Post on 19-Feb-2020
IntroductionModel Existence SpectraCoherent Aronszajn Trees
Kurepa TreesAmalgamation Spectra
Statue of Aristotle at the
Aristotle University of
Thessaloniki, Greece
I. Souldatos absoluteness of amalgamation
IntroductionModel Existence SpectraCoherent Aronszajn Trees
Kurepa TreesAmalgamation Spectra
Is the amalgamation property for Lω1,ω-sentencesabsolute for transitive models of ZFC?
Logic Seminar - UIC
2018 02 01
Ioannis (Yiannis) Souldatos
I. Souldatos absoluteness of amalgamation
IntroductionModel Existence SpectraCoherent Aronszajn Trees
Kurepa TreesAmalgamation Spectra
Lω1,ωSpectraMain Questions
This is joint work with D. Sinapova and joint work with D. Milovich.
I. Souldatos absoluteness of amalgamation
IntroductionModel Existence SpectraCoherent Aronszajn Trees
Kurepa TreesAmalgamation Spectra
Lω1,ωSpectraMain Questions
Preliminaries
De�nition
Lω1,ω = �rst-order formulas +∨
n∈ω φn +∧
n∈ω φn
Lω1,ω satis�es the Downward Lowenheim-Skolem Theorem,
but fails the Upward Lowenheim-Skolem.
I. Souldatos absoluteness of amalgamation
IntroductionModel Existence SpectraCoherent Aronszajn Trees
Kurepa TreesAmalgamation Spectra
Lω1,ωSpectraMain Questions
Spectra
De�nition
For an Lω1,ω-sentence φ, de�ne the properties
Model-existence at κ (ME(κ) for short) for �φ has a model of
size κ�
Amalgamation at κ (AP(κ) for short) for �ME(κ) + the
models of φ of size κ satisfy amalgamation�
The model existence spectrum of φ,ME-Spec(φ) = {κ|ME (κ)}The amalgamation spectrum of φ, AP-Spec(φ) = {κ|AP(κ)}
I. Souldatos absoluteness of amalgamation
IntroductionModel Existence SpectraCoherent Aronszajn Trees
Kurepa TreesAmalgamation Spectra
Lω1,ωSpectraMain Questions
Very Important Remark
The amalgamation notion is not unique. One needs to specify the
embedding relation.
For the rest of the talk we assume that the embedding relation ≺satis�es the following axioms (for an Abstract Elementary Class)
I. Souldatos absoluteness of amalgamation
IntroductionModel Existence SpectraCoherent Aronszajn Trees
Kurepa TreesAmalgamation Spectra
Lω1,ωSpectraMain Questions
Assume A,B,C are models of some φ.
1 A ≺ B implies A ⊂ B ;
2 ≺ is re�exive and transitive;3 If 〈Ai |i ∈ κ〉 is an increasing ≺- chain, then
(i)⋃
i∈IAi is a model of φ;
(ii) for each i ∈ I , Ai ≺⋃
i∈IAi ; and
(iii) if for each i ∈ I , Ai ≺ M, then⋃
i∈IAi ≺ M.
4 If A ≺ B , B ≺ C and A ⊂ B , then A ≺ B .
5 There is a Lowenheim-Skolem number LS such that if A ⊂ B
and B satis�es φ, then there is some A′ which satis�es φ,A ⊂ A′ ≺ B , and |A′| ≤ |A|+ LS .
I. Souldatos absoluteness of amalgamation
IntroductionModel Existence SpectraCoherent Aronszajn Trees
Kurepa TreesAmalgamation Spectra
Lω1,ωSpectraMain Questions
Main Questions
1 What is known for the model existence and amalgamation
spectra of Lω1,ω-sentences?2 What is absolute (for transitive models of ZFC)?
I. Souldatos absoluteness of amalgamation
IntroductionModel Existence SpectraCoherent Aronszajn Trees
Kurepa TreesAmalgamation Spectra
ClustersAbsoluteness
Observation
By Downward Lowenheim-Skolem, ME-Spec(φ) is downward
closed.
So, either ME-Spec(φ)=[ℵ0,ℵα] (right-closed spectrum) or
ME-Spec(φ)=[ℵ0,ℵα) (right-open spectrum)
De�nition
If ME-Spec(φ)=[ℵ0,ℵα], then φ characterizes ℵα.
I. Souldatos absoluteness of amalgamation
IntroductionModel Existence SpectraCoherent Aronszajn Trees
Kurepa TreesAmalgamation Spectra
ClustersAbsoluteness
Hanf Number
Theorem (Morley, López-Escobar)
The Hanf number for model-existence of Lω1,ω sentences is iω1 .
