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Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Independence, Continuity, Tree Properties
Andrés Villaveces
Universidad Nacional de Colombia - Bogotá
Universitat de Barcelona - Model Theory Seminar - October 2016
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Contents
Independence notions
A review of canonicity of independence notions
Two recent results
Continuous world/independence
Continuous Logic / Stability / Lindström
Continuous Independence
Obstacles to continuous Chatzidakis-Pillay
The Chatzidakis-Pillay Theorem - Simplicity
Continuous Generic Predicates over Hilbert Spaces
Continuity carries Interference
Generic independence, good and bad
The theory TN is bad: it has TP2
The theory TN is good: it is NTP1 (really, NSOP1)
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Independence notions
Let us consider the following independence notion:
Denition (coheir independence (v. Boney-Grossberg))
Let M ≺ N .
Ach|M
N
m
∀A− ⊂∗ A ∀N− ≺∗ N
∃B− ⊂ M so that B− ≡N− A−.
(By ⊂∗, ≺∗ I mean “small”...)
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Independence notions
Let us consider the following independence notion:
Denition (coheir independence (v. Boney-Grossberg))
Let M ≺ N .
Ach|M
N
m
∀A− ⊂∗ A ∀N− ≺∗ N
∃B− ⊂ M so that B− ≡N− A−.
(By ⊂∗, ≺∗ I mean “small”...)
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Independence notions
Let us consider the following independence notion:
Denition (coheir independence (v. Boney-Grossberg))
Let M ≺ N .
Ach|M
N
m
∀A− ⊂∗ A ∀N− ≺∗ N
∃B− ⊂ M so that B− ≡N− A−.
(By ⊂∗, ≺∗ I mean “small”...)
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Independence notions
Let us consider the following independence notion:
Denition (coheir independence (v. Boney-Grossberg))
Let M ≺ N .
Ach|M
N
m
∀A− ⊂∗ A ∀N− ≺∗ N
∃B− ⊂ M so that B− ≡N− A−.
(By ⊂∗, ≺∗ I mean “small”...)
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Coheir
I The previous was an example of an “abstract” independence
notion. In rst order, these presentations of independence
notions have been studied by Adler and others (and back to von
Neumann).
I The point is to compare tp(A/N ) and tp(A/M).
I In this case, the type tp(A/N ) is locally (∀A− ⊂∗ A) realizable
inside M (∃B− ⊂ M so that B− ≡N− A−
tp(B−/N−) = tp(A−/N−).
I A |chMN generalizes to tame, typeshort AECs Shelah’s
“non-forking”
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Coheir
I The previous was an example of an “abstract” independence
notion. In rst order, these presentations of independence
notions have been studied by Adler and others (and back to von
Neumann).
I The point is to compare tp(A/N ) and tp(A/M).
I In this case, the type tp(A/N ) is locally (∀A− ⊂∗ A) realizable
inside M (∃B− ⊂ M so that B− ≡N− A−
tp(B−/N−) = tp(A−/N−).
I A |chMN generalizes to tame, typeshort AECs Shelah’s
“non-forking”
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Coheir
I The previous was an example of an “abstract” independence
notion. In rst order, these presentations of independence
notions have been studied by Adler and others (and back to von
Neumann).
I The point is to compare tp(A/N ) and tp(A/M).
I In this case, the type tp(A/N ) is locally (∀A− ⊂∗ A) realizable
inside M (∃B− ⊂ M so that B− ≡N− A−
tp(B−/N−) = tp(A−/N−).
I A |chMN generalizes to tame, typeshort AECs Shelah’s
“non-forking”
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Coheir
I The previous was an example of an “abstract” independence
notion. In rst order, these presentations of independence
notions have been studied by Adler and others (and back to von
Neumann).
I The point is to compare tp(A/N ) and tp(A/M).
I In this case, the type tp(A/N ) is locally (∀A− ⊂∗ A) realizable
inside M (∃B− ⊂ M so that B− ≡N− A−
tp(B−/N−) = tp(A−/N−).
I A |chMN generalizes to tame, typeshort AECs Shelah’s
“non-forking”
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Landmarks: Canonicity of independence
I 1970: Shelah discovers the notion tp(a/B) forks over A (usually
A ⊂ B). He generalized “change in Morley Rank”, from ω-stable
theories.
