
is the quotientBoolean algebravia I
P
.
Hene, our situation isa speial ase of Zapletal's. 7
Proof. See x3.
3. P-measurability and P-Baireness
In thissetion,we shallprovethe propositionslistedinx2. Inorder
todoso,werstonsidertheonnetionbetweenP-measurabilityanda
property alledP-Baireness (whihwas impliitlyintrodued by F
eng-Magidor-Woodin [11℄). This onnetion will allow us to haraterize
I
P
in terms of Banah-Mazur games, whih plays an essential role in
the proof of Proposition 2.17.
Let P be a partial order. The Stone spae of P (denoted by St(P))
is the set of ultralters of P equipped with the topology generated by
fO
p
j p2Pg, where O
p
=fu2St(P)ju3pg.
Forexample,if Pis Cohenforing(C), then St(C) ishomeomorphi
to the Bairespae !
!.
Dense sets in P are the same as open dense subsets in St(P): if D
is a dense subset of P, then the set S
fO
p
j p 2 Dg is open dense in
St(P). Conversely, if U is anopen dense subsetof St (P), thenfp2Pj
O
p
Ug is adense open subset of P.
Next,wewilltalkaboutmeagernessandtheBairepropertyinSt (P).
The rst observation weshould make is that this is not nonsense:
Lemma 3.1. LetP be a partial order. Then for any p2St(P), O
p is
not meager.
Proof. Take any p2P and letfU
n
jn 2!g beaountable set ofopen
dense subsets of St (P). We would like to prove that the intersetion
T
n2! U
n
with O
p
is nonempty. But this is just the Rasiowa-Sikorsky
Theorem orndinga generiobjetG overaountable struture
on-tainingP with p2G.
7
In[31,Corollary2.1.5℄,Zapletalprovedamoregeneralresult. HisIisessentially
thesameasourI
P
andifweuseb
n =jx_
gen
(n)=1j(n2!)insteadofb
t (t2
<!
2)
forthe generatorsofC, thenZapletal's I is exatlythesameasourI
P
Beforedening P-Baireness, letussee theonnetion between Baire
measurable funtionsfrom St (P) tothe reals and P-names fora real.
LetX;Y betopologialspaes. Thenafuntion f: X !Y is Baire
measurableif for any open set U in Y, f 1
(U) has the Baire property
inX. Bairemeasurablefuntionsare thesameasontinuousfuntions
modulo meager sets: let X;Y be topologial spaes and assume Y is
seondountable. Then itis fairlyeasytosee that afuntion f: X !
Y is Baire measurable i there is a omeager set D in X suh that
f D is ontinuous.
There is a natural orrespondene between Baire measurable
fun-tions from St(P)to the reals and P-names for a real:
Lemma 3.2 (Feng-Magidor-Woodin). Let Pbea partial order.
(1)If f: St(P)! !
! isa Baire measurablefuntion, then
f =
(m;n);p)jO
p
nfu2St(P) jf(u)(m)=ng ismeager
is aP-name for areal.
(2)Let beaP-nameforareal. Dene f
asfollows. Foru2St(P)
and m;n 2!,
f
(u)(m)=n () (9p2u) p(m ) =n:
Then the domain of f
is omeager in St (P) and f
is ontinuous on
the domain. Hene it an beuniquely extended toa Bairemeasurable
funtion fromSt(P) to the reals modulo meagersets.
(3) If f: St(P) ! !
! is a Baire measurable funtion, then f
f and
f agree on a omeager set in St(P). Also, if is a P-name for a real,
then
f =.
Proof. See [11, Theorem 3.2℄.
Reall that we have dened a generi real x
G
from a generi objet
G for any strongly arboreal foring P. Let x_
G
be a anonial P-name
for x
G .
Example 3.3. Let P be strongly arboreal. Then f
_ x
G
(u)(m) = n
i there is a T in u suh that stem (T)(m) = n. Hene f
_ x
G (u) =
S
fstem(T)jT 2ug for u2dom() asweexpet.
Now we dene the property P-Baireness. Let P be a partial order
and Abeaset ofreals. ThenA is P-Baire iffor any Bairemeasurable
funtion f: St(P) ! !
!, f 1
(A) has the Baire property in St(P). It
is easy to see that every Borel set of reals is P-Baire for any P by the
same argumentas for the Baire property.
Example 3.4. LetC beCohen foring. Aset ofreals A is C-Baire i
Proof. Aswehaveseeninthebeginningofthis setion,St(C) is
home-omorphi to the Baire spae !
!. In the Baire spae, any G
Æ
omeager
set is homeomorphi to the whole spae. Hene we an replae Baire
measurable funtions by ontinuous funtions in the denition of C
-Baireness.
Before talking about the relation between P-measurability and
P-Baireness,letusmentionthe onnetionbetween P-Bairenessand
uni-versally Baireness. A set of reals A is universally Baire if for any
ompat Hausdor spae X and any ontinuous funtion f: X ! !
!,
f 1
(A)hastheBairepropertyinX. AsetofrealsAisuniversallyBaire
i A is P-Baire for any partial order P. (This is essentially proved in
[11℄.)
ReallthatI
P
isatehnialidealintroduedinDenition2.11whih
is the same asI
P
for most ases.
Lemma 3.5 (P-measurability vs. P-Baireness). Let P be a strongly
arboreal, proper foring and A be aset of reals. Then
(1)A is inI
P
i f 1
_ x
G
(A)is meager inSt(P), and
(2)Ais P-measurableif 1
_ x
G
(A)has the Baireproperty inSt (P). In
partiular, if A is P-Baire, then A isP-measurable. Hene every Borel
set if P-measurable.
Notethat P-measurability doesnot imply P-Bairenessin general. 8
Proof of Lemma 3.5. Let =f
_ x
G
for abuse of notation.
The followingare useful for the proof:
Claim 3.6. (a)ForT inP and u2dom(),if T 2u, then(u)2[T℄.
(b) For T in P, the onverse of (a) holds for omeager many u in
dom().