So, if φ has models of size iω1 , then φ has models in all
cardinalities.
Corollary
If ℵα is characterized by an Lω1,ω-sentence, then ℵα < iω1 .
I. Souldatos absoluteness of amalgamation
IntroductionModel Existence SpectraCoherent Aronszajn Trees
Kurepa TreesAmalgamation Spectra
ClustersAbsoluteness
Alephs
Theorem (Hjorth)
For every α < ω1, there exists an Lω1,ω-sentence φα that
characterizes ℵα.
Hjorth's theorem generalizes to
Theorem
Assume ℵβ is characterized by φβ . Then for every α < ω1, there
exists an Lω1,ω-sentence φβα that characterizes ℵβ+α.
I. Souldatos absoluteness of amalgamation
IntroductionModel Existence SpectraCoherent Aronszajn Trees
Kurepa TreesAmalgamation Spectra
ClustersAbsoluteness
ℵ0�� ��
ℵω1
�� �� �� �� ��iω1
clusters of characterizable cardinalsof length ω1
��������9��
HHHHj . . .
I. Souldatos absoluteness of amalgamation
IntroductionModel Existence SpectraCoherent Aronszajn Trees
Kurepa TreesAmalgamation Spectra
ClustersAbsoluteness
ℵ0�� ��
ℵω1
�� �� �� �� ��iω1
clusters of characterizable cardinalsof length ω1
��������9��
HHHHj . . .
Under GCH there is only one cluster.
I. Souldatos absoluteness of amalgamation
IntroductionModel Existence SpectraCoherent Aronszajn Trees
Kurepa TreesAmalgamation Spectra
ClustersAbsoluteness
ℵ0�� ��
ℵω1
�� �� �� �� ��iω1
clusters of characterizable cardinalsof length ω1
��������9��
HHHHj . . .
Under GCH there is only one cluster.
Under ¬GCH, it is consistent that there exist
non-characterizable cardinals below iω1 .
I. Souldatos absoluteness of amalgamation
IntroductionModel Existence SpectraCoherent Aronszajn Trees
Kurepa TreesAmalgamation Spectra
ClustersAbsoluteness
Absoluteness
Question
What is absolute?
Hjorth's theorem is in ZFC.
By Shoen�eld's absoluteness, model-existence in ℵ0 is absolute.
I. Souldatos absoluteness of amalgamation
IntroductionModel Existence SpectraCoherent Aronszajn Trees
Kurepa TreesAmalgamation Spectra
ClustersAbsoluteness
Theorem (Grossberg-Shelah)
The property that an Lω1,ω-sentence has arbitrarily large models is
absolute.
Corollary
Model-existence in any power ≥ iω1 is absolute.
I. Souldatos absoluteness of amalgamation
IntroductionModel Existence SpectraCoherent Aronszajn Trees
Kurepa TreesAmalgamation Spectra
ClustersAbsoluteness
Theorem (S.Friedman, Hyttinen, Koerwien)
1 Model-existence in ℵ1 is absolute.
2 Model-existence in ℵα, 1 < α, is not absolute.
In order to prove (2), consider the sentence that describes a binary
tree of height ω. It has models of size up to 2ℵ0 . Manipulating the
size of 2ℵ0 in various models of ZFC, we prove non-absoluteness of
model-existence in ℵα.
Question
What happens if we assume GCH?
I. Souldatos absoluteness of amalgamation
IntroductionModel Existence SpectraCoherent Aronszajn Trees
Kurepa TreesAmalgamation Spectra
ClustersAbsoluteness
Theorem (S.Friedman, Hyttinen, Koerwien)
Assuming the existence of uncountably many inaccessible cardinals,
model-existence in ℵα+2, α < ω1, is not absolute for models of
ZFC+GCH.
Proof (sketch)
There exists some Lω1,ω-sentence φα+2 that has a model of size
ℵα+2 i� there is an ℵα+1-Kurepa tree. Collapsing the inaccessibles
destroys all Kurepa trees, while maintaining GCH.
I. Souldatos absoluteness of amalgamation
IntroductionModel Existence SpectraCoherent Aronszajn Trees
Kurepa TreesAmalgamation Spectra
ClustersAbsoluteness
Theorem (S.Friedman, Hyttinen, Koerwien)
Let α < ω1 be a limit ordinal. Assuming the existence of a
supercompact cardinal, model-existence in ℵα+1 is not
absolute for models of ZFC+GCH.