I 1974: Lascar proved that nonforking independence is canonical
in superstable theories.
I 1984: Harnik-Harrington generalized this to stable theories.
I 1997: Kim-Pillay generalized this to simple theories, by using
the “independence theorem” (really, type amalgamation)
I 2015: Chernikov-Ramsey generalize Kim-Pillay to SOP1
theories
I 2016: Boney-Grossberg-Kolesnikov-VanDieren extend
Harnik-Harrington to AECs (satisfying NMM, AP, JEP)
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Landmarks: Canonicity of independence
I 1970: Shelah discovers the notion tp(a/B) forks over A (usually
A ⊂ B). He generalized “change in Morley Rank”, from ω-stable
theories.
I 1974: Lascar proved that nonforking independence is canonical
in superstable theories.
I 1984: Harnik-Harrington generalized this to stable theories.
I 1997: Kim-Pillay generalized this to simple theories, by using
the “independence theorem” (really, type amalgamation)
I 2015: Chernikov-Ramsey generalize Kim-Pillay to SOP1
theories
I 2016: Boney-Grossberg-Kolesnikov-VanDieren extend
Harnik-Harrington to AECs (satisfying NMM, AP, JEP)
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Landmarks: Canonicity of independence
I 1970: Shelah discovers the notion tp(a/B) forks over A (usually
A ⊂ B). He generalized “change in Morley Rank”, from ω-stable
theories.
I 1974: Lascar proved that nonforking independence is canonical
in superstable theories.
I 1984: Harnik-Harrington generalized this to stable theories.
I 1997: Kim-Pillay generalized this to simple theories, by using
the “independence theorem” (really, type amalgamation)
I 2015: Chernikov-Ramsey generalize Kim-Pillay to SOP1
theories
I 2016: Boney-Grossberg-Kolesnikov-VanDieren extend
Harnik-Harrington to AECs (satisfying NMM, AP, JEP)
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Landmarks: Canonicity of independence
I 1970: Shelah discovers the notion tp(a/B) forks over A (usually
A ⊂ B). He generalized “change in Morley Rank”, from ω-stable
theories.
I 1974: Lascar proved that nonforking independence is canonical
in superstable theories.
I 1984: Harnik-Harrington generalized this to stable theories.
I 1997: Kim-Pillay generalized this to simple theories, by using
the “independence theorem” (really, type amalgamation)
I 2015: Chernikov-Ramsey generalize Kim-Pillay to SOP1
theories
I 2016: Boney-Grossberg-Kolesnikov-VanDieren extend
Harnik-Harrington to AECs (satisfying NMM, AP, JEP)
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Landmarks: Canonicity of independence
I 1970: Shelah discovers the notion tp(a/B) forks over A (usually
A ⊂ B). He generalized “change in Morley Rank”, from ω-stable
theories.
I 1974: Lascar proved that nonforking independence is canonical
in superstable theories.
I 1984: Harnik-Harrington generalized this to stable theories.
I 1997: Kim-Pillay generalized this to simple theories, by using
the “independence theorem” (really, type amalgamation)
I 2015: Chernikov-Ramsey generalize Kim-Pillay to SOP1
theories
I 2016: Boney-Grossberg-Kolesnikov-VanDieren extend
Harnik-Harrington to AECs (satisfying NMM, AP, JEP)
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Landmarks: Canonicity of independence
I 1970: Shelah discovers the notion tp(a/B) forks over A (usually
A ⊂ B). He generalized “change in Morley Rank”, from ω-stable
theories.
I 1974: Lascar proved that nonforking independence is canonical
in superstable theories.
I 1984: Harnik-Harrington generalized this to stable theories.
