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.