Proper forcing
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| Format: | Buch |
| Sprache: | English |
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Berlin [u.a.]
Springer
1982
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| Schriftenreihe: | Lecture notes in mathematics
940 |
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| 100 | 1 | |a Shelah, Saharon |d 1945- |e Verfasser |0 (DE-588)120062755 |4 aut | |
| 245 | 1 | 0 | |a Proper forcing |c Saharon Shelah |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1982 | |
| 300 | |a XXIX, 496 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Lecture notes in mathematics |v 940 | |
| 650 | 4 | |a Forcing (Théorie des modèles) | |
| 650 | 7 | |a Forcing (modeltheorie) |2 gtt | |
| 650 | 7 | |a Modeltheorie |2 gtt | |
| 650 | 7 | |a Verzamelingen (wiskunde) |2 gtt | |
| 650 | 4 | |a Axiomatic set theory | |
| 650 | 4 | |a Forcing (Model theory) | |
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| 655 | 7 | |a Cohensche Zahl |2 gnd |9 rswk-swf | |
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| 830 | 0 | |a Lecture notes in mathematics |v 940 |w (DE-604)BV000676446 |9 940 | |
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Datensatz im Suchindex
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| _version_ | 1820806112060899328 |
| adam_text | ANNOTATED CONTENT
I FORCING. BASIC FACTS
§1. Introducing forcing 1
[We define generic sets, names for forcing a notion, and formulate
Cohen s theorems]
{2. The consistency of CH 9
[Our aim is to construct by forcing a model of ZFC were CH holds, first
we explain the problem of not collapsing cardinals, and second prove
that H, complete forcing notion does not add reals]
53. On the consistency H
[We construct a model of ZFC in which the Continuum Hypothesis fails;
define the c.c.c, prove that forcing with c.c.c. forcing preserves cardi¬
nalities and coflnalitiea, and prove also the A system lemma for finite
sets]
§4. More on the cardinality 2 ° and Cohen reals 21
[We construct for every cardinal X in V which satisfies X ° = X a model
V[G] such that V[G] (= 2M° = X. Also define Cohen reals]
SB. Equivalence of forcing notions, and canonical names 26
[Define when two forcing notions are equivalent. Introduce canonical
names and prove that for every P name t there is a canonical f name a
XIV
such that p t = a ]
§6. Random r^Mlfo «»oll»»p»niig nartHnalg miJ fWamnnrta 3
[Introduce random reals; Levy collapse. Prove that for regular
X d (Ko, X) satisfies the X c.c. For every uncountable regular X and a
stationary S C X define a forcing notion P which does not collapse S
such that Vp |= Q 1
D ITERATION OF FORCING
jl. The composition of two forcing notions 39
[Define composition of two notions and state the associativity lemma]
J2. Iterated forcing 44
[Define it. and prove that the c.c.c is preserved by FS iteration]
§3. Martin Axiom and few applications 48
[Prove that ZFC +2 $! + MA is consistent. Use MA to prove many
Simple uniformization properties]
54. The uniformization properties 57
[Here we deal with more general uniformization properties, we weaken
the demand of almost disjointness to a kind of tree]
}5. Maximal almost disjoint families of subsets of u 68
[A maximal almost disjoint (mad) subset of P(o) is a family of infinite
XV
subsets of a such that the intersection of any two members is finite and
maximal with this property. We prove using MA every mad set has cardi
M M
nality 2 °. The other direction: For every Hj is. X 2 ° there exists a
generic extension of V by c.c.c. forcing such that in it there exist mad
set of power X]
in PROPER FORCING
Jl. Introducing properness 73
[We define P is a proper forcing notion prove some definitions are
equivalent (and deal with the closed unbounded filter O (X))]
o
J2. More on properness 82
[We define p is (N.P) generic and deal more with equivalent
definitions of properness]
{3. Preservation of properness under CS iteration 90
[We prove the theorem mentioned in the title]
{4. Martin Axiom revisited 95
[We discuss the popularity of the c.c.c., whether we can replace it by a
more natural and weaker condition. We give a sufficient condition for a
CS iteration of length k to satisfy the k c.c. We prove the consistency
(assuming existence of an inaccessible cardinal) of ZFC + 2 =M, +
HA for forcing notions not destroying stationary subsets of Wi . We
show that the last demand cannot be replaced by not collapsing
XVI
cardinalities or cofinalities ]
100
§5. Aronszajn trees
[We define K Aronszajn, K Souslin. present existence theorems (for *
when = X**) and prove that under MA every Aronszajn tree is special]
§6. Maybe there is no «Z Aronazajn tree 104
u
[We prove the consistency of ZFC + 2 ° = M2 + there is no Hg Aronszajn
tree , the method collapsing successively all X,Hj X k (e a weakly
compact cardinal) treating every potential initial segment of an Hz
Aronszajn tree so that it cannot actually be so.]
