First Steps in Synthetic Computability Constructive Math Basic - PowerPoint PPT Presentation

First Steps in Some interesting sets Synthetic Computability ◮ The set of truth values: Andrej Bauer Introduction Ω = P 1 Constructive Math truth ⊤ = 1 , falsehood ⊥ = ∅ Basic Computability Theory ◮ The set of decidable truth values: Theorems for Free Enumerability Axiom Markov Principle � � � � p ∨ ¬ p Injectivity Axiom 2 = { 0 , 1 } = p ∈ Ω , Conclusion where we write 1 = ⊤ and 0 = ⊥ . ◮ The set of classical truth values: � � � � ¬¬ p = p p ∈ Ω Ω ¬¬ = . ◮ 2 ⊆ Ω ¬¬ ⊆ Ω .

First Steps in Decidable and classical sets Synthetic Computability ◮ A subset S ⊆ A is equivalently given by its Andrej Bauer characteristic map χ S : A → Ω , χ S ( x ) = ( x ∈ S ) . Introduction Constructive Math Basic Computability Theory Theorems for Free Enumerability Axiom Markov Principle Injectivity Axiom Conclusion

First Steps in Decidable and classical sets Synthetic Computability ◮ A subset S ⊆ A is equivalently given by its Andrej Bauer characteristic map χ S : A → Ω , χ S ( x ) = ( x ∈ S ) . Introduction ◮ A subset S ⊆ A is decidable if χ S : A → 2 , equivalently Constructive Math Basic Computability ∀ x ∈ A . ( x ∈ S ∨ x �∈ S ) . Theory Theorems for Free Enumerability Axiom Markov Principle Injectivity Axiom Conclusion

First Steps in Decidable and classical sets Synthetic Computability ◮ A subset S ⊆ A is equivalently given by its Andrej Bauer characteristic map χ S : A → Ω , χ S ( x ) = ( x ∈ S ) . Introduction ◮ A subset S ⊆ A is decidable if χ S : A → 2 , equivalently Constructive Math Basic Computability ∀ x ∈ A . ( x ∈ S ∨ x �∈ S ) . Theory Theorems for Free Enumerability Axiom ◮ A subset S ⊆ A is classical if χ S : A → Ω ¬¬ , Markov Principle Injectivity Axiom equivalently Conclusion ∀ x ∈ A . ( ¬ ( x �∈ S ) = ⇒ x ∈ S ) .

First Steps in The generic convergent sequence Synthetic Computability ◮ A useful set is the generic convergent sequence : Andrej Bauer � � Introduction a ∈ 2 N � N + = � ∀ k ∈ N . a k ≤ a k + 1 . Constructive Math Basic Computability Theory Theorems for Free Enumerability Axiom Markov Principle Injectivity Axiom Conclusion

First Steps in The generic convergent sequence Synthetic Computability ◮ A useful set is the generic convergent sequence : Andrej Bauer � � Introduction a ∈ 2 N � N + = � ∀ k ∈ N . a k ≤ a k + 1 . Constructive Math Basic Computability ◮ We have N ⊆ N + via n �→ λ k . ( k ≤ n ) . Theory Theorems for Free Enumerability Axiom Markov Principle Injectivity Axiom Conclusion

First Steps in The generic convergent sequence Synthetic Computability ◮ A useful set is the generic convergent sequence : Andrej Bauer � � Introduction a ∈ 2 N � N + = � ∀ k ∈ N . a k ≤ a k + 1 . Constructive Math Basic Computability ◮ We have N ⊆ N + via n �→ λ k . ( k ≤ n ) . Theory Theorems for Free ◮ But there is also ∞ = λ k . 0. Enumerability Axiom Markov Principle Injectivity Axiom Conclusion

First Steps in The generic convergent sequence Synthetic Computability ◮ A useful set is the generic convergent sequence : Andrej Bauer � � Introduction a ∈ 2 N � N + = � ∀ k ∈ N . a k ≤ a k + 1 . Constructive Math Basic Computability ◮ We have N ⊆ N + via n �→ λ k . ( k ≤ n ) . Theory Theorems for Free ◮ But there is also ∞ = λ k . 0. Enumerability Axiom Markov Principle ◮ N + can be thought of as the one-point Injectivity Axiom Conclusion compactification of N .

