Statue of Aristotle at the Aristotle University of Thessaloniki, Greece

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1 Statue of Aristotle at the Aristotle University of Thessaloniki, Greece

2 Is the amalgamation property for L ω1,ω-sentences absolute for transitive models of ZFC? Logic Seminar - UIC Ioannis (Yiannis) Souldatos

3 L ω 1,ω Spectra Main Questions This is joint work with D. Sinapova and joint work with D. Milovich.

4 L ω 1,ω Spectra Main Questions Preliminaries Denition L ω1,ω = rst-order formulas + n ω φ n + n ω φ n L ω1,ω satises the Downward Lowenheim-Skolem Theorem, but fails the Upward Lowenheim-Skolem.

5 L ω 1,ω Spectra Main Questions Spectra Denition For an L ω1,ω-sentence φ, dene 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)}

6 L ω 1,ω Spectra Main 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 satises the following axioms (for an Abstract Elementary Class)

7 L ω 1,ω Spectra Main Questions Assume A, B, C are models of some φ. 1 A B implies A B; 2 is reexive and transitive; 3 If A i i κ is an increasing - chain, then (i) i I Ai is a model of φ; (ii) for each i I, Ai (iii) if for each i I, Ai i I ; and Ai M, then i I M. Ai 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 satises φ, then there is some A which satises φ, A A B, and A A + LS.

8 L ω 1,ω Spectra Main Questions Main Questions 1 What is known for the model existence and amalgamation spectra of L ω1,ω-sentences? 2 What is absolute (for transitive models of ZFC)?

9 Clusters Absoluteness Observation By Downward Lowenheim-Skolem, ME-Spec(φ) is downward closed. So, either ME-Spec(φ)=[ℵ 0, ℵ α ] (right-closed spectrum) or ME-Spec(φ)=[ℵ 0, ℵ α ) (right-open spectrum) Denition If ME-Spec(φ)=[ℵ 0, ℵ α ], then φ characterizes ℵ α.

10 Clusters Absoluteness Hanf Number Theorem (Morley, López-Escobar) The Hanf number for model-existence of L ω1,ω sentences is ℶ ω1. So, if φ has models of size ℶ ω1, then φ has models in all cardinalities. Corollary If ℵ α is characterized by an L ω1,ω-sentence, then ℵ α < ℶ ω1.

11 Clusters Absoluteness Alephs Theorem (Hjorth) For every α < ω 1, there exists an L ω1,ω-sentence φ α that characterizes ℵ α. Hjorth's theorem generalizes to Theorem Assume ℵ β is characterized by φ β. Then for every α < ω 1, there exists an L ω1,ω-sentence φ β α that characterizes ℵ β+α.

12 Clusters Absoluteness ℵ 0 clusters of characterizable cardinals of length ω 1... ℵ ω1 ℶ ω1

13 Clusters Absoluteness ℵ 0 clusters of characterizable cardinals of length ω 1... ℵ ω1 ℶ ω1 Under GCH there is only one cluster.

14 Clusters Absoluteness ℵ 0 clusters of characterizable cardinals of length ω 1... ℵ ω1 ℶ ω1 Under GCH there is only one cluster. Under GCH, it is consistent that there exist non-characterizable cardinals below ℶ ω1.

15 Clusters Absoluteness Absoluteness Question What is absolute? Hjorth's theorem is in ZFC. By Shoeneld's absoluteness, model-existence in ℵ 0 is absolute.

16 Clusters Absoluteness Theorem (Grossberg-Shelah) The property that an L ω1,ω-sentence has arbitrarily large models is absolute. Corollary Model-existence in any power ℶ ω1 is absolute.

17 Clusters Absoluteness Theorem (S.Friedman, Hyttinen, Koerwien) 1 Model-existence in ℵ 1 is absolute. 2 Model-existence in ℵ α, 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 2 ℵ 0. Manipulating the size of 2 ℵ 0 in various models of ZFC, we prove non-absoluteness of model-existence in ℵ α. Question What happens if we assume GCH?

18 Clusters Absoluteness Theorem (S.Friedman, Hyttinen, Koerwien) Assuming the existence of uncountably many inaccessible cardinals, model-existence in ℵ α+2, α < ω 1, is not absolute for models of ZFC+GCH. Proof (sketch) There exists some L ω1,ω-sentence φ α+2 that has a model of size ℵ α+2 i there is an ℵ α+1 -Kurepa tree. Collapsing the inaccessibles destroys all Kurepa trees, while maintaining GCH.