(7. Closed unbounded sets otol can run away from many sets 109
[We prove the consistency of ZFC + 2 ° = Hz with if for i »,, A* C «i
has order type C Sup At, then for some closed unbounded C C «i , (V
i)[Sup (C C A) Sup At]
IV ON ORACLE C.C. AND /?(«) /FINITE HAS NO NON TRIVIAL AUTOMORPHISH
§0. Introduction 114
[The oracle c.c. method enables us to start with V ^ Q. extend the
set of reals o g times (by iterated forcing), in the intermediate stages
Q^ holds, and we omit types of power M, along the way.]
|1. On oracle chain condition 117
XVII
[One way to build forcing notions satisfying the Mj c.c, is by successive
countable approximations including promises to maintain the preden
sity of countably many subset, many times using the diamond, we for¬
malize a corresponding property (S c.c. M an oracle) and prove the
equivalence of some variants of the definition]
{2. The omitting type theorem 122
[We prove that if the intersection of H, Borel sets is empty and even if we
add a Cohen real it remains empty , (and O, ) then for some oracle
S, for every P satisfying the U c.c. in Vp the intersection of the Borel
Bets (reinterpreted) is still empty]
83 Iterations of jtf c.c 124
[We show that for Finite Support iteration Q =(Pt.Q :* asSt 2). if
Hi e V**1 ig an»roracle large enough for ((1^/^.9/) ¦] *). and Qt
satisfies the 54 c.c. then Pa = Lim Q satisfies the fio c.c. The first
three sections give the exact formulation of the aim stated in the intro¬
duction and prove that it works]
J4. Reduction of the Main Theorem to the Main Lemma 129
[We show how to apply the method described in {1 S3 in order to get a
model in which the Boolean algebra P(o)/ nite has no non trivial auto¬
morphism, i.e., one induces by permutations of o]
85. Proof of the Main Lemma 4.6 134
[The point missing in {4 is: if F is an automorphism of //(u)/finite, M an
XVIII
H^oracle, then there is forcing notion P satisfying the M c.c, and a P
name Y of a real such that in V**, for no I d, (V
A,B e/9(u)K)[x n A = B = Y n ^U) = ^ (#)] (even a Cohen forcing
does not introduce such a Y). We try to build such F.Kand prove that if
we always fail F is trivial]
V a PROFERNESS AND NOT ADDING REALS
{1. Iterations of forcing notions which does not add real 153
[We define what it means to be ^ complete e.g., if P c (Bl 2, ), E Qot
stationary and /„ C /B+i £ P, Sup (Dom /„) e E. We show that prop
erness + .F completeness are preserved by CS iteration and get
corresponding Axiom. Also introduce a form of MA which is consistent
with CH and prove using it a uniformization property which imply
existence of a non free Whitehead group]
52. (£ a,e) properness 162
[We introduce various variants of properness]
(3. a properness and (E,a) properness revisited 164
[We repeat the previous section in more details]
J4. Preservation of u properness + the ua bounding property 169
[P satisfies the u bounding property if [V/ e CI«)vP][3flr e ( ) ]
( A / (n) g (n)). We prove in great detail the theorem stated in the
XIX
title]
{5. What forcing can we iterate without adding reals 177
[We explain why not adding reals is not preserved by any kind of itera¬
tion, and suggest a remedy completeness]
§6. Specializing Aronszajn trees without adding reals 181
[We prove that every Aronszajn tree can be specialized by a nice forcing
o proper for every a G t and complete for some Hj completeness sys¬
tem together with the next section this gives a proof Gan(ZFC + Zk[k
inaccessble]) = Qm(ZFC + G.C.H. + SM) and with Chapter VHI a new
proof of Jensen s Con(ZFC + G.C.H. + SH).]
