andrés villaveces universidad nacional de colombia - bogotáindependence notions continuous...

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

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)

Let us consider the following independence notion:

Denition (coheir independence (v. Boney-Grossberg))

Let M ≺ N .

∀A− ⊂∗ A ∀N− ≺∗ N

∃B− ⊂ M so that B− ≡N− A−.

(By ⊂∗, ≺∗ I mean “small”...)

Let M ≺ N .

∀A− ⊂∗ A ∀N− ≺∗ N

Let M ≺ N .

∀A− ⊂∗ A ∀N− ≺∗ N

Let M ≺ N .

∀A− ⊂∗ A ∀N− ≺∗ N

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”

Coheir

Neumann).

tp(B−/N−) = tp(A−/N−).

“non-forking”

Coheir

Neumann).

tp(B−/N−) = tp(A−/N−).

“non-forking”

Coheir

Neumann).

tp(B−/N−) = tp(A−/N−).

“non-forking”

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)

theories.

theories

theories.

theories

theories.

theories

theories.

theories

theories.

theories

Map of the Universe (à la façon FO Mod Th)

from Gabriel Conant’s interactive website -

http://forkinganddividing.com/

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 |

2. [existence over models] M |= T implies a |MM for all a

3. [monotonicity] aa′ |Mbb′ implies a |

4. [symmetry] a |Mb⇔ b |

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.

Canonicity for |ch on AEC’s

I At most one abstract independence relation satises existence,

extension, uniqueness and local character (under NMM, AP,

I |ch equals non-forking if furthermore the AEC is tame and

typeshort

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

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.

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.

Metric structures

)where the Ri and the fj are (uniformly) continuous functions with

M is a bounded metric space.

Each function, relation must be

Metric structures

)where the Ri and the fj are (uniformly) continuous functions with

M is a bounded metric space. Each function, relation must be

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

Example

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).

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.

Interpretation

formulas.

Interpretation

formulas.

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.).

Stability Theory

I ω-stability,

spaces),

Stability Theory

I ω-stability,

spaces),

Stability Theory

I ω-stability,

spaces),

"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.

"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, aware of metric and continuity

Denition (ε-coheir / the simplest)

Let M ≺ N .

Ach,ε|M

∀A− ⊂∗ A ∀N− ≺∗ N

∃B− ⊂ M so that

d(tp(B−/N−), tp(A−/N−)) < ε.

Let M ≺ N .

Ach,ε|M

∀A− ⊂∗ A ∀N− ≺∗ N

d(tp(B−/N−), tp(A−/N−)) < ε.

Let M ≺ N .

Ach,ε|M

∀A− ⊂∗ A ∀N− ≺∗ N

d(tp(B−/N−), tp(A−/N−)) < ε.

Let M ≺ N .

Ach,ε|M

∀A− ⊂∗ A ∀N− ≺∗ N

d(tp(B−/N−), tp(A−/N−)) < ε.

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 |

Generic Predicates

If (k, P) |= TP , A,B,C ⊂ k, C = C,

AP|CB ⇔ A |

Generic Predicates

If (k, P) |= TP , A,B,C ⊂ k, C = C,

AP|CB ⇔ A |

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).

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.

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‖

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‖

Amalgamation

Let (H0, d0

N ) ⊂ (Hi, diN ) where i ∈ 1, 2 and let H1 | H0

Hilbert spaces with distance functions, all of them in

Gr = 〈H , . . . , dN 〉|dN (0) = r.

Let H3 = spanH1,H2 and let

N (v) = min

N (PH1(v))2 + ‖P

H2∩H⊥

(v)‖2,√

N (PH2(v))2 + ‖P

H1∩H⊥

(v)‖2

Then (Hi, diN ) ⊂ (H3, d3

N ) for i ∈ 1, 2, (H3, d3

N ) |= T0 and

(H3, d3

N ) ∈ Gr .

Fraïssé limits

Characterize

Td,0 = Thcont(lim−→Fr

for the class K0 of all nite dimensional models of T0 such that 0 is a

black point.

A typical configuration for the antecedent

And the new axiom, modulo ε and ϕ

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.

EC models - QE

(We take H with orthonormal basis u1, u2; N = 0, u0 + 1

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

Let H ′′ also have an orthonormal basis vi : i ∈ ω,N2 = x ∈ H : ‖x − v1‖ = 1

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.)

No amalgams, no QE - Interference

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.

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

z) ≥ 1 ∧ dN (x + y +1

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!)

TP2 - bad side

TN has TP2.

af + bn + 1

ϕ(x, y, z) : dN (x + y − 1

z) ≥ 1 ∧ dN (x + y +1

z) ≤ 0.

worse!)

TP2 - bad side

TN has TP2.

af + bn + 1

ϕ(x, y, z) : dN (x + y − 1

z) ≥ 1 ∧ dN (x + y +1

z) ≤ 0.

worse!)

NTP1... really NSOP1 - good side

TN has NSOP1.

Denition (SOP1)

ϕ(x; y) is SOP1 i there exists a tree of parameters (aη)η∈2<ω such

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 .

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.

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?

Thank you! Gràcies! ¡Gracias!

andrés villaveces universidad nacional de colombia - bogotáindependence notions continuous...

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