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The descriptive set theoretical complexity of theembeddability relation on uncountable models
Luca Motto Ros
Abteilung fur Mathematische LogikAlbert-Ludwigs-Universitat, Freiburg im Breisgau, Germany
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).
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 2 / 18
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}.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 3 / 18
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}.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 3 / 18
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}.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 3 / 18
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}.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 3 / 18
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}.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 3 / 18
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}.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 3 / 18
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}.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 3 / 18
(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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 4 / 18
(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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 4 / 18
(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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 4 / 18
(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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 4 / 18
(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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 4 / 18
(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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 4 / 18
(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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 4 / 18
(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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 4 / 18
(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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 4 / 18
(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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 4 / 18
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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 5 / 18
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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 5 / 18
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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 5 / 18
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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 5 / 18
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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 5 / 18
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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 5 / 18
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.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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 6 / 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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 6 / 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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 6 / 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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 6 / 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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 6 / 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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 6 / 18
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).
Analytic equivalence relations = symmetric analytic quasi-orders.
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 ′.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 7 / 18
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).
Analytic equivalence relations = symmetric analytic quasi-orders.
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 ′.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 7 / 18
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).
Analytic equivalence relations = symmetric analytic quasi-orders.
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 ′.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 7 / 18
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).
Analytic equivalence relations = symmetric analytic quasi-orders.
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 ′.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 7 / 18
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).
Analytic equivalence relations = symmetric analytic quasi-orders.
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 ′.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 7 / 18
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).
Analytic equivalence relations = symmetric analytic quasi-orders.
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 ′.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 7 / 18
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).
Analytic equivalence relations = symmetric analytic quasi-orders.
R ≤κB S ⇐⇒ ∃ κ+-Borel f : dom(R)→ dom(S) reducing R to S
R ∼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 ′.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 7 / 18
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).
Analytic equivalence relations = symmetric analytic quasi-orders.
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 ′.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 7 / 18
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).
Analytic equivalence relations = symmetric analytic quasi-orders.
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 ′.Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 7 / 18
Back to countable case
Theorem (S.Friedman-M.)
For every analytic q.o. R on 2ω there is an Lω1ω-sentence ϕ s.t.R ∼B v� Modωϕ.
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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 8 / 18
Back to countable case
Theorem (S.Friedman-M.)
For every analytic q.o. R on 2ω there is an Lω1ω-sentence ϕ s.t.R ∼B v� Modωϕ.
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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 8 / 18
Back to countable case
Theorem (S.Friedman-M.)
For every analytic q.o. R on 2ω there is an Lω1ω-sentence ϕ s.t.R ∼B v� Modωϕ.
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” 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.]
5 Use the Lopez-Escobar theorem to find an Lω1ω-sentence ϕ s.t.Sat(f (2ω)) = Modωϕ.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 8 / 18
Back to countable case
Theorem (S.Friedman-M.)
For every analytic q.o. R on 2ω there is an Lω1ω-sentence ϕ s.t.R ∼B v� Modωϕ.
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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 8 / 18
Back to countable case
Theorem (S.Friedman-M.)
For every analytic q.o. R on 2ω there is an Lω1ω-sentence ϕ s.t.R ∼B v� Modωϕ.
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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 8 / 18
Back to countable case
Theorem (S.Friedman-M.)
For every analytic q.o. R on 2ω there is an Lω1ω-sentence ϕ s.t.R ∼B v� Modωϕ.
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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 8 / 18
Back to countable case
Theorem (S.Friedman-M.)
For every analytic q.o. R on 2ω there is an Lω1ω-sentence ϕ s.t.R ∼B v� Modωϕ.
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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 8 / 18
Generalized DST: the uncountable case
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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 9 / 18
Generalized DST: the uncountable case
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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 9 / 18
Generalized DST: the uncountable case
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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 9 / 18
Generalized DST: the uncountable case
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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 9 / 18
Generalized DST: the uncountable case
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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 9 / 18
Generalized DST: the uncountable case
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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 9 / 18
Generalized DST: the uncountable case
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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 9 / 18
Generalized DST: the uncountable case
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).
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 10 / 18
Generalized DST: the uncountable case
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).
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 10 / 18
Generalized DST: the uncountable case
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).
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 10 / 18
Generalized DST: the uncountable case
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).
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 10 / 18
Generalized DST: the uncountable case
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).
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 10 / 18
Generalized DST: the uncountable case
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).
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 10 / 18
Generalized DST: the uncountable case
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).
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 10 / 18
Generalized DST: the uncountable case
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).
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 10 / 18
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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 11 / 18
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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 11 / 18
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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 11 / 18
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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 11 / 18
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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 11 / 18
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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 11 / 18
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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 11 / 18
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.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 .
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 12 / 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 .
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 12 / 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 .
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 12 / 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 .
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 12 / 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 .
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 12 / 18
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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 13 / 18
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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 13 / 18
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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 13 / 18
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).
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 14 / 18
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).
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 14 / 18
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).
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 14 / 18
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).
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 14 / 18
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).
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 14 / 18
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).
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 14 / 18
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).
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 14 / 18
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).
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 14 / 18
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� 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).
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 14 / 18
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� 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).
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 14 / 18
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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 15 / 18
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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 15 / 18
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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 15 / 18
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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 15 / 18
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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 16 / 18
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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 16 / 18
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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 16 / 18
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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 16 / 18
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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 16 / 18
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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 16 / 18
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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 16 / 18
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.
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 16 / 18
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.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.)
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 17 / 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.)
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 17 / 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.)
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 17 / 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.)
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 17 / 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.)
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 17 / 18
The end
Thank you for your attention!
Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 18 / 18