Luca Motto Ros - uniroma1.it · (Classical) Descriptive Set Theory Polish spaces= separable completely metrizable topological spaces. Example: 2! with the product topology (which

Embeddability as a “universal” quasi-order

Luca Motto Ros

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

[email protected]

Bertinoro, May 27 2011

Luca Motto Ros (Freiburg, Germany) Embeddability is universal Bertinoro, 27-05-2011 1 / 15

The framework

Framework: Establish connections between (basic) Model Theory forinfinitary 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 (basic) Model Theory and DST: thecountable case

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

Lopez-Escobar theorem: B ⊆ ModωL is Borel and s.t. 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 (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.


































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 strictly 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), works on the existence of universalgraphs, i.e. graphs in which every other graph of the same size embeds(Shelah, Komjath, Dzamonja, S.Friedman, Thompson, ...).

3 A generalization of the F-MR theorem 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 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), works on the existence of universalgraphs, i.e. graphs in which every other graph of the same size 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), works on the existence of universalgraphs, i.e. graphs in which every other graph of the same size 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), works on the existence of universalgraphs, i.e. graphs in which every other graph of the same size embeds(Shelah, Komjath, Dzamonja, S.Friedman, Thompson, ...).



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);

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);









































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.]



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.

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 is universal Bertinoro, 27-05-2011 8 / 15


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:

the canonical representation of R cannot be obtained with theargument used for the countable case!

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.



Bad news: the canonical representation of R cannot be obtained with theargument used for the countable case!









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 is universal Bertinoro, 27-05-2011 11 / 15

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).


The main result

Lemma


Proof.




Theorem (M.)



The main result

Lemma


Proof.




Theorem (M.)



Open problems (work in progress!)

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? Can one relax theassumption on κ to κ<κ = κ? In particular, what happens for ω1 (underCH)?

Some comments:

1 The main difficulty is that in the previous arguments we heavily usedthe fact that 2κ is κ-compact, which is equivalent to κ being weaklycompact: therefore we need to use different ideas!

2 The condition κ<κ = κ would be optimal: if κ<κ > κ then one couldfind counterexamples (e.g. if 2κ

+> 2κ).




Main open problems: Is weakly compactness of κ really necessary toobtain that v on ModκL is invariantly universal?

Can one relax theassumption on κ to κ<κ = κ? In particular, what happens for ω1 (underCH)?

Some comments:



+> 2κ).




Main open problems: Is weakly compactness of κ really necessary toobtain that v on ModκL is invariantly universal? Can one relax theassumption on κ to κ<κ = κ?

In particular, what happens for ω1 (underCH)?

Some comments:



+> 2κ).





Some comments:



+> 2κ).





Some comments:



+> 2κ).





Some comments:



+> 2κ).





Some comments:



+> 2κ).


The end

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Luca Motto Ros - uniroma1.it · (Classical) Descriptive Set Theory Polish spaces= separable completely metrizable topological spaces. Example: 2! with the product topology (which