The core model induction in a choiceless context [Elektronische Ressource] / Daniel Busche

The core model induction in a choiceless context [Elektronische Ressource] / Daniel Busche

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Daniel BuscheThe Core Model Inductionin a Choiceless ContextOctober 2007The Core Model Inductionin a Choiceless ContextDaniel BuscheInauguraldissertation zur Erlangung des akademischen Gradeseines Doktors der Naturwissenschaften durchden Fachbereich Mathematik und Informatikder Westf¨alischen Wilhelms-Universit¨at Munster¨October 2007Dekan:Prof. Dr. Dr.h.c. Joachim CuntzGutachter:Prof. Dr. Ralf Schindler,Prof. Dr. John R. Steel (University of California, Berkeley)Tag der mundli¨ chen Pruf¨ ung:19.12.2007Tag der Promotion:19.12.2007CONTENTS1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i2. Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Some definitions and notations . . . . . . . . . . . . . . . . . . 12.2 Defining the model . . . . . . . . . . . . . . . . . . . . . . . . 222.3 The core model induction . . . . . . . . . . . . . . . . . . . . 263. Every uncountable successor cardinal is weakly compact . . . . . . 333.1 The projective case . . . . . . . . . . . . . . . . . . . . . . . . 333.2 The inadmissible cases . . . . . . . . . . . . . . . . . . . . . . 423.3 The end-of-gap cases . . . . . . . . . . . . . . . . . . . . . . . 634. Every uncountable cardinal is singular . . . . . . . . . . . . . . . . 854.1 The projective case . . . . . . . . . . . . . . . . . . . . . . . . 864.2 The inadmissible cases . . . . . . . . . . . . . . . . . . . . . . 884.3 The end-of-gap cases . . . . .

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Daniel Busche
The Core Model Induction
in a Choiceless Context
October 2007The Core Model Induction
in a Choiceless Context
Daniel Busche
Inauguraldissertation zur Erlangung des akademischen Grades
eines Doktors der Naturwissenschaften durch
den Fachbereich Mathematik und Informatik
der Westf¨alischen Wilhelms-Universit¨at Munster¨
October 2007Dekan:
Prof. Dr. Dr.h.c. Joachim Cuntz
Gutachter:
Prof. Dr. Ralf Schindler,
Prof. Dr. John R. Steel (University of California, Berkeley)
Tag der mundli¨ chen Pruf¨ ung:
19.12.2007
Tag der Promotion:
19.12.2007CONTENTS
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
2. Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2.1 Some definitions and notations . . . . . . . . . . . . . . . . . . 1
2.2 Defining the model . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 The core model induction . . . . . . . . . . . . . . . . . . . . 26
3. Every uncountable successor cardinal is weakly compact . . . . . . 33
3.1 The projective case . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 The inadmissible cases . . . . . . . . . . . . . . . . . . . . . . 42
3.3 The end-of-gap cases . . . . . . . . . . . . . . . . . . . . . . . 63
4. Every uncountable cardinal is singular . . . . . . . . . . . . . . . . 85
4.1 The projective case . . . . . . . . . . . . . . . . . . . . . . . . 86
4.2 The inadmissible cases . . . . . . . . . . . . . . . . . . . . . . 88
4.3 The end-of-gap cases . . . . . . . . . . . . . . . . . . . . . . . 931. INTRODUCTION
In this paper we suppose that the universe V is a model of ZF. We want to
show that the axiom of determinacy is consistent relative to the hypotheses
“each uncountable successor cardinal is weakly compact and each uncount-
able limit cardinal is singular” and “each uncountable cardinal is singular”,
respectively.
It is natural first to ask if these hypotheses themselves are consistent.
Moti Gitik has shown that it is possible that all uncountable cardinals are
singular. He proved this in [Git80]:
Theorem 1.1. If “ZFC+∀α∈On∃κ>α κ is a strongly compact cardinal”
is consistent, then “ZF+∀α∈On cof(ℵ ) =ℵ ” is consistent, too.α 0
Note that if each uncountable cardinal is singular, then the axiom of
choice is violated badly. In this case neitherDC norAC hold, because theseω
kinds of choice force ω to be regular. So it is clear that we demand that V1
is a model of ZF, rather than a model of ZFC.