Proof of Claim 3.6. (a) Suppose T 2 u. We prove (u) n 2 T for
eah n2!. Fix anaturalnumbern. Thenby Example 3.3, there isa
T 0
in u suh that stem(T 0
) (u) n. Sine both T and T 0
are in u,
theyare ompatible,espeiallystem(T 0
)2T (otherwise[T℄\[T 0
℄=;).
Hene (u)n2T.
(b) Take any T inP. Then the set D =fT 0
2 Pj T 0
T or [T 0
℄\
[T℄ = ;g is dense in P. (Take any T 0
. If T 0
* T, then there is a
t 0
2 T 0
nT. By strong arborealness of P, T 0
t 0
2 P and [T 0
t
0℄\[T℄ =;.)
SineDisdense,the setfuju\D6=;gisdense openinSt(P). Hene
8
Forexample,ifAisa 1
2
(lightfae)setofrealsuniversalfor 1
2
(boldfae)sets
ofreals andifevery 1
2
(lightfae) setof realshastheBairepropertybutthere is
a 1
2
it suÆes to show that if u is in dom(), u\D 6= ; and (u) 2 [T℄,
then T 2u. Suppose T 2= u. Then sine u\D 6=;, there is a T 0
2 u
suh that [T 0
℄ \[T℄ = ;. By (a), (u) 2 [T 0
℄, hene (u) 2= [T℄, a
ontradition.
(1)Weprove the diretionfrom leftto right.
We rst show that 1
(A) is meager if A is in N
P
. If A is in N
P ,
then the set D =fT j [T℄\A =;g is dense in P. Hene the set of all
u 2 dom() with u\D 6=; is omeager. But if u is in the omeager
set, then there is a T 2 u\D and by Claim 3.6 (a), (u) 2 [T℄ and
[T℄\A =;, inpartiular (u)2= A. Therefore 1
(A) is meager.
We have seen that 1
(A) is meager assumingA is in N
P
. Sine I
P
isthe -idealgenerated bysets inN
P ,
1
(A)ismeagerforallA inI
P .
We show that 1
(A) is meager if A is in I
P
. Sine A is in I
P
,
the set D 0
= fT j [T℄\A 2 I
P
g is dense in P. We use the following
well-known fat:
Fat 3.7. LetX beatopologialspaeand Abe asubsetof X. Then
S
fU jU isopen and U \Ais meager g
\A ismeager.
Proof of Fat 3.7. See [20, Theorem 8.29℄.
SineD 0
isdense, S
fO
T
jT 2D 0
gisopen dense. Bytheabovefat,
it suÆes to prove that O
T \
1
(A) ismeager forany T in D 0
.
Take any T in D 0
. By the denition of D 0
, we know that [T℄\A is
in I
P
. Hene 1
([T℄\A) is meager in St(P). But by Claim 3.6 (a),
O
T \
1
(A) 1
([T℄\A). Therefore, O
T \
1
(A)is meager aswe
desired.
Next, we see the diretion from right to left. Suppose 1
(A) is
meager. Take any T in P and we will nd an extension T 0
of T suh
that [T 0
℄\A isinI
P
. Sine 1
(A)ismeager,then thereis asequene
hU
n
jn 2!i ofopen densesets inSt(P)suhthat T
n2! U
n \
1
(A)=
;. Foreahn 2!,letD
n
=fS 2PjO
S U
n
g. SineU
n
isopendense
inSt (P), D
n
is denseopen inP. We hoose asequene hA
n
jn 2!i of
maximal antihainssuh that A
n
D
n
, for eahelement S of A
n , the
length of stem(S) is greater than n, and A
n+1
renes A
n
, i.e., every
element of A
n+1
isbelowsome element inA
n .
Now we use the properness of P to treat eah A
n
as \ountable".
Let be a suÆiently large regular ardinal and X be a ountable
elementary substruture of H
suh that P;T;hA
n
j n 2 !i are in X.
By properness, there is an (X;P)-generi ondition T 0
below T. We
showthat [T 0
℄\A is inI
P
, whih willomplete the proofof (1).
Consider the set B = T
n2! S
f[S℄jS 2A
n
\Xgn S
n2!
f[S℄\[S 0
℄j
S;S 0
2A \X and S 6=S 0
whih ondition from A
n
ontains it for eah n. By the property of
hA
n
jn2!i,itwillgeneratealteromingfromelementsinA
n s. The
point is that any ultralter u extending that lter satises (u) = x,
the given element,and that uis inU
n
foreah n. This willplaya role
for the argument.
Now we laim [T 0
℄nB 2 I
P
and B \A = ;. We will be done if
we prove them. The fat that [T 0
℄nB 2I
P
follows from the fat that
fS j S 2 A
n
\Xg is predense below [T 0
℄ for eah n beause T 0
is
(X;P)-generi and from that [S℄\[S 0
℄ 2 I
P
for eah S;S 0
2 A
n \X
with S 6= S 0
beause A
n
is an antihain, and from that A
n
\X is
ountablefor eah n.
To prove B\A =;, take any element x fromB. As we mentioned
above, for eah n 2 !, there is a unique element S
n in A
n
\X with
x 2 [S
n
℄. Sine A
n+1 renes A n , S n+1 S n
for eah n. Hene the
set fS
n
j n 2!g generate a lter F
x
. Take any ultralter u extending
F
x
. We laim that (u)= x and u 2 U
n
for eah n. By the property
of hA
n
j n 2 !i, the length of stem(S
n
) is greater than n. Hene, by
Example 3.3, (u) is already deided to be x by S
n
s. The fat that
u 2 U
n
for eah n follows from the fat that S
n 2 A n D n and the
denition of D
n .
Sinewehaveassumedthat T n2! U n \ 1
(A)=;,xdoesnotbelong
to A beause x =(u)2 U
n
for eah n by Claim 3.6. Hene we have
seen B\A=; as wedesired.