First Steps in Enumerable & finite sets Synthetic Computability ◮ A is finite if there exist n ∈ N and an onto map Andrej Bauer e : { 1 , . . . , n } ։ A , called a listing of A . An element Introduction may be listed more than once. Constructive Math Basic Computability Theory Theorems for Free Enumerability Axiom Markov Principle Injectivity Axiom Conclusion

First Steps in Enumerable & finite sets Synthetic Computability ◮ A is finite if there exist n ∈ N and an onto map Andrej Bauer e : { 1 , . . . , n } ։ A , called a listing of A . An element Introduction may be listed more than once. Constructive Math ◮ A is enumerable (countable) if there exists an onto map Basic Computability e : N ։ 1 + A , called an enumeration of A . For Theory Theorems for Free inhabited A we may take e : N ։ A . Enumerability Axiom Markov Principle Injectivity Axiom Conclusion

First Steps in Enumerable & finite sets Synthetic Computability ◮ A is finite if there exist n ∈ N and an onto map Andrej Bauer e : { 1 , . . . , n } ։ A , called a listing of A . An element Introduction may be listed more than once. Constructive Math ◮ A is enumerable (countable) if there exists an onto map Basic Computability e : N ։ 1 + A , called an enumeration of A . For Theory Theorems for Free inhabited A we may take e : N ։ A . Enumerability Axiom Markov Principle ◮ A is infinite if there exists an injective a : N ֌ A . Injectivity Axiom Conclusion

Lawvere → Cantor First Steps in Synthetic Computability Andrej Bauer Theorem (Lawvere) If e : A → B A is onto then B has the fixed point property. Introduction Constructive Math Basic Computability Proof. Theory Theorems for Free Given f : B → B , there is x ∈ A such that Enumerability Axiom Markov Principle e ( x ) = λ y : A . f ( e ( y )( y )) . Then e ( x )( x ) = f ( e ( x )( x )) . Injectivity Axiom Conclusion

Lawvere → Cantor First Steps in Synthetic Computability Andrej Bauer Theorem (Lawvere) If e : A → B A is onto then B has the fixed point property. Introduction Constructive Math Basic Computability Proof. Theory Theorems for Free Given f : B → B , there is x ∈ A such that Enumerability Axiom Markov Principle e ( x ) = λ y : A . f ( e ( y )( y )) . Then e ( x )( x ) = f ( e ( x )( x )) . Injectivity Axiom Conclusion Corollary (Cantor) There is no onto map e : A ։ P A. Proof. P A = Ω A and ¬ : Ω → Ω does not have a fixed point.

First Steps in Non-enumerability of Cantor and Baire space Synthetic Computability Andrej Bauer Corollary Introduction 2 N and N N are not enumerable. Constructive Math Basic Computability Proof. Theory Theorems for Free 2 and N do not have the fixed-point property. Enumerability Axiom Markov Principle Injectivity Axiom Conclusion

First Steps in Non-enumerability of Cantor and Baire space Synthetic Computability Andrej Bauer Corollary Introduction 2 N and N N are not enumerable. Constructive Math Basic Computability Proof. Theory Theorems for Free 2 and N do not have the fixed-point property. Enumerability Axiom Markov Principle Injectivity Axiom We have proved our first synthetic theorem: there are no Conclusion effective enumerations of recursive sets and total recursive functions.

First Steps in Projection Theorem Synthetic Computability Andrej Bauer Recall: the projection of S ⊆ A × B is the set � � � Introduction � ∃ y ∈ B . � x , y � ∈ S x ∈ A . Constructive Math Basic Computability Theory Theorems for Free Enumerability Axiom Markov Principle Injectivity Axiom Conclusion

First Steps in Projection Theorem Synthetic Computability Andrej Bauer Recall: the projection of S ⊆ A × B is the set � � � Introduction � ∃ y ∈ B . � x , y � ∈ S x ∈ A . Constructive Math Basic Computability Theorem (Projection) Theory Theorems for Free Enumerability Axiom A subset of N is enumerable iff it is the projection of a decidable Markov Principle Injectivity Axiom subset of N × N . Conclusion Proof. If A is enumerated by e : N → 1 + A then A is the projection of the graph of e .

First Steps in Projection Theorem Synthetic Computability Andrej Bauer Recall: the projection of S ⊆ A × B is the set � � � Introduction � ∃ y ∈ B . � x , y � ∈ S x ∈ A . Constructive Math Basic Computability Theorem (Projection) Theory Theorems for Free Enumerability Axiom A subset of N is enumerable iff it is the projection of a decidable Markov Principle Injectivity Axiom subset of N × N . Conclusion Proof. If A is enumerated by e : N → 1 + A then A is the projection of the graph of e . If A is the projection of B ⊆ N × N , define e : N × N → 1 + A by e � m , n � = if � m , n � ∈ B then m else ⋆ .