Let α < ω1 be a limit ordinal greater than ω. Assuming the
existence of a supercompact cardinal, model-existence in ℵα is
not absolute for models of ZFC+GCH.
Proof (sketch)
There exists some Lω1,ω-sentence φα+1 that has a model of size
ℵα+1 i� there is a special ℵα+1-Aronszajn tree. Assuming a
supercompact cardinal, there is a forcing extension in which GCH is
true, but there are no special ℵα+1-Aronszajn trees.
I. Souldatos absoluteness of amalgamation
IntroductionModel Existence SpectraCoherent Aronszajn Trees
Kurepa TreesAmalgamation Spectra
ClustersAbsoluteness
The previous theorem leaves the question for ℵω open.
Theorem (D. Milovich,S.)
Model-existence in ℵω is not absolute for models of ZFC+GCH
The idea is to use coherent special Aronszajn trees.
I. Souldatos absoluteness of amalgamation
IntroductionModel Existence SpectraCoherent Aronszajn Trees
Kurepa TreesAmalgamation Spectra
Model ExistenceAmalgamation
De�nition
A κ-Aronszajn tree is a κ-tree with no branch of length κ.
A κ+-tree is special if it is the union of κ-many of its
antichains.
=∗ means equality of sets modulo a �nite set.
A tree of functions is coherent if for every s, t withdom(s) = dom(t), s =∗ t.
I. Souldatos absoluteness of amalgamation
IntroductionModel Existence SpectraCoherent Aronszajn Trees
Kurepa TreesAmalgamation Spectra
Model ExistenceAmalgamation
It follows from the de�nition that a coherent tree can not contain a
copy of 2≤ω.
Theorem (Konig)
If �λ holds, then there is a coherent special λ+-Aronszajn tree.
Theorem (Todor£evi¢)
After Levy collapsing a Mahlo to ω2, every special ℵ2-Aronszajntree contains a copy of 2<ω1 .
I. Souldatos absoluteness of amalgamation
IntroductionModel Existence SpectraCoherent Aronszajn Trees
Kurepa TreesAmalgamation Spectra
Model ExistenceAmalgamation
A drawback in working with Lω1,ω is that we can not characterize
well-orderings.
So, instead of working with (well-founded) trees, we have to work
with pseudo-trees.
De�nition
A pseudotree is a partial ordered set T such that each set of the
form ↓ x = {y |y <T x} is linearly ordered.
I. Souldatos absoluteness of amalgamation
IntroductionModel Existence SpectraCoherent Aronszajn Trees
Kurepa TreesAmalgamation Spectra
Model ExistenceAmalgamation
Theorem (Milovich, S.)
Given 1 ≤ α < ω1, there is an Lω1,ω-sentence φα satisfying the
following.
1 There is no model of φα of size greater than ℵα.2 If there is a coherent special ℵβ-Aronszajn tree for each
β < α, then φα has a model of size ℵα.3 If φα has a model of size ≥ ℵ2, then there is a coherent
pseudotree T with co�nality ω2 and an order embedding of a
special ℵ2-Aronszajn tree into T .
I. Souldatos absoluteness of amalgamation
IntroductionModel Existence SpectraCoherent Aronszajn Trees
Kurepa TreesAmalgamation Spectra
Model ExistenceAmalgamation
Corollary
Let 2 ≤ α < ω1.
In L, the sentence φα characterizes ℵα.After collapsing a Mahlo to ω, φα has no model of size ℵ2.
Corollary
Assuming a Mahlo cardinal, model-existence in ℵα, 2 ≤ α < ω1, isnot absolute for models of ZFC+GCH.
This not only answers the open question for α = ω, but it alsoimproves the large cardinal requirement from a supercompact to a
Mahlo.
I. Souldatos absoluteness of amalgamation
IntroductionModel Existence SpectraCoherent Aronszajn Trees
Kurepa TreesAmalgamation Spectra
Model ExistenceAmalgamation
Amalgamation
Consider the collection of all models of φα with the substructure
relation.
Theorem (Milovich, S.)
Assume 2 ≤ α < ω. If there are models of φα of size ℵα, thenAP-Spec(φα) = {ℵα}. Otherwise AP-Spec(φα) is empty.
Corollary
The following statements are not absolute for transitive models of
ZFC+GCH.
(a) The amalgamation spectrum of an Lω1,ω-sentence is empty.
(b) For �nite n ≥ 2 and φ an Lω1,ω-sentence, ℵn ∈ AP-Spec(φ).