$7. Iteration of (£ l) complete forcing notions 189
[We prove the limit of a CS iteration of ft each is o proper for every
a £ ,, and complete for some simpler completeness]
VIP POINTS AND PRESERVATION THEOREM
Jl. A general preservation theorem 195
[We present a way to prove preservation of («,l) properness + x for
properties x restricting our set of reals. Our hope is that this frame¬
work is easy to be applied to many properties]
{2. Three known properties 203
[We prove that the a bounding property, the Sacks property and the
XX
Laver property comes under the framework of §1; the Sacks and
Laver property appear first and most characteristically in the forcing
notions bearing the respective names]
$3. PP ( P point) property 209
[We introduce a new property (PP) which conies under the framework of
jl, and some variants of it]
J4. There may be no P point 213
[We present another proof of this theorem, using the preservation of the
/V property. This may serve as a preliminary test, whether our general
machinery simplifies and clarifies proofs]
85. There may be a unique Ramsey ultrafllter 221
[The main result is the consistency of ZFC + 2 ° = M2 +¦ there is a
unique Ramsey ultrafllter on a up to permutations of a . For this we
have to prove that D generates a Ramsey ultrafilter is preserved by
another application of Jl, and of course to work on each iterand]
VH THE SEPTEMBER NOTES ON PROPER FORCING
51 On the Hg c.c 233
[When we iterate M2 times forcings not adding real, (but not necessarily
M, complete) we suggest a condition called H2 e.c.c. so that if each ft
satisfies the Mj e.c.c, then P^ satisfies the Hg cc]
XXI
§2. The axioms 236
[We suggest some axioms whose consistency follows from the theorem on
preservation under iteration of various properties.]
§3. Applications of Ax n 241
[We prove several applications of an axiom consistent with G.C.H.]
§4. Applications of Ax I 253
[We prove some applications and mention others of an axiom consistent
with 2*° = Mz.]
85. An Example 255
[An example is given of a countable support iteration of length a of forc¬
ing not collapsing stationary subsets of alt but the limit collapse M,]
Vm THE OCTOBER NOTES ON PROPER FORCING
81. Mixed Iteration 258
[We prove that we can iterate Kg complete forcings and Mi complete
forcings satisfying a strong M2 chain condition, without collapsing H, and
«*]
S3. Chains conditions revisited 262
¦ [We suggest another condition, e pic, to ensure the limit of the itera¬
tion PK satisfies the jc c.c. The aim is e.g., we start with
V |= 3*° = Hi a 2*1 Mg , and use CS iteration Q of length ug , each
XXII
time dealing with all problems (there are 2T1) at once]
(3. The Axioms Revisited 266
[We discuss what axioms we can get according to the four possibilities of
the truth of 2*°= H, . 2*l =HZ but assuming always 2M°*«2]
(4. More on forcings not aHiting u sequences and on the diagonal arguments .... 269
a
[We prove e.g., that CH does not imply ?. by dealing with 2 complete
systems]
KSOXJSLIN HYPOTHESIS DOES NOT IMPLY EVERY ARONSZAJN TREE IS SPECIAL
jl. Free Unit 278
[We look at Boolean algebras generated by a set of sentences in
inflnitary prepositional calculus (mainly LUl,u). This enables us to define
free limit]
§2. Preservation by free limit 281
[We prove that an iteration in which we use £ei,B free limit at limit
stages, preserve properness]
}3. Aronszajn trees: various ways to specialize 285
[We introduce some new ways to specialize Aronszajn trees, and present
the old ones, as well as the connection between those properties]
XXIII
§4 Independence results 291
[Here are the main results. We use an iterated forcing Sst specializing
any Aronszajn tree. The problem is to make sure that some fixed tree T
will remain not special. They introduce such a property of forcing
(7 *,S) preserving and show that is is preserved in iteration. There is a
discussion of the problem and our strategy in the beginning of the sec¬
tion and discussion of open problems and how can the preservation
theorem be generalized]
X SEMI PROPER FORCINGS
JO. Introduction 304
{1. Iterated forcing with RCS (rerised countable support) 304
[The standard countable support iteration cannot be spoiled when
cofinalities are changed to a, we introduce the revised version suitable
for this case.]
82. Proper forcing revisited 313
[We define semi properness, and prove that it is strongly preserved by
RCS iteration.]
B3. Pseudo completeness 320
[We prove that a weakening of Mi completeness is strongly preserved by
RCS iteration.]
XXIV
J4. Specific forcings 326
[Ye deal with Prikry forcing. Namba forcing and generalizations which
are semi proper when we use Galvin filter.]
(S. Chain conditions and Avraham s problem 335
[We prove that under reasonable conditions the ic c.c. holds and get its
first application: a universe V in which for every A Ca% there is a count¬
able subset of of which does not belong to L(A).]
|B. Reflection properties of S§. Refining Abraham s problem and precipitous 338
ideals
[For some large cardinal k, by iteration we find a forcing notion P, such
that V* |= k = M, and A = 6 k : cf 6 = Mo , S regular in V) is station¬
ary . So we may make A large in some sense, as mentioned in the title.]
