The descriptive set theoretical complexity of the ...lc2011/Slides/MottoRos.pdf · The descriptive set theoretical complexity of the embeddability relation on uncountable models Luca

The descriptive set theoretical complexity of theembeddability relation on uncountable models

Luca Motto Ros

Abteilung fur Mathematische LogikAlbert-Ludwigs-Universitat, Freiburg im Breisgau, Germany

[email protected]

Barcelona — July 13, 2011

Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 1 / 18

The framework

Framework: Establish connections between (basic) Model Theory (MT)for infinitary languages and Descriptive Set Theory (DST).


Model Theory

L = graph language.

Infinitary logic Lκλ: negation, conjunctions and disjunctions of length < κ,(simultaneous) quantification over < λ variables.

∼= denotes isomorphism, v denotes embeddability.

Each L-structure of size µ can be identified (up to isomorphism) with anL-structure with domain µ.

ModµL = the space of all L-structures with domain µ.

For X ⊆ ModµL, Sat(X ) is the closure under ∼= of X .

Given an Lκλ-sentence ϕ, we set

Modµϕ = {x ∈ ModµL | x � ϕ}.


Model Theory

L = graph language.









Model Theory

L = graph language.









Model Theory

L = graph language.









Model Theory

L = graph language.









Model Theory

L = graph language.









Model Theory

L = graph language.









(Classical) Descriptive Set Theory

Polish spaces = separable completely metrizable topological spaces.

Example: 2ω with the product topology (which coincides with thetopology generated by Ns = {x ∈ 2ω | s ⊆ x} for s ∈ <ω2).

Bor(X ) = the minimal σ-algebra containing the open sets of X .

A ⊆ X is analytic if A = proj(C ) for C a closed subset of X × ωω.

Borel functions: f : X → Y s.t. f −1(U) ∈ Bor(X ) for open U ⊆ Y .

Analytic quasi-orders: R ⊆ B × B such that B ∈ Bor(X ) (for some PolishX ) and R is reflexive, transitive, and analytic. B = dom(R).

Analytic equivalence relations = symmetric analytic quasi-orders.

R ≤B S ⇐⇒ ∃ Borel f : dom(R)→ dom(S) s.t. x R y ⇐⇒ f (x) S f (y)R ∼B S ⇐⇒ R ≤B S ≤B R

Every two uncountable Borel subsets of Polish spaces are Borel isomorphic:hence w.l.o.g. we may assume dom(R) = 2ω.












































































R ≤B S ⇐⇒ ∃ Borel f : dom(R)→ dom(S) s.t. x R y ⇐⇒ f (x) S f (y)

R ∼B S ⇐⇒ R ≤B S ≤B R

























Connections between MT and DST: the countable case

ModωL is a Polish space (homeomorphic to 2ω).

Lopez-Escobar theorem: B ⊆ ModωL is Borel and B = Sat(B) iffB = Modωϕ for some Lω1ω sentence ϕ.

Therefore ∼=� Modωϕ and v� Modωϕ are examples of, respectively, ananalytic equivalence relation and an analytic quasi-order.

Theorem (Louveau-Rosendal)

For every analytic q.o. R on 2ω, R ≤B v� ModωL.

We abbreviate this statement with: v on ModωL is complete.

Theorem (S.Friedman-M.)

For every analytic q.o. R on 2ω there is an Lω1ω-sentence ϕ s.t.R ∼B v� Modωϕ.

We abbreviate this statement with: v on ModωL is invariantly universal.


































































We abbreviate this statement with: v on ModωL is invariantly universal.Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 5 / 18

Main goal and motivations

Goal: generalize, if possible, these connections to the uncountable setting.

Some motivations:

1 (S.Friedman-Hyttinen-Kulikov) For many uncountable cardinals κ, the(generalization of the) ≤B -relation between the isomorphism relationson models of size κ of first order theories is related to Shelah’sstability theory (highly nontrivial model theory!).