I 1997: Kim-Pillay generalized this to simple theories, by using
the “independence theorem” (really, type amalgamation)
I 2015: Chernikov-Ramsey generalize Kim-Pillay to SOP1
theories
I 2016: Boney-Grossberg-Kolesnikov-VanDieren extend
Harnik-Harrington to AECs (satisfying NMM, AP, JEP)
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Map of the Universe (à la façon FO Mod Th)
from Gabriel Conant’s interactive website -
http://forkinganddividing.com/
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Two extensions, inside and orthogonal to “the map”
Theorem (2015 - Chernikov, Ramsey - guaranteeing SOP1)
Let | be an Aut(M)-invariant independence relation on small subsetsof M (the monster M |= T) such that for all M ≺∗ M
1. [strong nite character] if a 6 |Mb then there is a formula
ϕ(x, b,m) ∈ tp(a/bM) such that for any a′ |= ϕ(x, b,m),a′ 6 |
Mb
2. [existence over models] M |= T implies a |MM for all a
3. [monotonicity] aa′ |Mbb′ implies a |
Mb
4. [symmetry] a |Mb⇔ b |
Ma
5. [independent amalgamation] c0 | M c1, b0 | M c0, b1 | M c1,b0 ≡M b1 implies there exists b with b ≡c0M b0, b ≡c1M b1.
Then, T is NSOP1.
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Two extensions, inside and orthogonal to “the map”
Theorem (2015 - Chernikov, Ramsey - guaranteeing SOP1)
Let | be an Aut(M)-invariant independence relation on small subsetsof M (the monster M |= T) such that for all M ≺∗ M
1. [strong nite character] if a 6 |Mb then there is a formula
ϕ(x, b,m) ∈ tp(a/bM) such that for any a′ |= ϕ(x, b,m),a′ 6 |
Mb
2. [existence over models] M |= T implies a |MM for all a
3. [monotonicity] aa′ |Mbb′ implies a |
Mb
4. [symmetry] a |Mb⇔ b |
Ma
5. [independent amalgamation] c0 | M c1, b0 | M c0, b1 | M c1,b0 ≡M b1 implies there exists b with b ≡c0M b0, b ≡c1M b1.
Then, T is NSOP1.
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Two extensions, inside and orthogonal to “the map”
Theorem (2015 - Chernikov, Ramsey - guaranteeing SOP1)
Let | be an Aut(M)-invariant independence relation on small subsetsof M (the monster M |= T) such that for all M ≺∗ M
1. [strong nite character] if a 6 |Mb then there is a formula
ϕ(x, b,m) ∈ tp(a/bM) such that for any a′ |= ϕ(x, b,m),a′ 6 |
Mb
2. [existence over models] M |= T implies a |MM for all a
3. [monotonicity] aa′ |Mbb′ implies a |
Mb
4. [symmetry] a |Mb⇔ b |
Ma
5. [independent amalgamation] c0 | M c1, b0 | M c0, b1 | M c1,b0 ≡M b1 implies there exists b with b ≡c0M b0, b ≡c1M b1.
Then, T is NSOP1.
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Two extensions, inside and orthogonal to “the map”
Theorem (2015 - Chernikov, Ramsey - guaranteeing SOP1)
Let | be an Aut(M)-invariant independence relation on small subsetsof M (the monster M |= T) such that for all M ≺∗ M
1. [strong nite character] if a 6 |Mb then there is a formula
ϕ(x, b,m) ∈ tp(a/bM) such that for any a′ |= ϕ(x, b,m),a′ 6 |
Mb
2. [existence over models] M |= T implies a |MM for all a
3. [monotonicity] aa′ |Mbb′ implies a |
Mb
4. [symmetry] a |Mb⇔ b |
Ma
5. [independent amalgamation] c0 | M c1, b0 | M c0, b1 | M c1,b0 ≡M b1 implies there exists b with b ≡c0M b0, b ≡c1M b1.
Then, T is NSOP1.
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Two extensions, inside and orthogonal to “the map”
Theorem (2015 - Chernikov, Ramsey - guaranteeing SOP1)
Let | be an Aut(M)-invariant independence relation on small subsetsof M (the monster M |= T) such that for all M ≺∗ M
1. [strong nite character] if a 6 |Mb then there is a formula
ϕ(x, b,m) ∈ tp(a/bM) such that for any a′ |= ϕ(x, b,m),a′ 6 |
Mb
2. [existence over models] M |= T implies a |MM for all a
3. [monotonicity] aa′ |Mbb′ implies a |
Mb
4. [symmetry] a |Mb⇔ b |
Ma
5. [independent amalgamation] c0 | M c1, b0 | M c0, b1 | M c1,b0 ≡M b1 implies there exists b with b ≡c0M b0, b ≡c1M b1.
