Equivalence Relations, Classification Problems, and Descriptive Set Theory Su Gao Department of Mathematics University of North Texas Institute of Mathematics, CAS September 9, 2015 Su Gao Equivalence Relations, Classification Problems, and Descriptive Set Theory An equivalence relation on a set X is a binary relation E ⊆ X × X such that I (x; x) 2 E, I if (x; y) 2 E then (y; x) 2 E, I if (x; y) 2 E and (y; z) 2 E then (x; z) 2 E, for all x; y; z 2 X . Equivalence Relations Su Gao Equivalence Relations, Classification Problems, and Descriptive Set Theory I (x; x) 2 E, I if (x; y) 2 E then (y; x) 2 E, I if (x; y) 2 E and (y; z) 2 E then (x; z) 2 E, for all x; y; z 2 X . Equivalence Relations An equivalence relation on a set X is a binary relation E ⊆ X × X such that Su Gao Equivalence Relations, Classification Problems, and Descriptive Set Theory I if (x; y) 2 E then (y; x) 2 E, I if (x; y) 2 E and (y; z) 2 E then (x; z) 2 E, for all x; y; z 2 X . Equivalence Relations An equivalence relation on a set X is a binary relation E ⊆ X × X such that I (x; x) 2 E, Su Gao Equivalence Relations, Classification Problems, and Descriptive Set Theory I if (x; y) 2 E and (y; z) 2 E then (x; z) 2 E, for all x; y; z 2 X . Equivalence Relations An equivalence relation on a set X is a binary relation E ⊆ X × X such that I (x; x) 2 E, I if (x; y) 2 E then (y; x) 2 E, Su Gao Equivalence Relations, Classification Problems, and Descriptive Set Theory Equivalence Relations An equivalence relation on a set X is a binary relation E ⊆ X × X such that I (x; x) 2 E, I if (x; y) 2 E then (y; x) 2 E, I if (x; y) 2 E and (y; z) 2 E then (x; z) 2 E, for all x; y; z 2 X . Su Gao Equivalence Relations, Classification Problems, and Descriptive Set Theory 1. Coset equivalence: if G is a group and H ≤ G, define −1 g1 ∼ g2 () g1 g2 2 H () g1H = g2H 2. Orbit equivalence: if G y X is an action of a group on a set, then define x1 ∼ x2 () 9g 2 G g · x1 = x2 Equivalence Relations Some well-known examples: Su Gao Equivalence Relations, Classification Problems, and Descriptive Set Theory () g1H = g2H 2. Orbit equivalence: if G y X is an action of a group on a set, then define x1 ∼ x2 () 9g 2 G g · x1 = x2 Equivalence Relations Some well-known examples: 1. Coset equivalence: if G is a group and H ≤ G, define −1 g1 ∼ g2 () g1 g2 2 H Su Gao Equivalence Relations, Classification Problems, and Descriptive Set Theory 2. Orbit equivalence: if G y X is an action of a group on a set, then define x1 ∼ x2 () 9g 2 G g · x1 = x2 Equivalence Relations Some well-known examples: 1. Coset equivalence: if G is a group and H ≤ G, define −1 g1 ∼ g2 () g1 g2 2 H () g1H = g2H Su Gao Equivalence Relations, Classification Problems, and Descriptive Set Theory Equivalence Relations Some well-known examples: 1. Coset equivalence: if G is a group and H ≤ G, define −1 g1 ∼ g2 () g1 g2 2 H () g1H = g2H 2. Orbit equivalence: if G y X is an action of a group on a set, then define x1 ∼ x2 () 9g 2 G g · x1 = x2 Su Gao Equivalence Relations, Classification Problems, and Descriptive Set Theory 3. Vitali set: Consider the cosets of Q in R. Using AC, find a set V that meets each coset at exactly one point. V is not Lebesgue measurable. 4. Measure equivalence: two measures are equivalent iff they are absolutely continuous to each other. µ ν () 8A (ν(A) = 0 ) µ(A) = 0) Equivalence Relations Examples in measure theory: Su Gao Equivalence Relations, Classification Problems, and Descriptive Set Theory V is not Lebesgue measurable. 4. Measure equivalence: two measures are equivalent iff they are absolutely continuous to each other. µ ν () 8A (ν(A) = 0 ) µ(A) = 0) Equivalence Relations Examples in measure theory: 3. Vitali set: Consider the cosets of Q in R. Using AC, find a set V that meets each coset at exactly one point. Su Gao Equivalence Relations, Classification Problems, and Descriptive Set Theory 4. Measure equivalence: two measures are equivalent iff they are absolutely continuous to each other. µ ν () 8A (ν(A) = 0 ) µ(A) = 0) Equivalence Relations Examples in measure theory: 3. Vitali set: Consider the cosets of Q in R. Using AC, find a set V that meets each coset at exactly one point. V is not Lebesgue measurable. Su Gao Equivalence Relations, Classification Problems, and Descriptive Set Theory µ ν () 8A (ν(A) = 0 ) µ(A) = 0) Equivalence Relations Examples in measure theory: 3. Vitali set: Consider the cosets of Q in R. Using AC, find a set V that meets each coset at exactly one point. V is not Lebesgue measurable. 