In 1985 Gitik has generalized his result by showing in [Git85]:
Theorem1.2. If “ZFC+∃κκ is an almost huge cardinal” is consistent, then
there is a model M of ZFC such that for every class A of M consisting of
nonlimit ordinals there exists a model N of ZFC such that its regular alephsA
are exactly {ℵ :α∈A∪{0}}.α
Alsothehypothesis“everyuncountablesuccessorcardinalisweaklycom-
pact and each uncountable limit cardinal is singular” is relatively consistent.
For this we use a paper of Arthur W. Apter. Apter has generalized Gitik’s
method in [Apt85] in 1985 to get the following result:
Theorem 1.3. Suppose V is a model of ZFC such that
1. V|= “κ is a 3-huge cardinal” and
2. V|= “A andB are disjoint subsets of the successor ordinals<κ with
A∪B ={α<κ :α is a successor ordinal}.”ii 1. Introduction
Then there is a symmetric submodel N of a generic extension of V suchA
that N is a model of ZF+¬AC (in fact of ZF+¬AC ), the ordinals of NA ω A
have height κ, and
1. for α∈A, N |= “ℵ is a Ramsey cardinal”,A α
2. for β ∈ B, N |= “ℵ is a singular Rowbottom cardinal which carriesA β
a Rowbottom filter”, and
3. for γ a limit ordinal, N |= “ℵ is a J´onsson cardinal which carries aA γ
J´onsson filter”.
Moreover, it follows from the construction of N that all limit cardinalsA
are singular, so all uncountable regular cardinals inN are Ramsey carA
in N .A
If we use this theorem with A ={α < κ : α is a successor ordinal} and
B =∅, then we get a model N of ZF which satisfiesA
1. every uncountable successor cardinal is a Ramsey cardinal and
2. every uncountable limit cardinal is singular.
Soweevenhavethatthereisamodelof“ZF+eachuncountablesuccessor
cardinalisRamsey”ratherthanamodelof“ZF+eachuncountablesuccessorl is weakly compact”, but since we only need weak compactness we
don’t demand more.
Our first result will be
Theorem 1.4. Let V be a model of ZF. Suppose that each uncountable
successor cardinal is weakly compact and each uncountable limit cardinal is
singular.
L(R)Then there is a cardinal μ and a set of ordinals X such that AD holds
V+Col(ω,<μ )
in HOD .X
and in Chapter 4 we will show the corresponding theorem
Theorem 1.5. Let V be a model of ZF in which each uncountable cardinal
is singular.
L(R)Then there is a cardinal μ and a set of ordinals X such that AD holds
HOD+ XCol(ω,<μ )
in HOD .Xiii
V+Remark. Note that we use the L´evy collapse ofμ in Theorem 1.4 whereas
HOD+ Xwe use the L´evy-collapse of μ in Theorem 1.5. In both cases the set
X is just a technical addition which ensures that certain structures do not
compute the cardinal successor of a special cardinal κ correctly.
Theorem 1.5 answers a question which arises from a corollary of the main
theoremApterprovedin[Apt96]. Hederived“ZF+AD”fromtheconsistency
of “ZF + each uncountable cardinal belowθ is singular”, whereθ is the least
ordinal onto which the set of reals cannot be mapped. So it is natural to
ask if it is possible to find a model of “ZF + each uncountable cardinal is
singular” from “ZF +AD”. The question has a negative answer, because
if V were a model of “ZF + each uncountable cardinal is singular”, then
the proof of our theorem would show that each set in HOD , and even inX
HOD+ Xthe generic extension HOD [g], where g is Col(ω,<μ )-generic, wouldX
] HOD [g]Xhave a sharp; in particular R would exist for R := R . But then
there would be an α such that J (R)≺L(R). Since L(R)|=AD, we wouldα
therefore have J (R) |= AD. But J (R) has set size, so in the end, if theα α
answer of the question were “yes”, we would have that “ZF+AD” implies
Con(ZF+AD), which contradicts G¨odel’s second incompleteness theorem.