2 The embeddability relation is a quite well-studied notion, e.g.:

κ = ω: v on linear orders is a wqo (Laver); v on graphs is extremelycomplicated (Louveau-Rosendal, S.Friedman-M.);κ > ω: “if κ is regular then (STATκ,⊆NSTAT) embeds into v on linearorders of size κ” (Baumgartner, 1972); works on the existence ofuniversal graphs, i.e. graphs in which every other graph of the samesize embeds (Shelah, Komjath, Dzamonja, S.Friedman, Thompson, ...).

3 A generalization of the L-R and F-MR theorems could allow to betterunderstand the embeddability relation on ModκL, e.g. we would havethat (P(κ),⊆∗) “Borel embeds” into v on (generalized) trees. Inparticular, we would have a generalization of Baumgartner’s result.




Some motivations:








Some motivations:








Some motivations:


2 The embeddability relation is a quite well-studied notion, e.g.:κ = ω: v on linear orders is a wqo (Laver); v on graphs is extremelycomplicated (Louveau-Rosendal, S.Friedman-M.);

κ > ω: “if κ is regular then (STATκ,⊆NSTAT) embeds into v on linearorders of size κ” (Baumgartner, 1972); works on the existence ofuniversal graphs, i.e. graphs in which every other graph of the samesize embeds (Shelah, Komjath, Dzamonja, S.Friedman, Thompson, ...).





Some motivations:


2 The embeddability relation is a quite well-studied notion, e.g.:κ = ω: v on linear orders is a wqo (Laver); v on graphs is extremelycomplicated (Louveau-Rosendal, S.Friedman-M.);κ > ω: “if κ is regular then (STATκ,⊆NSTAT) embeds into v on linearorders of size κ” (Baumgartner, 1972); works on the existence ofuniversal graphs, i.e. graphs in which every other graph of the samesize embeds (Shelah, Komjath, Dzamonja, S.Friedman, Thompson, ...).





Some motivations:


2 The embeddability relation is a quite well-studied notion, e.g.:κ = ω: v on linear orders is a wqo (Laver); v on graphs is extremelycomplicated (Louveau-Rosendal, S.Friedman-M.);κ > ω: “if κ is regular then (STATκ,⊆NSTAT) embeds into v on linearorders of size κ” (Baumgartner, 1972); works on the existence ofuniversal graphs, i.e. graphs in which every other graph of the samesize embeds (Shelah, Komjath, Dzamonja, S.Friedman, Thompson, ...).



Generalized DST: the uncountable case

Borκ(X ) = the smallest κ+-algebra containing all open sets of X .

κ+-Borel functions: Borκ(X )-measurable functions.

We are interested in spaces X which are κ+-Borel isomorphic to 2κ

endowed with the bounded topology, i.e. with the topology generated byNs = {x ∈ 2κ | s ⊆ x} for s ∈ <κ2. (Example: κκ.)

A ⊆ X is analytic if A = proj(C ) for C a closed subset of X × κκ.

Analytic quasi-orders: R ⊆ B × B such that B ∈ Borκ(X ) and R isanalytic, reflexive and transitive. B = dom(R).


R ≤κB S ⇐⇒ ∃ κ+-Borel f : dom(R)→ dom(S) reducing R to SR ∼κB S ⇐⇒ R ≤κB S ≤κB R

In general, a κ+-Borel subsets of X of size > κ need not be κ+-Borelisomorphic to 2κ, but: for every analytic q.o. R there is an analytic q.o. R ′

on 2κ s.t. R ∼κB R ′.






































































R ≤κB S ⇐⇒ ∃ κ+-Borel f : dom(R)→ dom(S) reducing R to S

R ∼κB S ⇐⇒ R ≤κB S ≤κB R


























on 2κ s.t. R ∼κB R ′.Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 7 / 18

Back to countable case



1 represent R as proj[T ], where T is a tree on 2× 2× ω of height ω“mirroring” the reflexivity and transitivity of R at all finite levels;

2 find a Borel reduction f of R into v� ModωL;3 ensure that x 6= y ⇒ f (x) 6∼= f (y), so that the “inverse” reduction

g : Sat(f (2ω))→ 2ω sending z to the unique x such that f (x) ∼= z iswell-defined (as a map), and reduces v� Sat(f (2ω)) to R;

4 show that Sat(f (2ω)) is Borel (which also implies that g is Borel);

[Sketch of the proof: each f (x) is rigid by construction, hence themap h : S∞ × 2ω → ModωL : (p, x) 7→ jL(p, f (x)) is injective. Since his Borel and range(h) = Sat(f (2ω)), this last set is Borel.]