First Steps in Semidecidable sets Synthetic Computability ◮ A semidecidable truth value p ∈ Ω is one of the form, Andrej Bauer for some d : N → 2 , Introduction Constructive Math p = ∃ n ∈ N . d ( n ) . Basic Computability Theory Theorems for Free Enumerability Axiom Markov Principle Injectivity Axiom Conclusion

First Steps in Semidecidable sets Synthetic Computability ◮ A semidecidable truth value p ∈ Ω is one of the form, Andrej Bauer for some d : N → 2 , Introduction Constructive Math p = ∃ n ∈ N . d ( n ) . Basic Computability Theory ◮ The set of semidecidable truth values: Theorems for Free � � Enumerability Axiom � � ∃ d ∈ 2 N . p = ∃ n ∈ N . d ( n ) Markov Principle p ∈ Ω Σ = . Injectivity Axiom Conclusion This is Rosolini’s dominance .

First Steps in Semidecidable sets Synthetic Computability ◮ A semidecidable truth value p ∈ Ω is one of the form, Andrej Bauer for some d : N → 2 , Introduction Constructive Math p = ∃ n ∈ N . d ( n ) . Basic Computability Theory ◮ The set of semidecidable truth values: Theorems for Free � � Enumerability Axiom � � ∃ d ∈ 2 N . p = ∃ n ∈ N . d ( n ) Markov Principle p ∈ Ω Σ = . Injectivity Axiom Conclusion This is Rosolini’s dominance . ◮ 2 ⊆ Σ ⊂ Ω .

First Steps in Semidecidable sets Synthetic Computability ◮ A semidecidable truth value p ∈ Ω is one of the form, Andrej Bauer for some d : N → 2 , Introduction Constructive Math p = ∃ n ∈ N . d ( n ) . Basic Computability Theory ◮ The set of semidecidable truth values: Theorems for Free � � Enumerability Axiom � � ∃ d ∈ 2 N . p = ∃ n ∈ N . d ( n ) Markov Principle p ∈ Ω Σ = . Injectivity Axiom Conclusion This is Rosolini’s dominance . ◮ 2 ⊆ Σ ⊂ Ω . ◮ A subset S ⊆ N is semidecidable if χ S : A → Σ .

Σ as a quotient of N + First Steps in Synthetic Computability ◮ Σ is a quotient of 2 N via taking countable joins: Andrej Bauer d ∈ 2 N is mapped to ∃ n ∈ N . d ( n ) . Introduction Constructive Math Basic Computability Theory Theorems for Free Enumerability Axiom Markov Principle Injectivity Axiom Conclusion

Σ as a quotient of N + First Steps in Synthetic Computability ◮ Σ is a quotient of 2 N via taking countable joins: Andrej Bauer d ∈ 2 N is mapped to ∃ n ∈ N . d ( n ) . Introduction ◮ Σ is a quotient of N + via the map q : N + → Σ , Constructive Math Basic defined by q ( t ) = ( t < ∞ ) . Computability Theory Theorems for Free Enumerability Axiom Markov Principle Injectivity Axiom Conclusion

Σ as a quotient of N + First Steps in Synthetic Computability ◮ Σ is a quotient of 2 N via taking countable joins: Andrej Bauer d ∈ 2 N is mapped to ∃ n ∈ N . d ( n ) . Introduction ◮ Σ is a quotient of N + via the map q : N + → Σ , Constructive Math Basic defined by q ( t ) = ( t < ∞ ) . Computability Theory ◮ If q ( t ) = s we say that t is a time at which s becomes Theorems for Free Enumerability Axiom true . Beware, t is not uniquely determined! Markov Principle Injectivity Axiom Conclusion

First Steps in Semidecidable subsets Synthetic Computability Andrej Bauer Theorem Introduction The enumerable subsets of N are precisely the semidecidable Constructive Math subsets of N . Basic Computability Theory Proof. Theorems for Free Enumerability Axiom By Projection Theorem, an enumerable A ⊆ N is the Markov Principle Injectivity Axiom projection of a decidable B ⊆ N × N . Then n ∈ A iff Conclusion ∃ m ∈ N . � n , m � ∈ B . Conversely, if A ∈ Σ N , by Number Choice there is d : N × N → 2 such that n ∈ A iff ∃ m ∈ N . d ( m , n ) .

First Steps in Semidecidable subsets Synthetic Computability Andrej Bauer Theorem Introduction The enumerable subsets of N are precisely the semidecidable Constructive Math subsets of N . Basic Computability Theory Proof. Theorems for Free Enumerability Axiom By Projection Theorem, an enumerable A ⊆ N is the Markov Principle Injectivity Axiom projection of a decidable B ⊆ N × N . Then n ∈ A iff Conclusion ∃ m ∈ N . � n , m � ∈ B . Conversely, if A ∈ Σ N , by Number Choice there is d : N × N → 2 such that n ∈ A iff ∃ m ∈ N . d ( m , n ) . The enumerable subsets of N : E = Σ N .