4. Measure equivalence: two measures are equivalent iff they are absolutely continuous to each other. Su Gao Equivalence Relations, Classification Problems, and Descriptive Set Theory Equivalence Relations Examples in measure theory: 3. Vitali set: Consider the cosets of Q in R. Using AC, find a set V that meets each coset at exactly one point. V is not Lebesgue measurable. 4. Measure equivalence: two measures are equivalent iff they are absolutely continuous to each other. µ ν () 8A (ν(A) = 0 ) µ(A) = 0) Su Gao Equivalence Relations, Classification Problems, and Descriptive Set Theory 5. Quotient space: If X is a topological space and ∼ an equivalence relation on X , then define I the quotient space: X =∼ = f [x]∼ : x 2 X g I the quotient map: π : X ! X = ∼ by π(x) = [x]∼ I the quotient topology: A ⊆ X = ∼ is open iff π−1(A) ⊆ X is open. Equivalence Relations Examples in topology: Su Gao Equivalence Relations, Classification Problems, and Descriptive Set Theory I the quotient map: π : X ! X = ∼ by π(x) = [x]∼ I the quotient topology: A ⊆ X = ∼ is open iff π−1(A) ⊆ X is open. Equivalence Relations Examples in topology: 5. Quotient space: If X is a topological space and ∼ an equivalence relation on X , then define I the quotient space: X =∼ = f [x]∼ : x 2 X g Su Gao Equivalence Relations, Classification Problems, and Descriptive Set Theory I the quotient topology: A ⊆ X = ∼ is open iff π−1(A) ⊆ X is open. Equivalence Relations Examples in topology: 5. Quotient space: If X is a topological space and ∼ an equivalence relation on X , then define I the quotient space: X =∼ = f [x]∼ : x 2 X g I the quotient map: π : X ! X = ∼ by π(x) = [x]∼ Su Gao Equivalence Relations, Classification Problems, and Descriptive Set Theory Equivalence Relations Examples in topology: 5. Quotient space: If X is a topological space and ∼ an equivalence relation on X , then define I the quotient space: X =∼ = f [x]∼ : x 2 X g I the quotient map: π : X ! X = ∼ by π(x) = [x]∼ I the quotient topology: A ⊆ X = ∼ is open iff π−1(A) ⊆ X is open. Su Gao Equivalence Relations, Classification Problems, and Descriptive Set Theory 6. The Henkin model: G¨odel'sCompleteness Theorem: Every consistent set of first-order sentences has a model. Henkin constructed a model using all first-order terms and defining t ∼ s () T ` t = s where T is a suitably constructed maximally consistent term-complete theory in an extended language with new constant symbols. Equivalence Relations Examples in logic: Su Gao Equivalence Relations, Classification Problems, and Descriptive Set Theory G¨odel'sCompleteness Theorem: Every consistent set of first-order sentences has a model. Henkin constructed a model using all first-order terms and defining t ∼ s () T ` t = s where T is a suitably constructed maximally consistent term-complete theory in an extended language with new constant symbols. Equivalence Relations Examples in logic: 6. The Henkin model: Su Gao Equivalence Relations, Classification Problems, and Descriptive Set Theory Henkin constructed a model using all first-order terms and defining t ∼ s () T ` t = s where T is a suitably constructed maximally consistent term-complete theory in an extended language with new constant symbols. Equivalence Relations Examples in logic: 6. The Henkin model: G¨odel'sCompleteness Theorem: Every consistent set of first-order sentences has a model. Su Gao Equivalence Relations, Classification Problems, and Descriptive Set Theory where T is a suitably constructed maximally consistent term-complete theory in an extended language with new constant symbols. Equivalence Relations Examples in logic: 6. The Henkin model: G¨odel'sCompleteness Theorem: Every consistent set of first-order sentences has a model. Henkin constructed a model using all first-order terms and defining t ∼ s () T ` t = s Su Gao Equivalence Relations, Classification Problems, and Descriptive Set Theory Equivalence Relations Examples in logic: 6. The Henkin model: G¨odel'sCompleteness Theorem: Every consistent set of first-order sentences has a model. Henkin constructed a model using all first-order terms and defining t ∼ s () T ` t = s where T is a suitably constructed maximally consistent term-complete theory in an extended language with new constant symbols. Su Gao Equivalence Relations, Classification Problems, and Descriptive Set Theory Example Classify square matrices up to similarity: A and B are similar iff there is a nonsingular matrix S such that A = S −1BS Two square matrices are similar iff they have the same Jordan normal form.