5 Use the Lopez-Escobar theorem to find an Lω1ω-sentence ϕ s.t.Sat(f (2ω)) = Modωϕ.
















2 find a Borel reduction f of R into v� ModωL;

3 ensure that x 6= y ⇒ f (x) 6∼= f (y), so that the “inverse” reductiong : Sat(f (2ω))→ 2ω sending z to the unique x such that f (x) ∼= z iswell-defined (as a map), and reduces v� Sat(f (2ω)) to R;









































4 show that Sat(f (2ω)) is Borel (which also implies that g is Borel);[Sketch of the proof: each f (x) is rigid by construction, hence themap h : S∞ × 2ω → ModωL : (p, x) 7→ jL(p, f (x)) is injective. Since his Borel and range(h) = Sat(f (2ω)), this last set is Borel.]




ModκL can be identified with 2κ (up to homeomorphism).

Want to show: under suitable hypothesis on κ, for every analytic q.o. R on2κ there is an Lκ+κ-sentence ϕ s.t. R ∼κB v� Modκϕ (briefly: v on ModκLis invariantly universal).

Good news:

1 it is not too difficult to find a κ+-Borel reduction f of R to v� ModκLs.t. x 6= y ⇒ f (x) 6∼= f (y) (so that the “inverse” reduction g iswell-defined), PROVIDED THAT R admits a canonical representationT , i.e. R = proj[T ] for T a tree on 2× 2× κ of height κ mirroringthe reflexivity and transitivity of R at all bounded levels;

2 if κ<κ = κ the generalized Lopez-Escobar theorem holds: for everyB ⊆ ModκL, B is κ+-Borel and Sat(B) = B iff there is anLκ+κ-sentence ϕ s.t. B = Modκϕ.

Remark: if κ<κ > κ both directions of Lopez-Escobar theorem can fail.





Good news:








Good news:








Good news:

1 it is not too difficult to find a κ+-Borel reduction f of R to v� ModκLs.t. x 6= y ⇒ f (x) 6∼= f (y) (so that the “inverse” reduction g iswell-defined),

PROVIDED THAT R admits a canonical representationT , i.e. R = proj[T ] for T a tree on 2× 2× κ of height κ mirroringthe reflexivity and transitivity of R at all bounded levels;







Good news:








Good news:








Good news:






Bad news:

there is a technical obstacle to get the canonicalrepresentation of R in the uncountable case (briefly discussed later)!

Very bad news: very few results of DST can be generalized to theuncountable case:

1 the κ-PSP can fail for closed sets;

2 no Luzin’s separation theorem (there are bianalytic sets which are notκ+-Borel);

3 injective κ+-Borel images of κ+-Borel sets need not be κ+-Borel;

4 even if Sat(f (2κ)) is proved to be κ+-Borel, the “inverse” reduction gneed not be κ+-Borel (it is just a function with bianalytic graph).



Bad news: there is a technical obstacle to get the canonicalrepresentation of R in the uncountable case (briefly discussed later)!









Very bad news:

very few results of DST can be generalized to theuncountable case:














































Weakly compact cardinals

Definition

An uncountable cardinal κ is weakly compact if κ→ (κ)22, i.e. if theRamsey theorem holds for κ.

Theorem

Let κ be an uncountable cardinal. TFAE:

1 κ→ (κ)22;

2 κ is inaccessible and has the tree property;

3 2κ is κ-compact (i.e. κ-Lindelof).

The second condition easily allows to find the required canonicalrepresentation for R, so it remains to show:

Sat(f (2κ)) is κ+-Borel;

the “inverse” reduction g is κ+-Borel.

This can be done in a completely different way w.r.t. the countable case.