19 Clusters Absoluteness Theorem (S.Friedman, Hyttinen, Koerwien) Let α < ω 1 be a limit ordinal. Assuming the existence of a supercompact cardinal, model-existence in ℵ α+1 is not absolute for models of ZFC+GCH. Let α < ω 1 be a limit ordinal greater than ω. Assuming the existence of a supercompact cardinal, model-existence in ℵ α is not absolute for models of ZFC+GCH. Proof (sketch) There exists some L ω1,ω-sentence φ α+1 that has a model of size ℵ α+1 i there is a special ℵ α+1 -Aronszajn tree. Assuming a supercompact cardinal, there is a forcing extension in which GCH is true, but there are no special ℵ α+1 -Aronszajn trees.

20 Clusters Absoluteness The previous theorem leaves the question for ℵ ω open. Theorem (D. Milovich,S.) Model-existence in ℵ ω is not absolute for models of ZFC+GCH The idea is to use coherent special Aronszajn trees.

21 Model Existence Amalgamation Denition 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 with dom(s) = dom(t), s = t.

22 Model Existence Amalgamation It follows from the denition 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 ω 2, every special ℵ 2 -Aronszajn tree contains a copy of 2 <ω 1.

23 Model Existence Amalgamation A drawback in working with L ω1,ω is that we can not characterize well-orderings. So, instead of working with (well-founded) trees, we have to work with pseudo-trees. Denition A pseudotree is a partial ordered set T such that each set of the form x = {y y < T x} is linearly ordered.

24 Model Existence Amalgamation Theorem (Milovich, S.) Given 1 α < ω 1, there is an L ω1,ω-sentence φ α satisfying the following. 1 There is no model of φ α of size greater than ℵ α. 2 If there is a coherent special ℵ β -Aronszajn tree for each β < α, then φ α has a model of size ℵ α. 3 If φ α has a model of size ℵ 2, then there is a coherent pseudotree T with conality ω 2 and an order embedding of a special ℵ 2 -Aronszajn tree into T.

25 Model Existence Amalgamation Corollary Let 2 α < ω 1. Corollary In L, the sentence φ α characterizes ℵ α. After collapsing a Mahlo to ω, φ α has no model of size ℵ 2. Assuming a Mahlo cardinal, model-existence in ℵ α, 2 α < ω 1, is not absolute for models of ZFC+GCH. This not only answers the open question for α = ω, but it also improves the large cardinal requirement from a supercompact to a Mahlo.

26 Model Existence Amalgamation Amalgamation Consider the collection of all models of φ α with the substructure relation. Theorem (Milovich, S.) Assume 2 α < ω. If there are models of φ α of size ℵ α, then AP-Spec(φ α ) = {ℵ α }. Otherwise AP-Spec(φ α ) is empty. Corollary The following statements are not absolute for transitive models of ZFC+GCH. (a) The amalgamation spectrum of an L ω1,ω-sentence is empty. (b) For nite n 2 and φ an L ω1,ω-sentence, ℵ n AP-Spec(φ).

27 Model Existence Amalgamation Remark The above examples can not be used to resolve the absoluteness question of ℵ α -amalgamation, for ω α < ω 1. The reason is the following lemma. Lemma Assume ω α < ω 1. The amalgamation spectrum of φ α is empty. The question for ℵ 1 -amalgamation remains open. Conjecture If φ is an L ω1,ω-sentence and F is the fragment generated by φ, then ℵ 1 -amalgamation under F is absolute for transitive models of ZFC.

28 Closure Properties Consistency Results Closure Properties Theorem The set of cardinals characterized by L ω1,ω-sentences is closed under 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 L ω1,ω-sentences.

29 Closure Properties Consistency Results Let C be the least set of cardinals that contains ℵ 0 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.

30 Closure Properties Consistency Results There exists an L ω1,ω-sentence φ that characterizes the maximum of 2 ℵ 0 and B. B = sup{κ there exists a Kurepa tree withκ many branches}. Reminder: A Kurepa tree has height ω 1, countable levels and more than ℵ 1 many branches. So, ℵ 1 < B 2 ℵ 1.