Then, T is NSOP1.
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Two extensions, inside and orthogonal to “the map”
Theorem (2015 - Chernikov, Ramsey - guaranteeing SOP1)
Let | be an Aut(M)-invariant independence relation on small subsetsof M (the monster M |= T) such that for all M ≺∗ M
1. [strong nite character] if a 6 |Mb then there is a formula
ϕ(x, b,m) ∈ tp(a/bM) such that for any a′ |= ϕ(x, b,m),a′ 6 |
Mb
2. [existence over models] M |= T implies a |MM for all a
3. [monotonicity] aa′ |Mbb′ implies a |
Mb
4. [symmetry] a |Mb⇔ b |
Ma
5. [independent amalgamation] c0 | M c1, b0 | M c0, b1 | M c1,b0 ≡M b1 implies there exists b with b ≡c0M b0, b ≡c1M b1.
Then, T is NSOP1.
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Canonicity for |ch on AEC’s
I At most one abstract independence relation satises existence,
extension, uniqueness and local character (under NMM, AP,
JEP)
I |ch equals non-forking if furthermore the AEC is tame and
typeshort
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Continuous Model Theory - Origins
I Remote origins of Continuous
Model Theory: von Neumann
(Continuous Geometry),
I Chang & Keisler (1966),
I Ben Yaacov, Berenstein, Henson,
Usvyatsov: a monograph that
summarizes stability theory for
continuous logic (around 2004)
I More recently, Boney, Caicedo,
Eagle, Iovino, etc. have
generalized continuous logic
I Hirvonen, Hyttinen - V.,
Zambrano: categoricity and
superstability in metric AECs
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Continuous predicates and functions
Denition (Interpreting in a metric structure)
Fix (M, d) a bounded metric space. A continuous n-ary predicate is a
uniformly continuous function
P : Mn → [0, 1].
A continuous n-ary function is a uniformly continuous function
f : Mn → M.
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Metric structures
Metric structures are of the form
M =(M, d, (fi)i∈I , (Rj)j∈J , (ak)k∈K
)
where the Ri and the fj are (uniformly) continuous functions with
values in [0, 1], the ak are distinguished elements of M .
M is a bounded metric space. Each function, relation must be
endowed with a modulus of uniform continuity.
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Metric structures
Metric structures are of the form
M =(M, d, (fi)i∈I , (Rj)j∈J , (ak)k∈K
)where the Ri and the fj are (uniformly) continuous functions with
values in [0, 1], the ak are distinguished elements of M .
M is a bounded metric space.
Each function, relation must be
endowed with a modulus of uniform continuity.
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Metric structures
Metric structures are of the form
M =(M, d, (fi)i∈I , (Rj)j∈J , (ak)k∈K
)where the Ri and the fj are (uniformly) continuous functions with
values in [0, 1], the ak are distinguished elements of M .
M is a bounded metric space. Each function, relation must be
endowed with a modulus of uniform continuity.
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Examples of FO metric structures
Example
I Any FO structure, endowed with the discrete metric.
I Banach algebras (bounding them).
I Hilbert spaces with inner product as a binary predicate.
I For a probability space (Ω,B, µ), construct a metric structureM based on
the usual measure algebra of (Ω,B, µ).
I Representations of C∗-algebras (Argoty, Berenstein, Ben Yaacov, V.).
I Farah, Hart: pathological properties of operator algebras
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Examples of FO metric structures
Example
I Any FO structure, endowed with the discrete metric.
I Banach algebras (bounding them).
I Hilbert spaces with inner product as a binary predicate.
I For a probability space (Ω,B, µ), construct a metric structureM based on
the usual measure algebra of (Ω,B, µ).
I Representations of C∗-algebras (Argoty, Berenstein, Ben Yaacov, V.).
I Farah, Hart: pathological properties of operator algebras
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Examples of FO metric structures
Example
I Any FO structure, endowed with the discrete metric.
