The descriptive set theoretical complexity of the embeddability relation on uncountable models Luca Motto Ros Abteilung f¨ ur Mathematische Logik Albert-Ludwigs-Universit¨ at, Freiburg im Breisgau, Germany [email protected]Barcelona — July 13, 2011 Luca Motto Ros (Freiburg, Germany) Embeddability on uncountable models Barcelona, 13-07-2011 1 / 18
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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
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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
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
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
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
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
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.
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.
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.
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).
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).
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).
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).
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).
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).
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).
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).
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).
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.)
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.)
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.)
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.)
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.)