(2) For left to right, we assume A is P-measurable. Then the set
D= fT 2Pj either [T℄\A2 I
P
or [T℄nA2I
P
g is dense. Then the
set U = S
fO
T
j T 2 Dg is open dense in St(P). Let U
1 = S fO T j
[T℄nA 2 I
P g, U 2 = S fO T
j [T℄\A 2 I
P
g. Then U = U
1 [U
2 . By
Lemma 2.13 (1), Lemma 3.1, Claim 3.6 (a), and (1) in this lemma,
U
1 \U
2
=;. Hene, it suÆes to show that U
1 n 1 (A), U 2 \ 1 (A)
are meager beause that willimplyU
1 4
1
(A)is meager.
WewillonlyseethatU
2 \
1
(A)ismeager. TheaseforU
1 n
1
(A)
beingmeagerissimilar. ByFat 3.7,itsuÆestoseethatO
T \
1
(A)
is meagerwhen[T℄\A2I
P
. Butif [T℄\A2I
P
,then O
T \ 1 (A) 1
([T℄\A) and 1
([T℄\A) ismeagerbyClaim3.6(a), Lemma2.13
(1), and (1) inthis lemma. Hene we are done.
Now we see the diretionfromright toleft. Assume 1
(A)has the
Baire property in St (P). Then there are open sets U
1 , U 2 suh that U 1 4 1 (A),U 2 4 1 ( !
!nA) aremeager. ByLemma3.1, U
1 \U 2 =; and U 1 [U 2
is open dense in St (P). Let D
i
= fT 2 P j O
T
U
i g
for i = 1;2. Then D
1 [ D
2
(2), it suÆes to prove that [T℄nA 2 I
P
for eah T in D
1
and that
[T℄\A 2I
P
for eah T in D
2 .
Weonlyprove[T℄nA2I
P
foreahT inD
1
. By(1)inthisLemma,it
isenoughtosee that 1
([T℄nA)ismeagerinSt(P). Butby Claim3.6
(b), 1
([T℄nA) is almostthe same as O
T n
1
(A). Sine T is inD
1 ,
by the denition of U
1 , O
T n
1
(A) is meager. This ompletes the
proof of (2).
Notethat if Psatises the ondition inLemma2.13 (5), then wedo
not need the properness of P for the proofs of Lemma3.5.
Now we are ready to prove Proposition 2.17 and Proposition 2.18.
Werst see the proof of Proposition2.18:
Proof of Proposition 2.18. First we see that the map i is well-dened,
i.e.,[T℄isnotinI
P
foreahT inP. IfitwereinI
P
,thenbyLemma3.5
(1), 1
([T℄) would be meager and O
T
1
([T℄) by Claim 3.6 (a).
Hene O
T
must be meager, whih ontradits Lemma 3.1. Therefore
[T℄is not in I
P
.
It islear that if T
1 T
2
,then i(T
1
)i(T
2
). To show the onverse,
assume T
1 T
2
and we prove that i(T
1
)i(T
2
). Sine T
1 T
2
, there
is a t 2T
1
whih is not in T
2
. By strong arborealness of P, (T
1 )
t 2 P
and [(T
1 )
t ℄ \[T
2
℄ = ;. Hene i((T
1 )
t
) i(T
2
). Sine (T
1 )
t
T
1 ,
i((T
1 )
t
)i(T
1
). Therefore, i(T
1
)i(T
2 ).
So it suÆes to see that i\P is dense in B=I
P
nf0g. Let B be a
Borelset whih isnot inI
P
. Wewillnd aT inP with[T℄nB 2I
P
.
SineeveryBorelsetisP-Baire,byLemma3.5(2),BisP-measurable.
Sine B is not in I
P
, there is a T suh that [T℄ nB 2 I
P
, hene
[T℄nB 2I
P
by Lemma2.13 (1), aswe desired.
Proof of Proposition 2.17. (1)Let =f
_ x
G
asintheproofofLemma3.5.
ByLemma 3.5,a set of realsA isinI
P
, i 1
(A)is meagerinSt (P).
Hene, it suÆes to showthat fj 1
(B
)is meagerg2 1
2 .
We willprove the following:
(?) 1
(B
) ismeager ()(8M 3) M:a .t.m. of ZFC
=) M \ 1
(B
)is meager"
:
Firstnotethattherighthandsidemakessensebeausethestatement
\P is a strongly arboreal foring" is 1
2
by the assumption that P is
provably 1
2
,sobydownwardabsoluteness,thisisalsotrueinM. Sine
the right hand side is 1
2
,it suÆes toshow the above equivalene.
Claim3.8. LetM beaountabletransitivemodelofZFCwith2M.
IfM \ 1
(B
)is meager", then forany T 2P M
(orP\M), thereis
a T 0
T suh that O
T 0
\ 1
(B
) ismeager inV.
Proof of Claim 3.8. Take any T in P M
. Sine P is provably 1
2 , P
M
,
M
and ? M
are subsets of P, and ? respetively. Hene, by strong
properness,there is a T 0
T suh that T 0
is (M;P)-generi.
We will show that T 0
satises the desired property. For that, we
willuse the unfolded Banah-Mazur game. Let U be a tree on !!,
reursiveinsuhthatB
=p[U℄holdsinanytransitivemodelofZFC
N with2N. ConsiderthefollowinggameG 0
: playerIandIIprodue
a dereasing sequene hS
n 0
T 0
j n 2 !i one by one and in addition,
playerII produes a real hy
n
jn 2!i. Player II wins if ((u);y)2 [U℄
for any u2 T
n2! O
Sn 0.
Notethat we may assumethat is dened for
any u 2 T
n2! O
Sn 0
and the value of only depends on the sequene
hS
n 0
jn 2!ibeausewe anarrange(u)= S
n2!
stem(S
n 0
)by strong
arborealness of P and Example 3.3.