(7. Strong preservation and propernesa 346
[We present some properties strongly preserved by RCS iteration; the
most important is a strengthening of not adding reals. This continues VI
II.]
JB. Friedman s problem 347
[We collapse some large c, by iterated forcing, which sometimes col¬
lapses (2 *)* to H,, sometimes change the cofinality of M2 to Ho, and
sometimes add a closed unbounded C C S of order type ax, where
S c Sif is stationary. We get a model V in which every stationary
S c S§ = 6 M2 : cf 6 =M0J contains a closed copy of Ui By stronger
hypothesis we get it for every stationary S Q S£. cf H. Mo]
XXV
XI CHANGING COF1NA1JTIES: EQUI CONSISTENCY RESULTS
Jl. The theorems 354
[Here we describe what kind of a condition on forcing notions we want.
Then we proceed to get consistency results. The proof uses RCS
iteration of length k, k a strongly inaccessible cardinal. In each step,
we allow Namba forcing. The consistency results are mostly from X but
here we use the minimal large cardinals required.]
52. The condition 359
[We describe here the conditions, and some helping definitions and con¬
ventions]
53. The preservation properties guaranteed by the S condition 362
[We prove that such a condition implies Hj is not collapsed, and (assum¬
ing CH) no real is added; and for it partitions theorems on trees]
|4. Forcing notions satisfying the S condition 366
[We show that Namba forcing, Nm satisfies the (H2} condition that Nm
and Nm are really different forcing notions that Nm, Nm may satisfy the
H4 C.C. (while 2 ° = Kj , 2 is large) We also prove Bi complete forcing
and a forcing notion shooting a closed unbounded subset of order type
«i through a stationary S c S§ satisfies our condition]
§5. Finite composition 372
[We prove that under suitable hypothesis, a composition of forcing satis¬
fying an S condition satisfies it. For this we prove a combinatorial
XXVI
theorem on trees]
§6. Preservation of the I condition by iteration 375
[Here we prove that if we iterate forcing notions satisfying our condi¬
tions, but enough times collapse the present 2|p| to Hi, the composite
forcing satisfies the condition. So usually we have large segments of car¬
dinals which we have to collapse by Mi complete forcings, but for
strongly inaccessible we can use Nm straight away (by 6.5)]
J7. Further independence results 388
[We prove the equlconsistency of ZFC + k Is Mahlo + H2 has the Fried¬
man property , and a further result using weakly compact cardinal. We
also prove the equiconsistency of ZFC + ic is 2 Mahlo and ZFC+ there
is the club of M2 consisting of regular cardinals of L]
XD IMPROPER FORCING
JO. Introduction 394
§1. When Namba forcing is semi proper, Chang s Conjecture and games 395
[We prove e.g., that if some {H^ semi proper forcing changes the
cofinality of H2 to a then Namba forcing is semi proper, and Chang s
Conjecture holds hence 0* e V.]
J2. Games and properness 400
[Equivalent definitions of variants of properness by games are given, and
XXVII
it is exemplified how the proofs of the preservation theorems in this
context look like]
§3. Amalgamating propemegs with the 5 condition 406
[We show how we can extend the results of the previous Chapter to more
forcing notions]
Xm THE STRONG COVERING LEMMA ND THE G.CJ1.
JO. Introduction 410
[Explanation of the history of the singular cardinals problem, and the
significance of the strong covering lemma and its relation to proper
forcing]
$1. The strong covering lemma: Definitions and implications 416
[Here we introduce the notions connected with the strong covering
lemma and notice some trivial connections. Note that if (ff,V) satisfies
the hypothesis of the strong covering lemma, then so does ( *, V) when¬
ever ircifcf]
J2. Proof of the strong covering lemma 420
[Here we prove the statement on games phrased above (and obviously
implying the strong covering lemma) by induction on a. We prove e.g.,
that if 0* ttL, then (L.V) satisfies the M3 covering lemma. This part is
somewhat harder than the rest of the paper.]
XXVIII
53. A counterexample 435
[We prove that we may extend L to V (by forcing) collapsing Mg only,
cl K(Mg) =Ki, so that the Kj covering lemma holds but the strong Hi
covering lemma fall.]
§4. When adding a real cannot destroy CH 437
[We deal with variants of can adding a real violate CH while preserving
cardinals , and prove that each implies an inner model with suitable
large cardinals. For getting sharper result we have to improve the
results of §2 getting e.g., that if A c az , A e V . H£W = M/ and 0* £ V
then (L[A].V) satisfies the strong X covering lemma for every X.]
