Forking and dimensions in pseudofinite structures Darío García University of Leeds. Logic Seminar - School of Mathematics. Leeds, UK. February 22, 2017
Model theory, forking and dimension
Model Theory: tame vs. wild structures Example (The quintessential example of a tame structure) Consider the theory of the complex field which can be effectively axiomatized by a finite number of axiom schemes: 1. Field axioms (finite in number). 2. ∀ x 1 , . . . , x n ∃ y ( y n + x 1 y n − 1 + · · · + x n = 0 ) , for n = 1 , 2 , 3 , . . . 3. 1 + · · · + 1 � = 0 , for n = 1 , 2 , 3 , . . . . � �� � n -times Example (The quintessential example of a wild structure) Gödel proved that Th ( Z , + , · ) cannot be effectively described in any reasonable way, so in contrast to the field of complex numbers, the ring of integers is wild. (But Z as ordered additive group is tame again!)
◮ We use here “tame” and “wild” very informally , to suggest the distinction between good and bad model-theoretic behavior. ◮ Shelah defined several dividing lines (stability, simplicity, NIP, NTP 2 , etc.) to distinguish “tame” from “wild” structures in terms of the (non-)existence of combinatorial configurations of their definable sets. ◮ Gödel’s work is often characterized as saying that those structures M for which Th ( M ) can be effectively axiomatizable are uninteresting. ◮ However, even in extremely “wild” subjects (such as number theory), the solution to difficult problems often uses illuminating explorations into tame territory!
◮ We use here “tame” and “wild” very informally , to suggest the distinction between good and bad model-theoretic behavior. ◮ Shelah defined several dividing lines (stability, simplicity, NIP, NTP 2 , etc.) to distinguish “tame” from “wild” structures in terms of the (non-)existence of combinatorial configurations of their definable sets. ◮ Gödel’s work is often characterized as saying that those structures M for which Th ( M ) can be effectively axiomatizable are uninteresting. ◮ However, even in extremely “wild” subjects (such as number theory), the solution to difficult problems often uses illuminating explorations into tame territory! model theory = geography of tame mathematics . E. Hrushovski c �
Model theory=geography of tame mathematics
Model theory=geography of tame mathematics Simple Rosy ( Z , + , · , < ) ZFC NTP 2 Pseudofinite fields Pseudoreal closed fields Random Graph Stable O-minimal ( Q , < ) ( Z , + , < ) ( Q p , + , · ) ( Z , +) NIP ( R , + , · , < ) Strongly minimal ( C , + , · , 0 , 1 ) A more detailed map at www. forkinganddividing. com (due to Gabriel Conant)
Dividing and forking. Definition Let φ ( x , b ) be a formula and A ⊆ M be a set of parameters. 1. We say that φ ( x , b ) divides over A if there is an infinite sequence � b i : i < ω � of elements such that: ◮ tp ( b i / A ) = tp ( b / A ) . ◮ The set of formulas { φ ( x , b i ) : i < ω } is k -inconsistent for some k < ω . 2. We say that a formula θ ( x ) (possibly with parameters) forks over A if θ ( x ) implies a finite disjunction of formulas that divide over A .
Example 1. In any theory T with infinite models, the formula φ ( x , b ) ≡ x = b divides over A whenever b �∈ acl ( A ) .
Example 1. In any theory T with infinite models, the formula φ ( x , b ) ≡ x = b divides over A whenever b �∈ acl ( A ) . It is enough to take a sequence � b i : i < ω � of different conjugates of b over A . The set of formulas { x = b i : i < ω } will be 2-inconsistent.
Example 1. In any theory T with infinite models, the formula φ ( x , b ) ≡ x = b divides over A = ∅ . 2. For the theory ACF of algebraically closed fields, the formula φ ( x , π ) ≡ x 2 = π divides over A = Q .
Example 1. In any theory T with infinite models, the formula φ ( x , b ) ≡ x = b divides over A = ∅ . 2. For the theory ACF of algebraically closed fields, the formula φ ( x , π ) ≡ x 2 = π divides over A = Q . If � π 1 , π 2 , . . . , � is a sequence of infinitely many distinct transcendental numbers, then we have: ◮ tp ( π i / Q ) = tp ( π/ Q ) . ◮ The set of formulas { x 2 = π i : i < ω } is 3-inconsistent. In fact, one can show that in ACF forking can be characterized by the algebraic formulas.