I Banach algebras (bounding them).
I Hilbert spaces with inner product as a binary predicate.
I For a probability space (Ω,B, µ), construct a metric structureM based on
the usual measure algebra of (Ω,B, µ).
I Representations of C∗-algebras (Argoty, Berenstein, Ben Yaacov, V.).
I Farah, Hart: pathological properties of operator algebras
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Examples of FO metric structures
Example
I Any FO structure, endowed with the discrete metric.
I Banach algebras (bounding them).
I Hilbert spaces with inner product as a binary predicate.
I For a probability space (Ω,B, µ), construct a metric structureM based on
the usual measure algebra of (Ω,B, µ).
I Representations of C∗-algebras (Argoty, Berenstein, Ben Yaacov, V.).
I Farah, Hart: pathological properties of operator algebras
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Examples of FO metric structures
Example
I Any FO structure, endowed with the discrete metric.
I Banach algebras (bounding them).
I Hilbert spaces with inner product as a binary predicate.
I For a probability space (Ω,B, µ), construct a metric structureM based on
the usual measure algebra of (Ω,B, µ).
I Representations of C∗-algebras (Argoty, Berenstein, Ben Yaacov, V.).
I Farah, Hart: pathological properties of operator algebras
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
The syntax
1. Terms: as usual.
2. Atomic formulas: d(t1, tn) and R(t1, · · · , tn), if the ti are terms.
Formulas are then interpreted as functions into [0, 1].
3. Connectives: continuous functions from [0, 1]n → [0, 1].Therefore, applying connectives to formulas gives new
formulas.
4. Quantiers: supx ϕ(x) (universal) and infx ϕ(x) (existential).
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Interpretation
The logical distance between ϕ(x) and ψ(x) is
supa∈M |ϕM(a)− ψM(a)|.The satisfaction relation is dened on conditions rather than on
formulas.
Conditions are expressions of the form ϕ(x) ≤ ψ(y), ϕ(x) ≤ ψ(y),
ϕ(x) ≥ ψ(y), etc.
Notice also that the set of connectives is too large, but it may be
“densely” and uniformly generated by 0, 1, x/2,.−: for every ε, for
every connective f (t1, · · · , tn) there exists a connective g(t1, · · · , tn)generated by these four by composition such that |f (~t)− g(~t)| < ε.
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Interpretation
The logical distance between ϕ(x) and ψ(x) is
supa∈M |ϕM(a)− ψM(a)|.The satisfaction relation is dened on conditions rather than on
formulas.
Conditions are expressions of the form ϕ(x) ≤ ψ(y), ϕ(x) ≤ ψ(y),
ϕ(x) ≥ ψ(y), etc.
Notice also that the set of connectives is too large, but it may be
“densely” and uniformly generated by 0, 1, x/2,.−: for every ε, for
every connective f (t1, · · · , tn) there exists a connective g(t1, · · · , tn)generated by these four by composition such that |f (~t)− g(~t)| < ε.
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Interpretation
The logical distance between ϕ(x) and ψ(x) is
supa∈M |ϕM(a)− ψM(a)|.The satisfaction relation is dened on conditions rather than on
formulas.
Conditions are expressions of the form ϕ(x) ≤ ψ(y), ϕ(x) ≤ ψ(y),
ϕ(x) ≥ ψ(y), etc.
Notice also that the set of connectives is too large, but it may be
“densely” and uniformly generated by 0, 1, x/2,.−: for every ε, for
every connective f (t1, · · · , tn) there exists a connective g(t1, · · · , tn)generated by these four by composition such that |f (~t)− g(~t)| < ε.
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Stability Theory
I Stability (Ben Yaacov, Iovino, etc.),
I Categoricity for countable languages (Ben Yaacov),
I ω-stability,
I Dependent theories (Ben Yaacov),
I Not much geometric stability theory: no analog to
Baldwin-Lachlan (no minimality, except some openings by
Usvyatsov and Shelah in the context of ℵ1-categorical Banach
spaces),
I NO simplicity!!! (Berenstein, Hyttinen, V.),
I Keisler measures, NIP (Hrushovski, Pillay, etc.).