Now it suÆes to show that playerII has a winningstrategy in this
game. Sine M \ 1
(B
) is meager", in M, player II has awinning
strategy in the game G whih is the same as G 0
exept that player
I an start from any ondition in P. The idea is to transfer to a
winning strategy for player II in G 0
in V. Instead of writing down a
winning strategy for player II in G 0
, we will desribe how to win the
game G 0
for player II asfollows:
I S
0 0
T 0
S
2 0
V
II (S
1 0
;y
0
) (S
3 0
;y
1 )
I S
0
S
2
M
II (S
1 ;y
0
) (S
3 ;y
1 )
We willonstrut sequenes hS
n
jn 2!i,hS
n 0
jn 2!i, hy
n
jn2!i
with the followingproperties:
hS
n 0
jn 2!i;hy
n
jn2!i
is arun in the game G 0
inV,
hS
n
jn2 !i;hy
n
jn2!i
isa run in the game G M
inV,
S
2n 0
is arbitrarilyhosenby playerI for eah n,
player II follows inG M
, and
S
2n+1 0
S
2n+1
for eah n.
Assuming we have onstruted the above sequenes, we prove that
player II wins in the game G 0
. First note that G M
((u);y)2 [U℄ for any u 2
n2! O
Sn
in V. But sine S
2n+1 S
2n+1
for eahn, ((u);y)2[U℄ for any u2 T n2! O S n 0
, heneplayer II wins
the game G 0
.
Wedesribehowtoonstrutthe abovesequenes. Suppose wehave
got h(S i 0 ;S i ;y i
)j i<2ni for some n. We willdeide S
2n 0 , S 2n+1 0 , S 2n , S 2n+1 and y n
. By the above properties, S
2n 0
is arbitrarily hosen by
playerI andS
2n+1 ,y
n
willbedeided bythe restand. Solet'sdeide
S 2n and S 2n+1 0 .
Let D be the set of all possible andidates for S
2n+1
by and the
previous play hS
i
ji <2ni;hy
i
ji <ni. Then in M, D is dense below
S
2n 1
(if it exists). Sine S
2n 0 S 2n 1 0 S 2n 1 and T 0
is (M;
P)-generi, D\M = D is predense belowS
2n 0
. Take an element from D
whih is ompatible with S
2n 0
and hoose S
2n
so that the element we
havetaken beomes S
2n+1
by and letS
2n+1 0
bea ommonextension
(inV)ofS
2n 0
andS
2n+1
. Thisnishestheonstrutionofthesequenes.
Claim3.8
Now letus prove the equivalene (?):
Suppose 1
(B
)ismeagerandassumethereisaountabletransitive
model M of ZFCwith 2M suhthat M \ 1
(B
) is not meager".
Wewillderiveaontradition. SineeveryBorelsetisP-Baire, 1
(B
)
has the Baire property. Hene there is a T 2 P M
suh that in M,
1
(B
) is omeager in O
T
. By Claim 3.6 (a), 1
([T℄nB
) is almost
inluded in O
T n
1
(B
), hene, in M, 1
([T℄n B
) is meager in
St(P). Now apply the laim for [T℄nB
. Then we get a T 0 T suh that O T 0 \ 1
([T℄nB
) is meager. But this meansthat O
T 0 isalmost inludedin 1 (B
). SineO
T 0
isnotmeagerbyLemma3.1, 1
(B
)is
not meager,whih ontraditsthe assumption that 1
(B
)is meager.
For the other diretion, by Fat 3.7, it suÆes to show that for any
T inP, thereisaT 0
T suhthat O
T 0 \ 1 (B
)ismeager. Sox any
T. Thenpik aountable transitivemodelM with ;T 2M. Thenby
Claim 3.8, there isa T 0
T suh that O
T 0 \
1
(B
)is meager,as we
desired.
(2) Let x be P-generi over M. Then the set G
x
= fT 2 P M
j x 2
[T℄g is a P M
-generi lter over M. We show that x 2= B
when is a
Borel ode in M with B
2I
P
.
Let be suh a Borelode. By (1)and the downward absoluteness
for 1
2
-formulas, M \B
2 I
P
". Let i M
be the dense embedding
fromP M to B=I P nf0g M
dened inProposition2.18 andi M
(G
x )
be the B=I
P
-generi lter over M indued by i M
and G
x
. Using
the fat that I
P
is a -ideal, it is routine to hek that B 2 i M
(G
i x 2 B for any Borel set B with a ode in M. But the left hand
sideof theaboveequivaleneimpliesM \B 2= I
P
",henebyupward
absoluteness for 1
2
-formulas, B 2= I
P
. Sine B
2 I
P
, x 2= B
as we
desired.
(3)Letx bea quasi-P-generireal over M and put G
x
=fT 2P M
j
x2[T℄g. Weshow that G
x is a P
M
-generilter over M.
We rst see that G
x
meets every maximal antihain of P M
in M.
Take any maximal antihain A of P M
in M. Sine P is provably ,
A is ountable in M. Nowonsider B = S
f[T℄jT 2Ag. ThenB is a
Borel set with a ode in M and M \ !
!nB 2 I
P
". By (1), this is
also true in V. Sine x is quasi-P-generi over M, x 2= B
, i.e., x is in
B. SoG
x
meets A.
Now we see that G
x
is alter. Take any two elements T
1 ;T
2 inG
x .
Wewillnd aommon extensionof T
1 , T
2 inG
x
. Consider D=fS 2
P j ([S℄\[T
1
℄ = ; and [S℄\ [T
2
℄ = ;)or (S T
1
and [S℄\ [T
2 ℄ =
;) or(S T
2
and [S℄\[T
2
℄ = ;) or(S T
1 ;T
2
)g in M. Then by
strong arborealness of P, D is dense in M. Hene G
x
meets D. Take
a ondition S from G
x
\D. Then only the last ase in D happens
beause S 2 G
x
() x 2 [S℄. Hene S T
1 ;T
2
. Therefore, G
x is a
P M
-generi lter over M.
There is a lose onnetion between foring absoluteness for P and
P-Baireness:
Theorem 3.9 (Castells). LetPbeapartialorder. Thenthe following
are equivalent:
(1) 1
3
-P-absoluteness holds, and
(2)every 1
2
-set of reals isP-Baire.
Proof. The argument is essentially the same as in [11, Theorem 3.1℄.
4. 1
3
-absoluteness
Now we givea preise statement of Theorem 1.3 and prove it. Also
we willprove related results.
Theorem 4.1. LetPbeastronglyarboreal,properforing. Then the
followingare equivalent:
(1) 1
3
-P-absoluteness holds, and
(2)every 1
2
-set of reals isP-measurable.