First Steps in The Topological View Synthetic Computability ◮ Σ is the Sierpinski space . Andrej Bauer Introduction Constructive Math Basic Computability Theory Theorems for Free Enumerability Axiom Markov Principle Injectivity Axiom Conclusion

First Steps in The Topological View Synthetic Computability ◮ Σ is the Sierpinski space . Andrej Bauer ◮ Σ is closed under finite meets, enumerable joins, and Introduction Constructive Math finite meets distribute over enumerable joins. Basic Computability Theory Theorems for Free Enumerability Axiom Markov Principle Injectivity Axiom Conclusion

First Steps in The Topological View Synthetic Computability ◮ Σ is the Sierpinski space . Andrej Bauer ◮ Σ is closed under finite meets, enumerable joins, and Introduction Constructive Math finite meets distribute over enumerable joins. Basic ◮ A σ -frame is a lattice with enumerable joins that Computability Theory distribute over finite meets. Theorems for Free Enumerability Axiom Markov Principle Injectivity Axiom Conclusion

First Steps in The Topological View Synthetic Computability ◮ Σ is the Sierpinski space . Andrej Bauer ◮ Σ is closed under finite meets, enumerable joins, and Introduction Constructive Math finite meets distribute over enumerable joins. Basic ◮ A σ -frame is a lattice with enumerable joins that Computability Theory distribute over finite meets. Theorems for Free Enumerability Axiom ◮ The topology of A is Σ A . Markov Principle Injectivity Axiom Conclusion

First Steps in Partial functions Synthetic Computability ◮ A partial function f : A ⇀ B is a function f : A ′ → B Andrej Bauer defined on a subset A ′ ⊆ A , called the domain of f . Introduction Constructive Math Basic Computability Theory Theorems for Free Enumerability Axiom Markov Principle Injectivity Axiom Conclusion

First Steps in Partial functions Synthetic Computability ◮ A partial function f : A ⇀ B is a function f : A ′ → B Andrej Bauer defined on a subset A ′ ⊆ A , called the domain of f . Introduction ◮ Equivalently, it is a function f : A → � B , where Constructive Math Basic � � � Computability � � ∀ x , y ∈ B . ( x ∈ s ∧ y ∈ s = s ∈ P B ⇒ x = y ) B = . Theory Theorems for Free Enumerability Axiom Markov Principle Injectivity Axiom Conclusion

First Steps in Partial functions Synthetic Computability ◮ A partial function f : A ⇀ B is a function f : A ′ → B Andrej Bauer defined on a subset A ′ ⊆ A , called the domain of f . Introduction ◮ Equivalently, it is a function f : A → � B , where Constructive Math Basic � � � Computability � � ∀ x , y ∈ B . ( x ∈ s ∧ y ∈ s = s ∈ P B ⇒ x = y ) B = . Theory Theorems for Free Enumerability Axiom ◮ The singleton map {−} : B → � B embeds B in � B . Markov Principle Injectivity Axiom Conclusion

First Steps in Partial functions Synthetic Computability ◮ A partial function f : A ⇀ B is a function f : A ′ → B Andrej Bauer defined on a subset A ′ ⊆ A , called the domain of f . Introduction ◮ Equivalently, it is a function f : A → � B , where Constructive Math Basic � � � Computability � � ∀ x , y ∈ B . ( x ∈ s ∧ y ∈ s = s ∈ P B ⇒ x = y ) B = . Theory Theorems for Free Enumerability Axiom ◮ The singleton map {−} : B → � B embeds B in � B . Markov Principle Injectivity Axiom ◮ For s ∈ � B , write s ↓ when s is inhabited. Conclusion

First Steps in Partial functions Synthetic Computability ◮ A partial function f : A ⇀ B is a function f : A ′ → B Andrej Bauer defined on a subset A ′ ⊆ A , called the domain of f . Introduction ◮ Equivalently, it is a function f : A → � B , where Constructive Math Basic � � � Computability � � ∀ x , y ∈ B . ( x ∈ s ∧ y ∈ s = s ∈ P B ⇒ x = y ) B = . Theory Theorems for Free Enumerability Axiom ◮ The singleton map {−} : B → � B embeds B in � B . Markov Principle Injectivity Axiom ◮ For s ∈ � B , write s ↓ when s is inhabited. Conclusion ◮ Which partial functions N → � N have enumerable graphs?