Definition


Theorem


1 κ→ (κ)22;









Definition


Theorem


1 κ→ (κ)22;









Definition


Theorem


1 κ→ (κ)22;









Definition


Theorem


1 κ→ (κ)22;









Definition


Theorem


1 κ→ (κ)22;









Definition


Theorem


1 κ→ (κ)22;









Definition


Theorem


1 κ→ (κ)22;






This can be done in a completely different way w.r.t. the countable case.Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 11 / 18

The main idea

1 Find a suitable Lκ+κ-sentence ψ s.t. f (2κ) ⊆ Modκψ (ψ essentially“describes” the common part of the structures f (x) for x ∈ 2κ);

2 classify Modκψ up to isomorphism with invariants in 2κ via someh : Modκψ → 2κ s.t. h ◦ f is continuous;

3 show that for every open U ⊆ 2κ, h−1(U) = ModκϕUfor some

Lκ+κ-sentence ϕU .

Lemma

Assume κ is weakly compact. Then there is an Lκ+κ-sentence ϕ s.t.Sat(f (2κ)) = Modκϕ.

Proof.

2κ is κ-compact: since h ◦ f is continuous, (h ◦ f )(2κ) is κ-compact andhence closed in 2κ (because 2κ is Hausdorff and κ is regular). LetU = 2κ \ (h ◦ f )(2κ): then h−1(U) = ModκϕU

, and hence it is enough to letϕ be ψ ∧ ¬ϕU .


The main idea





Lemma


Proof.




The main idea





Lemma


Proof.




The main idea





Lemma


Proof.




The main idea





Lemma


Proof.




The main result

Lemma

Assume κ is weakly compact. Then the “inverse” reduction g is κ+-Borel.

Proof.

Since κ<κ = κ, it is enough to show that g−1(Ns) is κ+-Borel for everys ∈ <κ2. Notice that for every A ⊆ 2κ, g−1(A) = Sat(f (A)). Each Ns isalso closed, hence κ-compact: using an argument similar to the one in theprevious lemma, find an Lκ+κ-sentence ϕs such that Sat(f (Ns)) = Modκϕs

.Then use the generalized Lopez-Escobar theorem.

Therefore we have shown:

Theorem (M.)

Let κ be a weakly compact cardinal. For every analytic q.o. R on 2κ thereis an Lκ+κ-sentence ϕ s.t. R ∼κB v� Modκϕ (i.e. v on ModκL is invariantlyuniversal). In particular, v� ModκL is also complete.


The main result

Lemma


Proof.




Theorem (M.)



The main result

Lemma


Proof.




Theorem (M.)



Some remarks

1 Which kind of structures are involved in the theorem?

κ = ω: combinatorial trees, i.e. connected acyclic graphs;κ > ω: generalized trees, i.e. partial orders in which the set ofpredecessors of every element is linearly ordered.

2 Our theorem extends Baumgartner’s result in two different directions.

STATκ is a proper coanalytic subset of 2κ, so (STATκ,⊆NSTAT) is notan analytic q.o. However, its minor variation S = (2κ,⊆NSTAT) is ananalytic q.o.

In this setup, Baumgartner’s result can be restated as:

S ≤κB v� ModκL.

This is now improved (for weakly compact cardinals) as follows:

in fact, there is an Lκ+κ-sentence ϕSTAT such that v� ModκϕSTATlooks

exactly like S (i.e. S ∼κB v� ModκϕSTAT);

the q.o. S is just an instantiation of a more general phenomenon,which involves all possible analytic q.o. (v� ModκL is complete).


Some remarks

1 Which kind of structures are involved in the theorem?κ = ω: combinatorial trees, i.e. connected acyclic graphs;

κ > ω: generalized trees, i.e. partial orders in which the set ofpredecessors of every element is linearly ordered.










Some remarks

1 Which kind of structures are involved in the theorem?κ = ω: combinatorial trees, i.e. connected acyclic graphs;κ > ω: generalized trees, i.e. partial orders in which the set ofpredecessors of every element is linearly ordered.










Some remarks











Some remarks



STATκ is a proper coanalytic subset of 2κ, so (STATκ,⊆NSTAT) is notan analytic q.o.

However, its minor variation S = (2κ,⊆NSTAT) is ananalytic q.o.