Proof. By Theorem 3.9, it suÆes to show that every 1
2
-set of reals
is P-measurable i every 1
is enough to see that every
2
-set of reals is P-Baire assuming every
1
2
-set of reals is P-measurable.
The followinglaim is the key point:
Claim 4.2. Let P be a strongly arboreal, proper foring and be a
P-name fora real. Then forany T in P, there is aT 0
T and aBorel
funtion g: [T 0
℄!R suh that T 0
=g(x_
G ).
Proof of Claim 4.2. This is a ombination of Proposition 2.18 in this
paperand [30, Proposition 2.3.1℄.
Nowtakeany 1
2
-setAandaBairemeasurablefuntionffromSt(P)
to the reals. We show that f 1
(A) has the Baire property. It suÆes
toshowthat fT jO
T \f
1
(A) ismeager orO
T nf
1
(A)is meagergis
dense inP.
SotakeanyT inPandwewillndanextensionSofT withtheabove
property. By the above laim, there is a T 0
T and a Borel funtion
g: [T 0
℄ ! R suh that T 0
f
=g(x_
G
), where
f
is the P-name for a
real dened in Lemma 3.2(1). Hene, by Lemma 3.2(3), f =gÆf
_ x
G
almost everywhere in O
T 0
. Sine g 1
(A) is 1
2
, it is P-measurable by
the assumption. ByLemma 3.5(2), f 1
_ x
G (g
1
(A))=(gÆf
_ x
G )
1
(A) has
the Baire property. Hene f 1
(A) has the Baire property in O
T 0. In
partiular, there is an S T 0
suh that either O
S \f
1
(A) is meager
or O
S nf
1
(A) is meageras we desired.
Theorem 4.3. Let P be a strongly arboreal, proper foring. Assume
the following:
fj isa Borel ode and B
2I
P
g2 1
2
: ()
Then the followingare equivalent:
(1) 1
3
-P-absoluteness holds,
(2)every 1
2
-set of reals is P-measurable, and
(3) for any real a and T 2 P, there isa quasi-P-generi real x2[T℄
over L[a℄.
Proof. Wehaveseentheequivalenebetween(1)and(2). Wewillshow
the diretionfrom (1) to (3)and the diretionfrom (3)to (2).
For(1)to(3),takearealaandT inP. Wewillndaquasi-P-generi
realxoverL[a℄withx2[T℄. Butbytheassumption(),thestatement
\Thereisaquasi-P-generirealxoverL[a℄withx2[T℄"is 1
3
andthis
is true in ageneri extension V[G℄with T 2 Gby the same argument
as in Proposition 2.17. (Although P might not be provably 1
2 as we
assumedinProposition2.17,weuseditonlytoseeM B
2I
P
when
B 2I
absolutenesswithoutusingPbeingprovably
2
.) Hene by
3
-foring
absoluteness, the statement isalsotrue in V aswe desired.
For (3) to (2), take any 1
2
-set A and we will show that A is
P-measurable. Take any T inP.
Case 1: ! L[a℄
1
<! V
1
for any real a.
Pik a real a with T 2L[a℄. Bythe assumption,the set of alldense
sets of P in L[a℄ isountable inV. Hene the set of all P-generi reals
over L[a℄ is of measure one w.r.t. I
P
, (i.e., the omplement of that set
is in I
P
). The rest is astandard Solovay argumentto prove regularity
properties in Solovay models. (Atually, every 1
2
-set of reals is
P-measurable inthis ase.)
Case 2: ! L[a℄
1
=! V
1
for some real a.
The argument is basially the same as in [7, Proposition 2.1℄. Pik
a real a with T 2L[a℄ and suh that ! L[a℄
1
=! V
1
and A is 1
2
(a). The
idea istodeompose [T℄\Aand [T℄nA intoBorelsets inan absolute
way between L[a℄ and V, and a Borelset ontaininga quasi-P-generi
real over L[a℄ must be I
P
-positive and below that Borel set we will
nd anextension of T asa witnessfor P-measurabilityof A.
Sine [T℄\A and [T℄nA are 1
2
(a) sets, there are Shoeneld trees
U
1
and U
2
in L[a℄ for [T℄ \A and [T℄nA respetively. From these
trees, we an naturally deompose [T℄\A and [T℄nA into !
1 -many
Borel sets as in [22, 2F.1-2F.3℄,i.e., there are sequenes h
j <!
1 i,
hd
j < !
1
i of Borel odes in L[a℄ suh that [T℄\A = S
<!1 B
and [T℄nA = S
<!
1 B
d
. The point is that the above equations are
absolutebetween L[a℄and V beausethosetwosequenes onlydepend
onU
1 ;U
2
and !
1
and ! L[a℄
1
=! V
1
as weassumed.
By assumption, there is a quasi-P-generi real x over L[a℄ with x 2
[T℄. Hene there is an < !
1
suh that either x 2 B
or x 2 B
d .
Withoutlossofgenerality,wemayassumex2B
. Sine
isinL[a℄,
by the denition of quasi-P-generiity, B
is not in I
P
. Sine every
Borelset isP-Baire,itisP-measurableby Lemma3.5(2). Hene there
isaonditionT 0
suh that[T 0
℄nB
2I
P
. SineB
[T℄\A,T 0
T
and [T 0
℄nA2I
P
,as we desired.
Theorem 4.4. Let P bea strongly arboreal, proper foring. Assume
fj isa Borel ode and B
2I
P
g2 1
2
; ()
and
I
P
is Borelgenerated orI
P =N
P
(1)every
2
-set of reals isP-measurable, and
(2)for any real a, R nfxjx is quasi-P-generiover L[a℄g2I
P
.
Proof. For (1) to (2), take any real a and we show that A = fx j
x is quasi-P-generiover L[a℄g is of measure one w.r.t. I
P
. Suppose
not. Then !
!nA2= I
P
. Bytheassumption(), !
!nAis 1
2
. Soby(1),
it isP-measurable. Hene there is a T inP suh that [T℄n( !