First Steps in Σ -partial functions Synthetic Computability Andrej Bauer Proposition Introduction f : N → � N has an enumerable graph iff f ( n ) ↓ ∈ Σ for all Constructive Math n ∈ N . Basic Computability Theory Theorems for Free Enumerability Axiom Markov Principle Injectivity Axiom Conclusion

First Steps in Σ -partial functions Synthetic Computability Andrej Bauer Proposition Introduction f : N → � N has an enumerable graph iff f ( n ) ↓ ∈ Σ for all Constructive Math n ∈ N . Basic Computability Theory Define the lifting operation Theorems for Free Enumerability Axiom � � � Markov Principle s ∈ � � s ↓ ∈ Σ Injectivity Axiom A ⊥ = A . Conclusion For f : A → B define f ⊥ : A ⊥ → B ⊥ to be � � � � x ∈ s f ⊥ ( s ) = f ( x ) .

First Steps in Σ -partial functions Synthetic Computability Andrej Bauer Proposition Introduction f : N → � N has an enumerable graph iff f ( n ) ↓ ∈ Σ for all Constructive Math n ∈ N . Basic Computability Theory Define the lifting operation Theorems for Free Enumerability Axiom � � � Markov Principle s ∈ � � s ↓ ∈ Σ Injectivity Axiom A ⊥ = A . Conclusion For f : A → B define f ⊥ : A ⊥ → B ⊥ to be � � � � x ∈ s f ⊥ ( s ) = f ( x ) . A Σ -partial function is a function f : A → B ⊥ .

First Steps in Domains of Σ -partial functions Synthetic Computability Andrej Bauer Proposition Introduction A subset is semidecidable iff it is the domain of a Σ -partial Constructive Math function. Basic Computability Theory Theorems for Free Proof. Enumerability Axiom Markov Principle A semidecidable subset S ∈ Σ A is the domain of its Injectivity Axiom Conclusion characteristic map χ S : A → Σ = 1 ⊥ . If f : A → B ⊥ is Σ -partial then its domain is the set � � � � f ( x ) ↓ x ∈ A , which is obviously semidecidable.

First Steps in The Single-Value Theorem Synthetic Computability Andrej Bauer A selection for R ⊆ A × B is a partial map f : A ⇀ B such that, for every x ∈ A , Introduction Constructive Math ∃ y ∈ B . R ( x , y ) = ⇒ f ( x ) ↓ ∧ R ( x , f ( x )) . Basic Computability Theory This is like a choice function, expect it only chooses when Theorems for Free Enumerability Axiom there is something to choose from. Markov Principle Injectivity Axiom Conclusion

First Steps in The Single-Value Theorem Synthetic Computability Andrej Bauer A selection for R ⊆ A × B is a partial map f : A ⇀ B such that, for every x ∈ A , Introduction Constructive Math ∃ y ∈ B . R ( x , y ) = ⇒ f ( x ) ↓ ∧ R ( x , f ( x )) . Basic Computability Theory This is like a choice function, expect it only chooses when Theorems for Free Enumerability Axiom there is something to choose from. Markov Principle Injectivity Axiom Conclusion Theorem (Single Value) Every open relation R ∈ Σ N × N has a Σ -partial selection.

First Steps in Axiom of Enumerability Synthetic Computability Andrej Bauer Axiom (Enumerability) Introduction There are enumerably many enumerable sets of numbers. Constructive Math Basic Let W : N ։ E be an enumeration. Computability Theory Theorems for Free Enumerability Axiom Markov Principle Injectivity Axiom Conclusion

First Steps in Axiom of Enumerability Synthetic Computability Andrej Bauer Axiom (Enumerability) Introduction There are enumerably many enumerable sets of numbers. Constructive Math Basic Let W : N ։ E be an enumeration. Computability Theory Theorems for Free Proposition Enumerability Axiom Markov Principle Injectivity Axiom Σ and E have the fixed-point property. Conclusion Proof. By Lawvere, W : N ։ E = Σ N ∼ = Σ N × N ∼ = E N .

Enumerability of N → N ⊥ First Steps in Synthetic Computability Andrej Bauer Proposition Introduction N → N ⊥ is enumerable. Constructive Math Basic Computability Proof. Theory Theorems for Free Let V : N ։ Σ N × N be an enumeration. By Single-Value Enumerability Axiom Markov Principle Theorem and Number Choice, there is ϕ : N → ( N → N ⊥ ) Injectivity Axiom Conclusion such that ϕ n is a selection of V n . The map ϕ is onto, as every f : N → N ⊥ is the only selection of its graph.