I. Souldatos absoluteness of amalgamation
IntroductionModel Existence SpectraCoherent Aronszajn Trees
Kurepa TreesAmalgamation Spectra
Model ExistenceAmalgamation
Remark
The above examples can not be used to resolve the absoluteness
question of ℵα-amalgamation, for ω ≤ α < ω1. The reason is the
following lemma.
Lemma
Assume ω ≤ α < ω1. The amalgamation spectrum of φα is empty.
The question for ℵ1-amalgamation remains open.
Conjecture
If φ is an Lω1,ω-sentence and F is the fragment generated by φ,then ℵ1-amalgamation under ≺F is absolute for transitive models
of ZFC.
I. Souldatos absoluteness of amalgamation
IntroductionModel Existence SpectraCoherent Aronszajn Trees
Kurepa TreesAmalgamation Spectra
Closure PropertiesConsistency Results
Closure Properties
Theorem
The set of cardinals characterized by Lω1,ω-sentences is closedunder
1 successor;
2 countable unions;
3 countable products;
4 powerset.
Many of the results are true even for the set of cardinals
characterized by complete Lω1,ω-sentences.
I. Souldatos absoluteness of amalgamation
IntroductionModel Existence SpectraCoherent Aronszajn Trees
Kurepa TreesAmalgamation Spectra
Closure PropertiesConsistency Results
Let C be the least set of cardinals that contains ℵ0 and is closed
under (1)-(4), e.g. successor, countable unions, countable products,
and powerset.
Theorem (S.)
C is also closed for powers
Question (S.)
Does C contain all characterizable cardinals?
Consistently yes, e.g. under GCH.
Theorem (Sinapova,S.)
Consistently no.
I. Souldatos absoluteness of amalgamation
IntroductionModel Existence SpectraCoherent Aronszajn Trees
Kurepa TreesAmalgamation Spectra
Closure PropertiesConsistency Results
There exists an Lω1,ω-sentence φ that characterizes the
maximum of 2ℵ0 and B.
B = sup{κ|there exists a Kurepa tree
withκ many branches}.
Reminder: A Kurepa tree has height ω1, countable levels and
more than ℵ1 many branches.
So, ℵ1 < B ≤ 2ℵ1 .
I. Souldatos absoluteness of amalgamation
IntroductionModel Existence SpectraCoherent Aronszajn Trees
Kurepa TreesAmalgamation Spectra
Closure PropertiesConsistency Results
Manipulating the size of Kurepa trees we can produce a variety of
consistency result.
Theorem (Sinapova, S.)
If ZFC is consistent, then so are the following:
ZFC+ (ℵω1 = B < 2ℵ0)
ZFC+ (2ℵ0 < ℵω1 = B < 2ℵ1) +�B is a maximum�, i.e. there
exists a Kurepa tree of size ℵω1�.Assuming the consistency of ZFC+�a Mahlo exists�, then the
following is also consistent:
ZFC+ (2ℵ0 < B = 2ℵ1) + �2ℵ1 is weakly inaccessible + �for
every κ < 2ℵ1 there is a Kurepa tree with exactly κ-many
maximal branches, but no Kurepa tree has exactly 2ℵ1-many
branches.�
I. Souldatos absoluteness of amalgamation
IntroductionModel Existence SpectraCoherent Aronszajn Trees
Kurepa TreesAmalgamation Spectra
Closure PropertiesConsistency Results
Reminder: A cardinal λ is weakly inaccessible if it is a regular
limit, i.e. λ = ℵλ.
Corollary
There is an Lω1,ω-sentence φ for which it is consistent that
ME-Spec(φ) = [ℵ0, 2ℵ0 ];CH (or ¬CH ) + �2ℵ1 is a regular cardinal greater than ℵ2� +�ME-Spec(φ) = [ℵ0, 2ℵ1 ]�;2ℵ0 < ℵω1 + �ME-Spec(φ) = [ℵ0,ℵω1 ]; and2ℵ0 < 2ℵ1 + �2ℵ1 is weakly inaccessible� +
�ME-Spec(φ) = [ℵ0, 2ℵ1).
I. Souldatos absoluteness of amalgamation
IntroductionModel Existence SpectraCoherent Aronszajn Trees
Kurepa TreesAmalgamation Spectra
Closure PropertiesConsistency Results
Remarks
1 Although the characterization of 2ℵ0 and 2ℵ1 was known
before, this is the �rst example that can consistently
characterize either.