!nA)=
[T℄\A 2I
P
. We show that this annot happen.
Case 1: I
P
isBorelgenerated, i.e.,for any N inI
P
there isaBorelset
B 2I
P
suhthat N B.
Sine [T℄ \A 2 I
P
, there is a Borel set B [T℄ in I
P
suh that
[T℄\AB. Let be aBorel ode forB. ByTheorem 4.3, thereis a
quasi-P-generi real x over L[a;℄ with x2 [T℄. Sine B 2 I
P
, x 2= B.
But this is impossible beause x is also quasi-P-generi over L[a℄ and
hene x2[T℄\AB.
Case 2: I
P =N
P .
In this ase, [T℄\A is P-null, hene there is a T 0
T suh that
[T 0
℄\A = ;. By Theorem 4.3, there is a quasi-P-generi real x over
L[a℄ with x2[T 0
℄. Hene x2[T 0
℄\A, aontradition.
For(2)to (1),takeany 1
2
-set A. We showthat A isP-measurable.
Let T be in P. We will nd an extension T 0
of T approximating A as
in the denition of P-measurability. If [T℄\A2 I
P
, we are done. So
we assume[T℄\A2= I
P
.
Case 1: ! L[a℄
1
<! V
1
for any real a.
As in(3) to(2) in Theorem 4.3,in this ase, every 1
2
-set ofreals is
P-measurable by a standard Solovay argument.
Case 2: ! L[a℄
1
=! V
1
for some real a.
Let a be a real suh that [T℄\A is 1
2
(a) and ! L[a℄
1
=! V
1
. Then we
haveaShoeneldtree inL[a℄for [T℄\A and weget an!
1
-manyBorel
deompositionof [T℄\A into Borelsets fB
j <!
1
g with
2L[a℄
for eah as in the proof of Theorem 4.3. Sine [T℄\A 2= I
P
and
the set of quasi-P-generi reals over L[a℄ is of measure one w.r.t. I
P
by (2), there is a quasi-P-generi real x overL[a℄ with x2[T℄\A, so
there is an suh that x2B
.
The rest is the same as in the proof for (3) to (2) in Theorem 4.3.
Sine
2 L[a℄ and x is quasi-P-generi over L[a℄, B
= 2 I
P
. Sine
Borel set is P-measurable, there is a T 0
in P suh that [T 0
℄nB
2I
P .
Sine B
[T℄\A, T 0
T and [T 0
℄nA2I
P
5.
4
-absoluteness
It is natural to try to generalize the relationship up to the one
be-tween 1
4
-foringabsolutenessandtheregularitypropertiesfor 1
3 -sets
of reals and 1
3
-sets of reals. But these analogues annotbeproved in
ZFC. 9
In this setion,with anadditionalassumption (sharpsfor sets),
we willprove the analogues of x4.
Theorem 5.1. LetP beastronglyarboreal,proper, 1
2
foring.
Sup-pose every set has a sharp. Then either 1
2
-determinay holds or the
followingare equivalent:
(1) 1
4
-P-absoluteness holds, and
(2)every 1
3
-set of reals is P-measurable.
Proof. For(1)to(2), the argumentisthe same asfor (1)to(2) in[11,
Theorem 3.1℄. What we should hek is that we get the absolute tree
representation for 1
3
-sets between V and V P
. The rest is exatly the
same.
Forsuhtree representation, Feng-Magidor-Woodin usedShoeneld
treesfor 1
2
-sets. Withthe help ofsharpsfor sets, nowweuse
Martin-Solovay trees for 1
3
-sets. By [13, Theorem 2.1℄, it suÆes to see that
u V
2 =u
V P
2
for the absoluteness of Martin-Solovay trees between V and
V P
. Butthis istrue assumingeveryset hasasharpand Pbeingproper
by [25, Theorem 2.1.9,Example 3.2.7℄.
For (2) to (1), rst note that we may assume that every 1
3 -set is
P-Baire by the same argument for (2) to (1) in Theorem 4.1. The
argument isthe same as for in[11, Theorem 3.1℄. What we need isto
uniformizea 1
2
-relationby a 1
3
-funtion(in[11, Theorem3.1℄, F
eng-Magidor-Woodin uniformized a 1
1
-relation by a 1
2
-funtion). The
rest is exatlythe same. Butsuh uniformizationis possible assuming
the failureof 1
2
-determinay.
The author would like to thank Hugh Woodin for pointing out the
followingfat tohim:
Theorem 5.2. Suppose every real has a sharp. Then either 1
2
-determinay holds or 1
3
has the uniformization property, i.e., any
1
3
-relationan be uniformizedby a 1
3
-funtion. 10
Proof. ItsuÆes toshowthat every 1
2
-relationanbeuniformizedby
a 1
3
-funtion. Suppose 1
2
-determinay fails. Then there is a real a
0
9
Start from L and add !
1
-many Cohen reals, then in this model, 1
4
-foring
absoluteness for Cohen foring holds but there is a 1
2
-set of reals without the
Baireproperty.
10
Sine 1
2
-determinay impliesthat 1
3
has the uniformization property, this
fatstatesthedihotomyoftheuniformizationpropertyfor 1
and 1
suh that for eah real a
T a
0 ,
2
(a)-determinay fails, where
T is
the Turing order.
Case 1: for any real a
T a
0 ,a
y
exists.
In thisase, by the result ofSteel, K
a is
1
3
-orretfor any a
T a
0 ,
where K
a
isthe Mithell-Steelore model. 11
For eah a
T a
0
, let<
a
bethe anonial good 1
3
(a)-well-ordering
on the reals in K
a
. Given a real b and a 1
2
(b)-relation R , dene the
uniformization f as follows:
f(x)=y () y is the rst <
hx;a
0 ;bi
-element with (x;y)2R ,
where hx;a
0
;bi is the real oding x;a
0
and b. For eah x 2 dom(R ),
suha y alwaysexists beause K
hx;a
0 ;bi
is 1
3
-orret. Sof uniformizes
R and regarding the fat that <
a
is a good 1
3
(a)-well-ordering in K
a
for eaha
T a
0
, itis easyto see that f is 1
3 .