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Stability Theory
I Stability (Ben Yaacov, Iovino, etc.),
I Categoricity for countable languages (Ben Yaacov),
I ω-stability,
I Dependent theories (Ben Yaacov),
I Not much geometric stability theory: no analog to
Baldwin-Lachlan (no minimality, except some openings by
Usvyatsov and Shelah in the context of ℵ1-categorical Banach
spaces),
I NO simplicity!!! (Berenstein, Hyttinen, V.),
I Keisler measures, NIP (Hrushovski, Pillay, etc.).
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Stability Theory
I Stability (Ben Yaacov, Iovino, etc.),
I Categoricity for countable languages (Ben Yaacov),
I ω-stability,
I Dependent theories (Ben Yaacov),
I Not much geometric stability theory: no analog to
Baldwin-Lachlan (no minimality, except some openings by
Usvyatsov and Shelah in the context of ℵ1-categorical Banach
spaces),
I NO simplicity!!! (Berenstein, Hyttinen, V.),
I Keisler measures, NIP (Hrushovski, Pillay, etc.).
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Stability Theory
I Stability (Ben Yaacov, Iovino, etc.),
I Categoricity for countable languages (Ben Yaacov),
I ω-stability,
I Dependent theories (Ben Yaacov),
I Not much geometric stability theory: no analog to
Baldwin-Lachlan (no minimality, except some openings by
Usvyatsov and Shelah in the context of ℵ1-categorical Banach
spaces),
I NO simplicity!!! (Berenstein, Hyttinen, V.),
I Keisler measures, NIP (Hrushovski, Pillay, etc.).
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
"Continuous Model Theory" beyond First Order
Several contexts, some unexplored so far.
1. Metric Abstract Elementary Classes (Hirvonen, Hyttinen -
ω-stability, V. Zambrano - superstability, domination, notions of
independence): an amalgam of the power of Abstract
Elementary Classes with metric ideas.
2. Continuous Lω1ω . So far, no published results as such. There are
however “Lindström theorems” for Continuous First Order due
to Caicedo/Iovino and Eagle.
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
"Continuous Model Theory" beyond First Order
Several contexts, some unexplored so far.
1. Metric Abstract Elementary Classes (Hirvonen, Hyttinen -
ω-stability, V. Zambrano - superstability, domination, notions of
independence): an amalgam of the power of Abstract
Elementary Classes with metric ideas.
2. Continuous Lω1ω . So far, no published results as such. There are
however “Lindström theorems” for Continuous First Order due
to Caicedo/Iovino and Eagle.
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Independence, aware of metric and continuity
Denition (ε-coheir / the simplest)
Let M ≺ N .
Ach,ε|M
N
m
∀A− ⊂∗ A ∀N− ≺∗ N
∃B− ⊂ M so that
d(tp(B−/N−), tp(A−/N−)) < ε.
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Independence, aware of metric and continuity
Denition (ε-coheir / the simplest)
Let M ≺ N .
Ach,ε|M
N
m
∀A− ⊂∗ A ∀N− ≺∗ N
∃B− ⊂ M so that
d(tp(B−/N−), tp(A−/N−)) < ε.
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Independence, aware of metric and continuity
Denition (ε-coheir / the simplest)
Let M ≺ N .
Ach,ε|M
N
m
∀A− ⊂∗ A ∀N− ≺∗ N
∃B− ⊂ M so that
d(tp(B−/N−), tp(A−/N−)) < ε.
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Independence, aware of metric and continuity
Denition (ε-coheir / the simplest)
Let M ≺ N .
Ach,ε|M
N
m
∀A− ⊂∗ A ∀N− ≺∗ N
∃B− ⊂ M so that
d(tp(B−/N−), tp(A−/N−)) < ε.
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Generic Predicates
Theorem (Chatzidakis-Pillay)
Let T0 = ACF0, LP = L ∪ P, P a new symbol for a unary predicate.TP = T0∪“P is a generic predicate”- again the model companion ofT0∪“P is a predicate”. Then TP is a simple theory.
The existence of TP is ensured as T0 eliminates ∃∞.