First Steps in The Law of Excluded Middle Fails Synthetic Computability The Law of Excluded Middle says 2 = Ω . Andrej Bauer Introduction Constructive Math Basic Computability Theory Theorems for Free Enumerability Axiom Markov Principle Injectivity Axiom Conclusion

First Steps in The Law of Excluded Middle Fails Synthetic Computability The Law of Excluded Middle says 2 = Ω . Andrej Bauer Introduction Corollary Constructive Math The Law of Excluded Middle is false. Basic Computability Theory Theorems for Free Proof. Enumerability Axiom Markov Principle Among the sets 2 ⊆ Σ ⊆ Ω only the middle one has the Injectivity Axiom Conclusion fixed-point property, so 2 � = Σ � = Ω .

� First Steps in Focal sets Synthetic Computability ◮ A focal set is a set A together with a map ǫ A : A ⊥ → A Andrej Bauer such that ǫ A ( { x } ) = x for all x ∈ A : Introduction Constructive Math {−} � A ⊥ A Basic Computability � � � � Theory � � � � ǫ A � � Theorems for Free � � � � Enumerability Axiom � � Markov Principle A Injectivity Axiom Conclusion The focus of A is ⊥ A = ǫ A ( ⊥ ) .

� First Steps in Focal sets Synthetic Computability ◮ A focal set is a set A together with a map ǫ A : A ⊥ → A Andrej Bauer such that ǫ A ( { x } ) = x for all x ∈ A : Introduction Constructive Math {−} � A ⊥ A Basic Computability � � � � Theory � � � � ǫ A � � Theorems for Free � � � � Enumerability Axiom � � Markov Principle A Injectivity Axiom Conclusion The focus of A is ⊥ A = ǫ A ( ⊥ ) . ◮ A lifted set A ⊥ is always focal (because lifting is a monad with whose unit is {−} ).

First Steps in Enumerable focal sets Synthetic Computability ◮ Enumerable focal sets, known as Erˇ sov complete sets , Andrej Bauer have good properties. Introduction Constructive Math Basic Computability Theory Theorems for Free Enumerability Axiom Markov Principle Injectivity Axiom Conclusion

First Steps in Enumerable focal sets Synthetic Computability ◮ Enumerable focal sets, known as Erˇ sov complete sets , Andrej Bauer have good properties. Introduction ◮ A flat domain A ⊥ is focal. It is enumerable if A is Constructive Math decidable and enumerable. Basic Computability Theory Theorems for Free Enumerability Axiom Markov Principle Injectivity Axiom Conclusion

First Steps in Enumerable focal sets Synthetic Computability ◮ Enumerable focal sets, known as Erˇ sov complete sets , Andrej Bauer have good properties. Introduction ◮ A flat domain A ⊥ is focal. It is enumerable if A is Constructive Math decidable and enumerable. Basic Computability ◮ If A is enumerable and focal then so is A N : Theory Theorems for Free Enumerability Axiom e N ǫ N Markov Principle ϕ � � N N ⊥ � � A N A � � A N Injectivity Axiom N ⊥ ⊥ Conclusion

First Steps in Enumerable focal sets Synthetic Computability ◮ Enumerable focal sets, known as Erˇ sov complete sets , Andrej Bauer have good properties. Introduction ◮ A flat domain A ⊥ is focal. It is enumerable if A is Constructive Math decidable and enumerable. Basic Computability ◮ If A is enumerable and focal then so is A N : Theory Theorems for Free Enumerability Axiom e N ǫ N Markov Principle ϕ � � N N ⊥ � � A N A � � A N Injectivity Axiom N ⊥ ⊥ Conclusion ◮ Some enumerable focal sets are Σ N , 2 N N N ⊥ , ⊥ .

First Steps in Multi-valued functions Synthetic Computability ◮ A multi-valued function f : A ⇒ B is a function Andrej Bauer f : A → P B such that f ( x ) is inhabited for all x ∈ A . Introduction Constructive Math Basic Computability Theory Theorems for Free Enumerability Axiom Markov Principle Injectivity Axiom Conclusion

First Steps in Multi-valued functions Synthetic Computability ◮ A multi-valued function f : A ⇒ B is a function Andrej Bauer f : A → P B such that f ( x ) is inhabited for all x ∈ A . Introduction ◮ This is equivalent to having a total relation R ⊆ A × B . Constructive Math The connection between f and R is Basic Computability � � � Theory � R ( x , y ) y ∈ B f ( x ) = Theorems for Free Enumerability Axiom Markov Principle R ( x , y ) ⇐ ⇒ y ∈ f ( x ) . Injectivity Axiom Conclusion

First Steps in Multi-valued functions Synthetic Computability ◮ A multi-valued function f : A ⇒ B is a function Andrej Bauer f : A → P B such that f ( x ) is inhabited for all x ∈ A . Introduction ◮ This is equivalent to having a total relation R ⊆ A × B . Constructive Math The connection between f and R is Basic Computability � � � Theory � R ( x , y ) y ∈ B f ( x ) = Theorems for Free Enumerability Axiom Markov Principle R ( x , y ) ⇐ ⇒ y ∈ f ( x ) . Injectivity Axiom Conclusion ◮ A fixed point of f : A ⇒ A is x ∈ A such that x ∈ f ( x ) .