Some remarks











Some remarks











Some remarks











Some remarks






This is now improved (for weakly compact cardinals) as follows:in fact, there is an Lκ+κ-sentence ϕSTAT such that v� ModκϕSTAT

looksexactly like S (i.e. S ∼κB v� ModκϕSTAT

);



Some remarks






This is now improved (for weakly compact cardinals) as follows:in fact, there is an Lκ+κ-sentence ϕSTAT such that v� ModκϕSTAT

looksexactly like S (i.e. S ∼κB v� ModκϕSTAT

);the q.o. S is just an instantiation of a more general phenomenon,which involves all possible analytic q.o. (v� ModκL is complete).


Some remarks

About the new technique used in the proof:

1 it is based on preliminary work (joint with A. Andretta) on κ-Souslinquasi-order on 2ω;

2 it cannot be used in the countable case, because the formula ψ whichappears in the proof must express well-foundness of certain parts ofthe generalized tree: so, as it is often the case in Model Theory, thecountable/uncountable cases must be treated separately and withdifferent methods!

3 the Lκ+κ-sentence ϕ that one obtains at the end of the proof is suchthat all models of ϕ have size κ (i.e. Modκϕ contains an isomorphiccopy of every model of ϕ): so all these sentences characterize κ.


Some remarks






Some remarks






Some remarks






Open problems

Weakly compact cardinals may not exists! Their existence is a (quiteweak) large cardinal assumption.

Main open problems: Is weakly compactness of κ really necessary toobtain that v on ModκL is invariantly universal (or at least complete)?Can one relax the assumption on κ to κ<κ = κ? In particular, whathappens for ω1 (under CH)?

The condition κ<κ = κ would be optimal for invariant universality.

Example

Assume κ<κ > κ and 2κ+> 2κ (e.g. let κ be a singular cardinal in a model

of GCH). Then there are at least 2κ+-many ≤κB -incomparable analytic

(and clopen) q.o., while there can be only 2κ-many Lκ+κ-sentences: inparticular, there is an analytic q.o. R such that R 6∼κB v� Modκϕ for everyLκ+κ-sentence ϕ, i.e. v on ModκL is not invariantly universal.

The counterexample remains valid even if we allow arbitrary reductions.


Open problems


Main open problems: Is weakly compactness of κ really necessary toobtain that v on ModκL is invariantly universal (or at least complete)?

Can one relax the assumption on κ to κ<κ = κ? In particular, whathappens for ω1 (under CH)?


Example






Open problems


Main open problems: Is weakly compactness of κ really necessary toobtain that v on ModκL is invariantly universal (or at least complete)?Can one relax the assumption on κ to κ<κ = κ?

In particular, whathappens for ω1 (under CH)?


Example






Open problems




Example






Open problems




Example






Open problems




Example


of GCH).

Then there are at least 2κ+-many ≤κB -incomparable analytic




Open problems




Example



(and clopen) q.o., while there can be only 2κ-many Lκ+κ-sentences:

inparticular, there is an analytic q.o. R such that R 6∼κB v� Modκϕ for everyLκ+κ-sentence ϕ, i.e. v on ModκL is not invariantly universal.



Open problems




Example






Open problems




Example




The counterexample remains valid even if we allow arbitrary reductions.Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 16 / 18

Work in progress and other problems

The main difficulty to attack the previous open problems is that in theargument sketched above we heavily used (twice) the fact that 2κ isκ-compact, which is equivalent to κ being weakly compact: therefore wenecessarily need to use different ideas!

Partial result (joint with H. Mildenberger): if κ<κ = κ, then v on ModκLis complete, i.e. for every analytic q.o. R on 2κ, R ≤κB v� ModκL.

Other open problems:

1 Is it possible to replace generalized trees with linear orders in theconstructions above?

2 Is the condition κ<κ = κ necessary to get completeness of v onModκL? In particular, can v� ModκL be complete when κ is a singularcardinal? (Note that this situation is not forbidden by the previouscounterexample.)





















2 Is the condition κ<κ = κ necessary to get completeness of v onModκL? In particular, can v� ModκL be complete when κ is a singularcardinal?

(Note that this situation is not forbidden by the previouscounterexample.)









The end

Thank you for your attention!


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The descriptive set theoretical complexity of the ...lc2011/Slides/MottoRos.pdf · The descriptive set theoretical complexity of the embeddability relation on uncountable models Luca