If (k, P) |= TP , A,B,C ⊂ k, C = C,
AP|CB ⇔ A |
CB
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Generic Predicates
Theorem (Chatzidakis-Pillay)
Let T0 = ACF0, LP = L ∪ P, P a new symbol for a unary predicate.TP = T0∪“P is a generic predicate”- again the model companion ofT0∪“P is a predicate”. Then TP is a simple theory.
The existence of TP is ensured as T0 eliminates ∃∞.
If (k, P) |= TP , A,B,C ⊂ k, C = C,
AP|CB ⇔ A |
CB
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Generic Predicates
Theorem (Chatzidakis-Pillay)
Let T0 = ACF0, LP = L ∪ P, P a new symbol for a unary predicate.TP = T0∪“P is a generic predicate”- again the model companion ofT0∪“P is a predicate”. Then TP is a simple theory.
The existence of TP is ensured as T0 eliminates ∃∞.
If (k, P) |= TP , A,B,C ⊂ k, C = C,
AP|CB ⇔ A |
CB
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Our aim: an analog of Chatzidakis-Pillay, TP
We will next look at structures of the form
(H,+, 0, 〈〉, dN )
where dN (x) “measures” the distance to a set of “black points” which
we call N (H).
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
TP and simplicity
This will correspond in a natural way, once we go to the model
companions, to a generic “predicate” no longer dividing the Hilbert
space into two complementary areas, but rather a “generic grey
spot” with shades of grey between black and white. . . in a generic
way. Our result also intended to produce the rst simple unstable
theories in Continuous Model Theory.
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Basic definitions
We consider structures of the form
(H,+, 0, 〈〉, dN )
The function dN is therefore additional to the structure of Hilbert
Spaces. Let THilbert be the theory of Hilbert spaces, and let
T0 := THilbert ∪ Ax1,Ax2:1. Ax1: supx infy max|dN (x)− ‖x − y‖|, dN (y) = 0
2. Ax2: supx supy dN (x) ≤ dN (y) + ‖x − y‖
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Basic definitions
We consider structures of the form
(H,+, 0, 〈〉, dN )
The function dN is therefore additional to the structure of Hilbert
Spaces. Let THilbert be the theory of Hilbert spaces, and let
T0 := THilbert ∪ Ax1,Ax2:1. Ax1: supx infy max|dN (x)− ‖x − y‖|, dN (y) = 0
2. Ax2: supx supy dN (x) ≤ dN (y) + ‖x − y‖
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Amalgamation
Let (H0, d0
N ) ⊂ (Hi, diN ) where i ∈ 1, 2 and let H1 | H0
H2 be
Hilbert spaces with distance functions, all of them in
Gr = 〈H , . . . , dN 〉|dN (0) = r.
Let H3 = spanH1,H2 and let
d3
N (v) = min
√d1
N (PH1(v))2 + ‖P
H2∩H⊥
0
(v)‖2,√
d2
N (PH2(v))2 + ‖P
H1∩H⊥
0
(v)‖2
.
Then (Hi, diN ) ⊂ (H3, d3
N ) for i ∈ 1, 2, (H3, d3
N ) |= T0 and
(H3, d3
N ) ∈ Gr .
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Fraïssé limits
Characterize
Td,0 = Thcont(lim−→Fr
(K0))
for the class K0 of all nite dimensional models of T0 such that 0 is a
black point.
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
A typical configuration for the antecedent
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
And the new axiom, modulo ε and ϕ
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
EC models
Theorem (Berenstein and V.)
(M, dN ) is an existentially closed model of T0 if and only if(M, dN ) |= TN .
However,
Theorem (Berenstein and V.)
TN does not have elimination of quantiers.
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
EC models - QE
(We take H with orthonormal basis u1, u2; N = 0, u0 + 1
4u1,
dN (x) = min1, d(x,N ). Then (H1, d1
N ) |= T0. Let a = u0,
b = u0 − 1
4u1 and c = u0 + 1
4u1. Then d′N (b) = 1
2.
Let H ′′ also have an orthonormal basis vi : i ∈ ω,N2 = x ∈ H : ‖x − v1‖ = 1
4, P
span(v1)(x) = v1 ∪ 0 y
d′′N (x) = min1, d(x,N2).