First Steps in Recursion Theorem Synthetic Computability Andrej Bauer Theorem (Recursion Theorem) Introduction Every f : A ⇒ A on enumerable focal A has a fixed point. Constructive Math Basic Computability Proof. Theory Theorems for Free Let e : N ։ A be an enumeration, and ǫ : A ⊥ → A a focal map. For Enumerability Axiom every k ∈ N there exists m ∈ N such that e ( m ) ∈ f ( e ( k )) . By Number Markov Principle Injectivity Axiom Choice there is a map c : N → N such that e ( c ( k )) ∈ f ( e ( k )) for every Conclusion k ∈ N . It suffices to find k such that e ( c ( k )) = e ( k ) since then x = e ( k ) is a fixed point for f . For every m ∈ N there is n ∈ N such that ǫ ( e ⊥ ( c ⊥ ( ϕ m ( m )))) = e ( n ) . By Number Choice there is g : N → N such that ǫ ( e ⊥ ( c ⊥ ( ϕ m ( m )))) = e ( g ( m )) for every m ∈ N . There is j ∈ N such that g = ϕ j . Let k = g ( j ) . Then e ( k ) = e ( g ( j )) = ǫ ( e ⊥ ( c ⊥ ( ϕ j ( j )))) == e ( c ( g ( j ))) = e ( c ( k )) .

First Steps in Classical Recursion Theorem Synthetic Computability Andrej Bauer Corollary (Classical Recursion Theorem) Introduction For every f : N → N there is n ∈ N such that ϕ f ( n ) = ϕ n . Constructive Math Basic Computability Proof. Theory Theorems for Free In Recursion Theorem, take the enumerable focal set N N Enumerability Axiom ⊥ Markov Principle and the multi-valued function Injectivity Axiom � � Conclusion � � ∃ n ∈ N . g = ϕ n ∧ h = ϕ f ( n ) h ∈ N N F ( g ) = . ⊥ There is g such that g ∈ F ( g ) . Thus there exists n ∈ N such that ϕ n = g = h = ϕ f ( n ) .

First Steps in Markov Principle Synthetic Computability ◮ If a binary sequence a ∈ 2 N is not constantly 0, does it Andrej Bauer contain a 1? Introduction Constructive Math Basic Computability Theory Theorems for Free Enumerability Axiom Markov Principle Injectivity Axiom Conclusion

First Steps in Markov Principle Synthetic Computability ◮ If a binary sequence a ∈ 2 N is not constantly 0, does it Andrej Bauer contain a 1? Introduction ◮ For p ∈ Σ , does p � = ⊥ imply p = ⊤ ? Constructive Math Basic Computability Theory Theorems for Free Enumerability Axiom Markov Principle Injectivity Axiom Conclusion

First Steps in Markov Principle Synthetic Computability ◮ If a binary sequence a ∈ 2 N is not constantly 0, does it Andrej Bauer contain a 1? Introduction ◮ For p ∈ Σ , does p � = ⊥ imply p = ⊤ ? Constructive Math Basic ◮ Is Σ ⊆ Ω ¬¬ ? Computability Theory Theorems for Free Enumerability Axiom Markov Principle Injectivity Axiom Conclusion

First Steps in Markov Principle Synthetic Computability ◮ If a binary sequence a ∈ 2 N is not constantly 0, does it Andrej Bauer contain a 1? Introduction ◮ For p ∈ Σ , does p � = ⊥ imply p = ⊤ ? Constructive Math Basic ◮ Is Σ ⊆ Ω ¬¬ ? Computability Theory ◮ For x ∈ N + , if x � = ∞ is x = k for some k ∈ N ? Theorems for Free Enumerability Axiom Markov Principle Injectivity Axiom Conclusion

First Steps in Markov Principle Synthetic Computability ◮ If a binary sequence a ∈ 2 N is not constantly 0, does it Andrej Bauer contain a 1? Introduction ◮ For p ∈ Σ , does p � = ⊥ imply p = ⊤ ? Constructive Math Basic ◮ Is Σ ⊆ Ω ¬¬ ? Computability Theory ◮ For x ∈ N + , if x � = ∞ is x = k for some k ∈ N ? Theorems for Free Enumerability Axiom Markov Principle Injectivity Axiom Axiom (Markov Principle) Conclusion A binary sequence which is not constantly 0 contains a 1 .