Case 2: there is areal a
T a
0
suh that a y
doesnot exist.
Thenthere isareal a
1
T a
0
suhthat forany real a
T a
1 ,a
y
does
not exist. By the result of Dodd-Jensen in [10℄, K
a is
1
3
-orret for
any a
T a
1
, whereK
a
isthe Dodd-Jensenore model. The restis the
same as Case 1.
Theorem 5.3. Let P be a strongly arboreal, strongly proper,
prov-ably 1
2
foring. Suppose every set has a sharp. Then either 1
2
-determinay holds orthe following are equivalent:
(1) 1
3
-P-absoluteness holds,
(2)every 1
3
-set of reals is P-measurable, and
(3) for any real a and any T 2 P, there is a quasi-P-generi real
x2[T℄over K
a
, where
K
a =
(
the Mithell-Steelore model if a y
exists,
the Dodd-Jensen ore model otherwise.
Proof. In Theorem 5.1, we have seen the equivalene between (1) and
(2). Weshowthe diretionfrom(1)to(3)andthe onefrom(3)to(2).
For (1) to (3), all we need is that the statement \there is a
quasi-P-generi real x over K
a
with x 2 [T℄" is 1
4
for eah real a and eah
11
In[28,Theorem7.9℄,Steelassumedtheexisteneoftwomeasurableardinals.
We an replae the lower measurable by a y
and the greater measurable by a y
#
.
(Reent development of inner model theory even allows one to omit this sharp.
T 2 P. But this is true by Proposition 2.17 (1)and the fat that the
set of reals in K
a is
1
3
(a) in V.
Theargumentfor(3)to(2)isbasiallythesameastheonein
Theo-rem4.3. Forsimpliity,weassumethefailureof 1
2
-determinay,hene
there is noinner modelwith a Woodin ardinal. The ase for the
fail-ure of 1
2
(a)-determinay for a real a an be dealt with in the same
way.
Case 1. ! K
a
1 <!
V
1
for any real a.
AsinTheorem 4.3,wean onlude thatevery 1
3
-setof reals(even
1
3
-setofreals) isP-measurableby using 1
3
-orretnessforK
a
. Tosee
1
3
-orretness for K
a
, we need the ase distintion whether a y
exists
or not. Ifa y
does not exist, this is due to Dodd-Jensen in [10℄. When
a y
exists, this is due toSteel. 11
Case 2. ! K
a
1 =!
V
1
for some real a.
We need the absolute deomposition of 1
3
-sets into Borel sets
be-tween K
a
and V for some real a. The following result is essential; its
proof was ommuniated to usby RalfShindler:
Theorem 5.4 (Shindler). Suppose there is no inner model with a
Woodin ardinal. Then if u Ka
2 <u
V
2
for any real a, then ! Ka
1 <!
V
1 for
any real a.
Proof. For simpliity, we only prove ! K
1 < !
V
1
assuming u Ka
2 < u
V
2 for
eah real a. To derive a ontradition, we assume ! K
1 = !
V
1
. The
followingis the rst point:
Claim5.5. ThemouseKj!
1
isuniversalforountablemie,i.e.,M
Kj!
1
forany ountable mouse M, where
is the mouse order.
Proof of Claim 5.5. Suppose thereisaountablemouseM withM >
Kj!
1
. Coiterate them and let T;U be the resulting trees for M and
Kj!
1
respetively.
Case 1: lh(T)is ountable.
Sine M >
Kj!
1
, U doesnot have a drop. But then the last model
ofU annotbeaninitialsegmentofthelastmodelofT sinethelength
of T is ountable, a ontradition.
Case 2: lh(T)is unountable.
SineM >
Kj!
1
,U doesnot haveadrop. IfU wasnon-trivial,then
the nal model of U would be non-sound and ould not be a proper
initialsegment of the nal model of T. Hene U is trivialand Kj!
1 is
aninitialsegmentof thenal modelof T. But thismeans !
1
isalimit
of ritial points of embeddings via T, hene !
1
is inaessible in K,
ontraditing the assumption ! K
=! V
By the same argument, we an prove that K
a j!
1
is universal for
ountablea-mie foreah real a. We now have twoases:
Case 1: There isa real a suh that a {
doesnot exist.
This asewastaken areof by Steeland Welh. In [27, Lemma3.6℄,
they assumed u
2 =!
2
, whih is stronger than u K a 2 <u V 2
for eah real
a, and proved there is a ountable mouse stronger than Kj!
1 w.r.t.
mouse order. But assuming! K
1 =!
V
1
and the non-existene of 0 {
, we
anrun theirsameargumentonlyassumingu K
2 <u
V
2
and getthe same
onlusion. Furthermore,wean easilyrelativize this argumenttoK
a .
Hene assuming ! K 1 = ! V 1 (even ! K a 1 = ! V 1
) and the non-existene of
a {
, if u Ka
2 < u
V
2
, then there is an a-mouse stronger than K
a j!
1 w.r.t.
mouse order,whih ontradits the a-relativized version of Claim 5.5.
Case 2: for any real a, a {
exists.
This ase is new. Sine u K
2 < u
V
2
, there is a real a suh that u K 2 < (! + 1 ) L[a℄
. The idea istouse a y
(that exists sine a {
exists) and linearly
iterateitwiththelowermeasure ina y
with length!
1
. Thenthe height
ofthelastmodelisbiggerthanu K 2 sineu K 2 <(! + 1 ) L[a℄
. Nowwerestrit
this linear iteration map to K in a y
onstruted up to the point with
the top measure. The point is this is an iteration map on it and the
nal model of this iteration has height bigger than u K
2
. Sine it is a
ountablemouse,byClaim5.5,wegetaountablemouseinKwiththe
sameproperty,whihyieldsaontraditionbyastandardboundedness
argument.
Wewilldisussthisideaindetail. Letibethelineariterationmapof
a y
derivedfromtheiteratedultrapowerstartingfromthelowermeasure
in it with length !
1
. Then the target N of i has height bigger than
u K 2 sine u K 2 < (! + 1 ) L[a℄
and the ritial point of i goes to !