Then (span(a), d1
N span(a))F∼= (span(v1), d2
N span(v1)). But (H ′, d′N )and (H ′′, d′′N ) have no amalgam over the common part: if they had,
we would have d(F(b), v1 + 1
4vi) = d(b, v1 + 1
4vi) < 1
2for (some)
i > 1, i.e. d1
N (b) < 1
2, contradiction.)
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
No amalgams, no QE - Interference
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
TN , among the few theories both TP2 and NTP1
Denition (TP2)
A formula ϕ(x; y) has TP2 if there exists k < ω and there exists a
matrix of tuples
a00 a01 · · · a0i · · ·a10 a11 · · · a1i · · ·...
aα0 aα1 · · · aαi · · ·...
rows (ϕ(x, aα,i) | i < ω is k-inconsistent for each α)
functions across going down for each f ∈ ωωϕ(x, aα,f (α)) | α < ω is consistent.
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
TP2 - bad side
Theorem (Hyttinen, V.)
TN has TP2.
Proof (sketch) Let 〈af | f : ω → ω〉 ∪ 〈bn | n < ω〉 ∪ 〈cn,i | i, n < ω〉be an orthonormal basis of a Hilbert space.
Let the “black points” consist of
af + bn + 1
2cn,f (i) | f : ω → ω, n, i < ω and let
ϕ(x, y, z) : dN (x + y − 1
2
z) ≥ 1 ∧ dN (x + y +1
2
z) ≤ 0.
This formula witnesses TP2. (Originally we had a proof of failure of
simplicity using Casanovas/Shelah’s characterization but this is far
worse!)
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
TP2 - bad side
Theorem (Hyttinen, V.)
TN has TP2.
Proof (sketch) Let 〈af | f : ω → ω〉 ∪ 〈bn | n < ω〉 ∪ 〈cn,i | i, n < ω〉be an orthonormal basis of a Hilbert space.
Let the “black points” consist of
af + bn + 1
2cn,f (i) | f : ω → ω, n, i < ω and let
ϕ(x, y, z) : dN (x + y − 1
2
z) ≥ 1 ∧ dN (x + y +1
2
z) ≤ 0.
This formula witnesses TP2. (Originally we had a proof of failure of
simplicity using Casanovas/Shelah’s characterization but this is far
worse!)
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
TP2 - bad side
Theorem (Hyttinen, V.)
TN has TP2.
Proof (sketch) Let 〈af | f : ω → ω〉 ∪ 〈bn | n < ω〉 ∪ 〈cn,i | i, n < ω〉be an orthonormal basis of a Hilbert space.
Let the “black points” consist of
af + bn + 1
2cn,f (i) | f : ω → ω, n, i < ω and let
ϕ(x, y, z) : dN (x + y − 1
2
z) ≥ 1 ∧ dN (x + y +1
2
z) ≤ 0.
This formula witnesses TP2. (Originally we had a proof of failure of
simplicity using Casanovas/Shelah’s characterization but this is far
worse!)
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
NTP1... really NSOP1 - good side
Theorem (Hyttinen, V.)
TN has NSOP1.
Denition (SOP1)
ϕ(x; y) is SOP1 i there exists a tree of parameters (aη)η∈2<ω such
that
I for every η ∈ 2ω
, ϕ(x; aηn) | n < ω is consistent
I if η_0 E ν ∈ 2<ω
then ϕ(x; aη_1), ϕ(x; aν) is inconsistent.
T is NSOP1 if NO formula has SOP1 in T .
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
NTP1... really NSOP1 - good side
Sketch of proof:
I We adapt Chernikov-Ramsey (characterization of NSOP1 in
terms of an independence notion) to the continuous setting.
I We prove that the independence property | ∗ satises the ve
properties.
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
estions
I Why does Chatzidakis-Pillay fail so badly in the continuous
setting?
I Why the “polarization” of dividing lines in the continuous case?
I The interference and the tree property TP2 both seem to be
connected with the non-triviality of the metric. Is there a
deeper model-theoretic/logical reason for this?
Independence notions Continuous world/independence Obstacles to continuous Chatzidakis-Pillay Generic independence, good and bad
Thank you! Gràcies! ¡Gracias!