31 Closure Properties Consistency 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+ (ℵ ω1 = B < 2 ℵ 0 ) ZFC+ (2 ℵ 0 < ℵ ω1 = B < 2 ℵ 1 ) +B is a maximum, i.e. there exists a Kurepa tree of size ℵ ω1. Assuming the consistency of ZFC+a Mahlo exists, then the following is also consistent: ZFC+ (2 ℵ 0 < B = 2 ℵ 1 ) + 2 ℵ 1 is weakly inaccessible + for every κ < 2 ℵ 1 there is a Kurepa tree with exactly κ-many maximal branches, but no Kurepa tree has exactly 2 ℵ 1 -many branches.

32 Closure Properties Consistency Results Reminder: A cardinal λ is weakly inaccessible if it is a regular limit, i.e. λ = ℵ λ. Corollary There is an L ω1,ω-sentence φ for which it is consistent that ME-Spec(φ) = [ℵ 0, 2 ℵ 0 ]; CH (or CH ) + 2 ℵ 1 is a regular cardinal greater than ℵ 2 + ME-Spec(φ) = [ℵ 0, 2 ℵ 1 ]; 2 ℵ 0 < ℵ ω1 + ME-Spec(φ) = [ℵ 0, ℵ ω1 ]; and 2 ℵ 0 < 2 ℵ ℵ 1 is weakly inaccessible + ME-Spec(φ) = [ℵ 0, 2 ℵ 1 ).

33 Closure Properties Consistency Results Remarks 1 Although the characterization of 2 ℵ 0 and 2 ℵ 1 was known before, this is the rst example that can consistently characterize either. 2 If 2 ℵ 0 = ℵ ω1, then ℵ ω1 is characterizable. This is the rst example that consistently with 2 ℵ 0 < ℵ ω1 characterizes ℵ ω1. Moreover, the proof works for many other cardinals besides ℵ ω1. 3 If κ = sup n ω κ n and φ n characterizes κ n, then n φ n has spectrum [ℵ 0, κ). All previous examples of sentences with right-open spectra where of the form n φ n. This is the rst example of a sentence with a right-open spectrum where the right endpoint has uncountable conality (indeed it is a regular cardinal).

34 Closure Properties Consistency Results Recall that C is the least set that contains ℵ 0 and is closed under successor, countable unions, countable products, and powerset. Corollary If ZFC is consistent then so is ZFC+ (2 ℵ 0 < ℵ ω1 = B < 2 ℵ 1 ) + B is not in the set C described above. This refuted a minimalist view of characterizable cardinals.

35 First Examples Hanf Number 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.

36 First Examples Hanf 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 L ω1,ω-sentence φ n with ME-Spec(φ) = [ℵ 0, ℵ n ] and AP-Spec(φ) = [ℵ 0, ℵ n 2] {ℵ n }. The reason that amalgamation holds in ℵ n is that all models are maximal.

37 First Examples Hanf Number The work on Kurepa trees yields the following results for amalgamation: Theorem (Sinapova, S.) There is an L ω1,ω-sentence φ for which it is consistent that AP-Spec(φ) = [ℵ 1, 2 ℵ 0 ]; CH (or CH ) + 2 ℵ 1 is a regular cardinal greater than ℵ 2 + AP-Spec(φ) = [ℵ 1, 2 ℵ 1 ]; 2 ℵ 0 < ℵ ω1 + AP-Spec(φ) = [ℵ 1, ℵ ω1 ]; and 2 ℵ 0 < 2 ℵ ℵ 1 is weakly inaccessible + AP-Spec(φ) = [ℵ 1, 2 ℵ 1 ).

38 First Examples Hanf Number Theorem (J. Baldwin, W. Boney) Let φ be an L ω1,ω-sentence, κ a strongly compact cardinal. If AP-Spec(φ) contains a conal 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 dened precisely).

39 First Examples Hanf Number Open Questions Is it consistent that for an L ω1,ω-sentence φ, AP-Spec(φ) is a conal subset of the rst measurable? Is ℵ 1 -amalgamation absolute? Is it consistent that for an L ω1,ω-sentence φ, AP-Spec(φ) equals [ℵ 0, ℵ ω1 ) (right-open)?

40 First Examples Hanf Number References Thank you! Copy of these slides can be found at Questions?