85. Bound on 2* for Ma singular 444
[We give bound to (Ha)ct* in the way we deal with scales and cofinalities
of ultraproducts. This section can be read alone].
56. Concluding remarks and questions 453
[This section continues {0. We make some remarks giving some claims
we can prove but their value is not clear, and discussing the open ques¬
tions, and explain how to get simpler proofs for weaker theorems.]
XIV ON WEAK DIAMONDS AND THE P0WEK OF EXT
50. Introduction 461
Jl. Unif strong negation of the weak diamonds fR 463
XXIX
[Introduce a generalization, of the negation of the weak diamond (i.e.,
e MM
#u ) and prove it from an appropriate replacement of 2 ° 2 ].
§2. On the power of Ext and Whitehead problem 474
§3 Weak diamond for M2 assuming CH 485
[We prove that every ladder system 5 = (i)j:Je5f) when ij« is con¬
tinuous cannot be uniformized assuming 2T0 =Mi],
REFERENCES 492
|
| any_adam_object | 1 |
| author | Shelah, Saharon 1945- |
| author_GND | (DE-588)120062755 |
| author_facet | Shelah, Saharon 1945- |
| author_role | aut |
| author_sort | Shelah, Saharon 1945- |
| author_variant | s s ss |
| building | Verbundindex |
| bvnumber | BV000043967 |
| callnumber-first | Q - Science |
| callnumber-label | QA3 |
| callnumber-raw | QA3 |
| callnumber-search | QA3 |
| callnumber-sort | QA 13 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SI 850 |
| ctrlnum | (OCoLC)8890854 (DE-599)BVBBV000043967 |
| dewey-full | 510 511/.8 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 510 - Mathematics 511 - General principles of mathematics |
| dewey-raw | 510 511/.8 |
| dewey-search | 510 511/.8 |
| dewey-sort | 3510 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| genre | Cohensche Zahl gnd Cohensche Zahlen gnd |
| genre_facet | Cohensche Zahl Cohensche Zahlen |
| id | DE-604.BV000043967 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-23T09:51:35Z |
| institution | BVB |
| isbn | 3540115935 0387115935 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-000012808 |
| oclc_num | 8890854 |
| open_access_boolean | |
| owner | DE-12 DE-91G DE-BY-TUM DE-384 DE-355 DE-BY-UBR DE-20 DE-824 DE-29T DE-19 DE-BY-UBM DE-706 DE-83 DE-11 DE-188 |
| owner_facet | DE-12 DE-91G DE-BY-TUM DE-384 DE-355 DE-BY-UBR DE-20 DE-824 DE-29T DE-19 DE-BY-UBM DE-706 DE-83 DE-11 DE-188 |
| physical | XXIX, 496 S. |
| publishDate | 1982 |
| publishDateSearch | 1982 |
| publishDateSort | 1982 |
| publisher | Springer |
| record_format | marc |
| series | Lecture notes in mathematics |
| series2 | Lecture notes in mathematics |
| spellingShingle | Shelah, Saharon 1945- Proper forcing Lecture notes in mathematics Forcing (Théorie des modèles) Forcing (modeltheorie) gtt Modeltheorie gtt Verzamelingen (wiskunde) gtt Axiomatic set theory Forcing (Model theory) Mengenlehre (DE-588)4074715-3 gnd Forcing (DE-588)4154978-8 gnd Kardinalzahl (DE-588)4163318-0 gnd |
| subject_GND | (DE-588)4074715-3 (DE-588)4154978-8 (DE-588)4163318-0 |
| title | Proper forcing |
| title_auth | Proper forcing |
| title_exact_search | Proper forcing |
| title_full | Proper forcing Saharon Shelah |
| title_fullStr | Proper forcing Saharon Shelah |
| title_full_unstemmed | Proper forcing Saharon Shelah |
| title_short | Proper forcing |
| title_sort | proper forcing |
| topic | Forcing (Théorie des modèles) Forcing (modeltheorie) gtt Modeltheorie gtt Verzamelingen (wiskunde) gtt Axiomatic set theory Forcing (Model theory) Mengenlehre (DE-588)4074715-3 gnd Forcing (DE-588)4154978-8 gnd Kardinalzahl (DE-588)4163318-0 gnd |
| topic_facet | Forcing (Théorie des modèles) Forcing (modeltheorie) Modeltheorie Verzamelingen (wiskunde) Axiomatic set theory Forcing (Model theory) Mengenlehre Forcing Kardinalzahl Cohensche Zahl Cohensche Zahlen |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000012808&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000676446 |
| work_keys_str_mv | AT shelahsaharon properforcing |