1
and N
has a ardinal bigger than !
1
and a 2 N. Let K a
y
j be the K in a y
onstruted up to , the ritial pointof the top measure in a y
. Then
K a
y
jis a mouse and we allitM.
We laim that if we restrit i to M, then it is an iteration map on
M. Sine i is from a linear iteration of ultrapowers via measures, by
applyingtheresultofShindler[24℄ineahultrapowerintheiteration,
we an prove that the restrition of i to M is an iterationwith length
!
1
(whihitselfmightbequiteompliated). Moreover,thenalmodel
of this iteration has height greater than u K
2
beause i maps greater
orequal to(! +
1 )
L[a℄
. Letusallthe tree of this iterationT and letM
bethe -thiterate viaT and i T ; :M !M
bethe indued mapsfor
!
1 .
SineM isaountablemouse,byClaim5.5,thereisan
0 <! 1 suh that M
there is an iterationfrom Kj
0
with length !
1
suh that the height of
the nal modelisgreater thanu K
2
. (Notethat there mightbe adrop.)
Coiterate Kj
0
and M and let : M !N bethe resulted map. Note
that there is nodrop fromthe M-side beause M
Kj
0 .
We will onstrut hN
j !
1 i, h : M ! N
j !
1 i, and hi U ; : N !N
j !
1
i with the following properties:
(1)The diagrams belowall ommute,
(2)M N M +1
for eah ,
(3)N
is the diret limitof N
( < ) for limit , and
(4)i U
;+1 and
+1
aretheresulted mapsbythe omparisonbetween
N
and M
+1
for eah .
Kj 0 // /o /o /o
N =N
0 i U 0;1 // N 1 i U 1;2 // // N i U ;+1 // // N ! 1
M =M
0 = 0 OO i T 0;1 // M 1 1 OO i T 1;2 // // M OO i T ;+1 // // M ! 1 ! 1 OO
The above properties uniquely speify hN
j !
1 i, h : M ! N
j !
1
i, and hi U ; : N !N
j !
1
i. Hene it suÆes to
hek (1) and (2) above for this onstrution.
For(1), itsuÆes toshowthat i U ;+1 Æ = +1 Æi T ;+1
foreah .
By the Dodd-Jensen Lemma (e.g., in [32, Theorem 9.2.10℄), any two
simple iteration maps from a mouse to a mouse are the same. By (2)
for ,
, +1 , i T ;+1
,and i U
;+1
are allsimple iterationmaps. Hene
weget thedesiredommutativity. (2)followsfromthefatthatallthe
maps onstruted beforeare simpleiteration maps.
SinetheheightofN
!1
isgreaterthanorequaltothatofM
!1
,there
is aniteration fromKj
0
with length !
1
whose nal modelhas height
greaterthan u K
2
, aswedesired.
Sine Kj
0
is in K and
0
is ountable in K, there is a real x in K
oding Kj
0
. We show that the height of N
!
1
is less than (! +
1 )
L[x℄
. In
L[x℄, weollapse! V
1
withthe foringColl(!;! V
1
). Letg: !!! V
1 bea
generisurjetionoverL[x℄. SineKj
0
isodedbyxandthelengthof
iterationis! V
1
whih isountable witnessed by g, by the boundedness
lemma in L[x℄[g℄, the height of N
!1
is less than ! L[x℄[g℄ 1 = (! + 1 ) L[x℄ , as
desired. Sine xis in K, (! + 1 ) L[x℄ <u K 2
and hene the heightof N
!
1 is
less than u K
2
. But the height wasgreater than u K
2
. Contradition!
Now by the assumption and the above fat, there is a real a suh
that ! K a 1 = ! V 1 and u K a 2 = u V 2
. By [13, Theorem 2.1℄, the
Martin-Solovay trees for 1
-sets are absolute between K
a
trees are on !u
!
and u
!
is absolute between K
a
and V, we get the
absolute deomposition of 1
3
-sets into Borel sets between K
a
and V
as wedesired. The rest is exatlythe same asin Theorem 4.3.
Theorem5.6. LetPbeastronglyarboreal,stronglyproper,provably
1
2
foring. Suppose every set has a sharp. Assume
I
P
is Borelgenerated orI
P =N
P
: ()
Then either 1
2
-determinayholds or the followingare equivalent:
(1)Every 1
3
-set of reals isP-measurable, and
(2)foranyreal a,Rnfx jx is quasi-P-generi overK
a g2I
P
,where
K
a =
(
the Mithell-Steelore model if a y
exists,
the Dodd-Jensen ore model otherwise.
Proof. The argumentis exatlythe sameas Theorem 4.4by replaing
L[a℄ by K
a
and using the analogous fats about K
a
we have already
stated.
6. Appliations
In this setion,wemention two appliationsof our theorems to
par-tiular ases. One willbe proved here and the other is in[9℄.
Brendle-Halbeisen-Lowe [7℄proved the following:
Proposition 6.1 (Brendle-Halbeisen-Lowe). Let V be Silver foring.
Suppose for any real a there is a quasi-V -generi real over L[a℄. Then
every 1
2
-set of reals isV -measurable. 12
Question 6.2(Brendle-Halbeisen-Lowe). Doesthe onverseof
Propo-sition 6.1hold? 13
We answerthe abovequestion positively:
Proposition6.3. Assumeevery 1
2
-setofrealsisV -measurable. Then
for any real a, there is aquasi-V -generi real over L[a℄.
Proof. Sine Silver foring is strongly arboreal and proper, by
Theo-rem 4.3, itsuÆes toshowthat the set ofBorelodes with B
2I
V
is
1
2
. We use the followingfat:
Fat 6.4 (Zapletal). LetG bethe graph on !
2 onneting two binary
sequenes if they dier in exatly one plae. Let I be the -ideal
generatedby BorelG-invariantsets(i.e.,Borelsetsin !
2suhthat any
12
See[7, Proposition 2.1℄. Regarding I
V =N
V
, it is easy to hek that Silver
measurabilityin theirsense oinideswithourV-measurability.