We will use the so-called core model induction to prove our theorems.
The core model induction, developed by W. Hugh Woodin and enhanced by
John R. Steel, is an induction along theJ (R)-hierarchy ofL(R). The goal isα
to show that at each stage α the axiom of determinacy holds true in J (R),α
L(R)
i.e. J (R)|=AD, so that in the end one gets AD .α
In the induction we don’t show explicitly that J (R) is a model of theα
axiom of determinacy, we rather show by induction that for allα a condition
?called (W ) holds. This condition demands that if there is a set of reals Uα
such that there are scales on U and R\U, whose associated sequences of
prewellorderings are both inJ (R), then there are structures, called Woodinα
mice, which are “correct” for that level of the J (R)-hierarchy, i.e. the ex-α
istence of these mice ensures that J (R) satisfies AD. We use Steel’s coreα
model theory to build these mice.
?Since (W ) only mentions sets of reals such that there are sequences ofα
?prewellorderingsinJ (R)comingfromascale, weonlyneedtoprove(W )α α+1
for thoseα for which there are a setU ⊆R and scales onU andR\U whose
associated sequences of prewellorderings are new in J (R), i.e. there is noα+1
scale (resp. no sequence of associated prewellorderings) on U in J (R). Weα
?call such an ordinal critical. Ifα is not critical, then (W ) follows trivially.α+1iv 1. Introduction
Descriptivesettheoryisusedtohandlethesecriticalordinals. Forthiswe
L(R)introduce the concept of Σ -gaps. First letθ beθ computed in L(R), i.e.1
the least ordinalγ such that there is no surjectionf:R→γ withf ∈L(R).
A Skolem hull argument then yields:
L(R)θ is the least γ such that Pow(R)∩L(R)⊆J (R)γ
? L(R)Soweonlyneedtoprove(W )forα<θ ,sinceeachsubsetoftherealsα
L(R)inL(R) appears beforeθ . Now a Σ -gap is a maximal interval [α,β] such1
that J (R) is a Σ -substructure of J (R) for statements with parameters inα 1 β
L(R)
R. One can show that these gaps partition θ . It follows that each scale
(resp.theassociatedsequenceofprewellorderings),showsupwithinaΣ -gap1
[α,β]. In [Ste83] John R. Steel has analyzed precisely at which levelsα there
are new scales. For this he used the concept of Σ -gaps.1
It turns out that the induction consists of various cases. The base for
? ? 1the induction is the case (W ) ⇒ (W ). Thus, in this case we show that0 1
? ?projectivedeterminacyholds. Thekindofproving(W )⇒ (W )forα> 0α α+1
depends on the sort of α:
21. αbeginsaΣ -gap,isR-inadmissible andsuccessorofacriticalordinal.1
2. α begins a Σ -gap, isR and has uncountable cofinality.1
3. α begins a Σ -gap, isR-inadmissible and has countable cofinality.1
4. α begins a Σ -gap, is R and successor of a non-critical1
ordinal.
5. α ends a weak Σ -gap.1
The difference in handling these cases is that in the first three cases we
work with ordinary premice as for example introduced in [Steb]. The cases
fourandfivearedifferent,sinceweworkwithso-calledhybrid premice,which
are premice with an additional predicate for some iteration strategy.
Our paper follows [Ste05], in which John R. Steel uses the core model
induction to show the following result [Ste05, Theorem 0.1]:
Theorem 1.6. Suppose there is a singular strong limit cardinal κ such that
fails; then AD holds in L(R).κ
1 ?(W ) holds trivially.0
2
R-admissibility is just the translation of the concept of admissibility from the L- to
the L(R)-context.