Example 1. In any theory T with infinite models, the formula φ ( x , b ) ≡ x = b divides over A = ∅ . 2. For the theory ACF of algebraically closed fields, the formula φ ( x , π ) ≡ x 2 = π divides over A = Q . In fact, one can show that in ACF forking can be characterized by the algebraic formulas. 3. In the theory DLO of dense linear orders, the formula φ ( x ; ab ) ≡ a < x < b divides over A = ∅ .
b b b Example 1. In any theory T with infinite models, the formula φ ( x , b ) ≡ x = b divides over A = ∅ . 2. For the theory ACF of algebraically closed fields, the formula φ ( x , π ) ≡ x 2 = π divides over A = Q . In fact, one can show that in ACF forking can be characterized by the algebraic formulas. 3. In the theory DLO of dense linear orders, the formula φ ( x ; ab ) ≡ a < x < b divides over A = ∅ . ( ) ( ) ( ) a 1 b 1 a 2 b 2 a 3 b 3 The set of formulas { a i < x < b i : i < ω } is 2-inconsistent.
Example 1. In any theory T with infinite models, the formula φ ( x , b ) ≡ x = b divides over A = ∅ . 2. For the theory ACF of algebraically closed fields, the formula φ ( x , π ) ≡ x 2 = π divides over A = Q . In fact, one can show that in ACF forking can be characterized by the algebraic formulas. 3. In the theory DLO of dense linear orders, the formula φ ( x ; ab ) ≡ a < x < b divides over A = ∅ . 4. In the theory T E of an equivalence relation with infinitely many infinite classes, the formula φ ( x , b ) ≡ xEb divides over A = ∅ .
Example 1. In any theory T with infinite models, the formula φ ( x , b ) ≡ x = b divides over A = ∅ . 2. For the theory ACF of algebraically closed fields, the formula φ ( x , π ) ≡ x 2 = π divides over A = Q . In fact, one can show that in ACF forking can be characterized by the algebraic formulas. 3. In the theory DLO of dense linear orders, the formula φ ( x ; ab ) ≡ a < x < b divides over A = ∅ . 4. In the theory T E of an equivalence relation with infinitely many infinite classes, the formula φ ( x , b ) ≡ xEb divides over A = ∅ . If � b i : i < ω � is a sequence of element in different equivalence classes, then the set of formulas { xEb i : i < ω } is 2-inconsistent (by the transitivity of E ).
Non-Forking independence Definition Given a tuple a , we say that a is independent from B over A if there is no formula φ ( x , b ) ∈ tp ( a / B ) that forks over A . We denote this by a | B ⌣ A ◮ The concept of forking and forking-independence played a crucial role in the Theory of Classification developed by Shelah, especially for stable theories. ◮ Even today, a recurrent theme in model theory is characterize forking in certain known structures in terms of combinatorial or algebraic invariants. ◮ The notion of non-forking independence generalizes several classic notions of independence (algebraic independence, linear independence, among others).
b b b Forking and dimension. Example Consider the formula φ ( x , y ; b ) ≡ y = x + b defined in C 2 . φ ( x , b 3 ) . . . φ ( x , b 2 ) 2-inconsistent φ ( x , b 1 ) b 3 b 2 b 1
This formula divides over ∅ . Intuitively, the set by φ ( x , y ; b ) is small as it defines a set of dimension 1 inside a space of dimension 2.
This formula divides over ∅ . Intuitively, the set by φ ( x , y ; b ) is small as it defines a set of dimension 1 inside a space of dimension 2. Question In those examples when there is a established notion of dimension, can forking-independence be detected by changes (decreasing) of this dimension? As the examples of the complex numbers suggest, an ideal answer to this question would be the following:
This formula divides over ∅ . Intuitively, the set by φ ( x , y ; b ) is small as it defines a set of dimension 1 inside a space of dimension 2. Question In those examples when there is a established notion of dimension, can forking-independence be detected by changes (decreasing) of this dimension? As the examples of the complex numbers suggest, an ideal answer to this question would be the following: c � | B ⇔ dim ( c / B ) < dim ( c / A ) ⌣ A ⇔ There is a set X = φ ( M ; b ) such that c ∈ X and dim ( X ) < dim ( Y ) for every A -definable set Y containing c .
Example 1. In ACF , the notion of dimension is transcendence degree: dim ( a / A ) := transc . deg ( Q ( A , a ) / Q ( A )) . We have √ π � | ⌣ Q Q ( π ) , but also dim ( √ π/ Q ) = transc . deg Q ( √ π ) / Q � � = 1 < dim ( √ π/ Q ( π )) = 0 .
b b b 2. In the theory T E , there is a combinatorial notion of dimension that can be defined recursively by the following rule: Definition The dimension of a point is equal to zero, and dim ( X ) ≥ n + 1 if and only if there are infinitely many disjoints sets Y i contained in X with dim ( Y i ) ≥ n . dim = 2
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