2 If 2ℵ0 = ℵω1 , then ℵω1 is characterizable. This is the �rst
example that consistently with 2ℵ0 < ℵω1 characterizes ℵω1 .Moreover, the proof works for many other cardinals besides
ℵω1 .3 If κ = supn∈ω κn and φn characterizes κn, then
∨nφn has
spectrum [ℵ0, κ). All previous examples of sentences with
right-open spectra where of the form∨
nφn. This is the �rst
example of a sentence with a right-open spectrum where the
right endpoint has uncountable co�nality (indeed it is a regular
cardinal).
I. Souldatos absoluteness of amalgamation
IntroductionModel Existence SpectraCoherent Aronszajn Trees
Kurepa TreesAmalgamation Spectra
Closure PropertiesConsistency Results
Recall that C is the least set that contains ℵ0 and is closed under
successor, countable unions, countable products, and powerset.
Corollary
If ZFC is consistent then so is ZFC+ (2ℵ0 < ℵω1 = B < 2ℵ1) + �Bis not in the set C described above�.
This refuted a �minimalist� view of characterizable cardinals.
I. Souldatos absoluteness of amalgamation
IntroductionModel Existence SpectraCoherent Aronszajn Trees
Kurepa TreesAmalgamation Spectra
First ExamplesHanf Number
Amalgamation Spectra
Remarks
Unlike model-existence, amalgamation spectra are not
downwards closed.
However, most of the known examples are either initial or
co-initial segments of the model-existence spectra.
We lack theorems of the form �If X is an amalgamation
spectrum, then Y(X) is also an amalgamation spectrum�.
I. Souldatos absoluteness of amalgamation
IntroductionModel Existence SpectraCoherent Aronszajn Trees
Kurepa TreesAmalgamation Spectra
First ExamplesHanf Number
The �rst example of an amalgamation spectrum that is not an
interval is the following:
Theorem (J. Baldwin, M. Koerwien. C.Laskowski)
Let 1 ≤ n < ω. There exists an Lω1,ω-sentence φn with
ME-Spec(φ) = [ℵ0,ℵn] and AP-Spec(φ) = [ℵ0,ℵn−2] ∪ {ℵn}.
The reason that amalgamation holds in ℵn is that all models are
maximal.
I. Souldatos absoluteness of amalgamation
IntroductionModel Existence SpectraCoherent Aronszajn Trees
Kurepa TreesAmalgamation Spectra
First ExamplesHanf Number
The work on Kurepa trees yields the following results for
amalgamation:
Theorem (Sinapova, S.)
There is an Lω1,ω-sentence φ for which it is consistent that
AP-Spec(φ) = [ℵ1, 2ℵ0 ];CH (or ¬CH ) + �2ℵ1 is a regular cardinal greater than ℵ2� +�AP-Spec(φ) = [ℵ1, 2ℵ1 ]�;2ℵ0 < ℵω1 + �AP-Spec(φ) = [ℵ1,ℵω1 ]; and2ℵ0 < 2ℵ1 + �2ℵ1 is weakly inaccessible� +
�AP-Spec(φ) = [ℵ1, 2ℵ1).
I. Souldatos absoluteness of amalgamation
IntroductionModel Existence SpectraCoherent Aronszajn Trees
Kurepa TreesAmalgamation Spectra
First ExamplesHanf Number
Theorem (J. Baldwin, W. Boney)
Let φ be an Lω1,ω-sentence, κ a strongly compact cardinal. If
AP-Spec(φ) contains a co�nal subset of κ, then AP-Spec(φ)contains [κ,∞).
So, the �rst strongly compact is greater or equal to �the Hanf
number for amalgamation� (this needs to be de�ned precisely).
I. Souldatos absoluteness of amalgamation
IntroductionModel Existence SpectraCoherent Aronszajn Trees
Kurepa TreesAmalgamation Spectra
First ExamplesHanf Number
Open Questions
Is it consistent that for an Lω1,ω-sentence φ, AP-Spec(φ) is aco�nal subset of the �rst measurable?
Is ℵ1-amalgamation absolute?
Is it consistent that for an Lω1,ω-sentence φ, AP-Spec(φ)equals [ℵ0,ℵω1) (right-open)?
I. Souldatos absoluteness of amalgamation
IntroductionModel Existence SpectraCoherent Aronszajn Trees
Kurepa TreesAmalgamation Spectra
First ExamplesHanf Number
References
Thank you!
Copy of these slides can be found at
http://souldatosresearch.wordpress.com/
Questions?
I. Souldatos absoluteness of amalgamation