First Steps in Post’s Theorem Synthetic Computability Andrej Bauer Theorem (Post) Introduction A subset is decidable if, and only if, it and its complement are Constructive Math both semidecidable. Basic Computability Theory Proof. Theorems for Free Enumerability Axiom Clearly, a decidable proposition is semidecidable and so Markov Principle Injectivity Axiom is its complement. If p and ¬ p are semidecidable then so Conclusion is p ∨ ¬ p . By Markov Principle p ∨ ¬ p ∈ Σ ⊆ Ω ¬¬ , hence p ∨ ¬ p = ¬¬ ( p ∨ ¬ p ) = ¬ ( ¬ p ∧ ¬¬ p ) = ¬⊥ = ⊤ , as required.

First Steps in Topological Exterior and Creative Sets Synthetic Computability ◮ The exterior of an open set is the largest open set Andrej Bauer disjoint from it. Introduction Constructive Math Basic Computability Theory Theorems for Free Enumerability Axiom Markov Principle Injectivity Axiom Conclusion

First Steps in Topological Exterior and Creative Sets Synthetic Computability ◮ The exterior of an open set is the largest open set Andrej Bauer disjoint from it. Introduction ◮ An open set U ∈ Σ A is creative if it is without exterior: Constructive Math for every V ∈ Σ A such that U ∩ V = ∅ there is Basic Computability V ′ ∈ Σ A such that U ∩ V ′ = ∅ and V ′ \ V is inhabited. Theory Theorems for Free Enumerability Axiom Markov Principle Theorem Injectivity Axiom Conclusion There exists a creative subset of N . Proof. � � � � n ∈ W n The familiar K = n ∈ N is creative. Given any V ∈ E with V = W k and K ∩ V = ∅ , we have n �∈ V , so we can take V ′ = V { k } .

First Steps in Immune and Simple Sets Synthetic Computability ◮ A set is immune if it is neither finite nor infinite. Andrej Bauer Introduction Constructive Math Basic Computability Theory Theorems for Free Enumerability Axiom Markov Principle Injectivity Axiom Conclusion

First Steps in Immune and Simple Sets Synthetic Computability ◮ A set is immune if it is neither finite nor infinite. Andrej Bauer ◮ A set is simple if it is open and its complement is Introduction Constructive Math immune. Basic Computability Theorem Theory Theorems for Free Enumerability Axiom There exists a closed subset of N which is neither finite nor Markov Principle infinite. Injectivity Axiom Conclusion Proof. ˛ n > 2 m ∧ n ∈ W m ˘ ˛ ¯ Following Post, consider P = � m , n � ∈ N × N , and let f : N → N ⊥ be a selection for P by Single-Value Theorem. Then ˛ ∃ m ∈ N . f ( m ) = n ˘ ˛ ¯ S = n ∈ N is the complement of the set we are looking for. Because f ( m ) > 2 m the set N \ S cannot be finite. For any infinite enumerable set U ⊆ N \ S with U = W m , we have f ( m ) ↓ , f ( m ) ∈ W m = U , and f ( m ) ∈ S , hence U is not contained in N \ S .

First Steps in Inseparable sets Synthetic Computability Andrej Bauer Theorem Introduction There exists an element of Plotkin’s 2 N ⊥ that is inconsistent Constructive Math with every maximal element of 2 N ⊥ . Basic Computability Theory Theorems for Free Proof. Enumerability Axiom Markov Principle Because 2 ⊥ is focal and enumerable, 2 N ⊥ is as well. Let Injectivity Axiom ψ : N ։ 2 N ⊥ be an enumeration, and let t : 2 ⊥ → 2 ⊥ be the Conclusion isomorphism t ( x ) = ¬ ⊥ x which exchanges 0 and 1. Consider a ∈ 2 N ⊥ defined by a ( n ) = t ( ψ n ( n )) . If b ∈ 2 N ⊥ is maximal with b = ψ k , then a ( k ) = ¬ ψ k ( k ) = ¬ b ( k ) . Because a ( k ) and b ( k ) are both total and different they are inconsistent. Hence a and b are inconsistent.

First Steps in Berger’s Lemma Synthetic Computability Andrej Bauer Lemma (Berger) Introduction If U : A ⇒ Σ is a multi-valued open set, and x : N + → A such Constructive Math that U ( x ∞ ) = {⊤} then there is k ∈ N for which ⊤ ∈ U ( x k ) . Basic Computability Theory Theorems for Free Proof. Enumerability Axiom Markov Principle For every y ∈ A there is p ∈ N + such that ( p < ∞ ) ∈ U ( y ) . Injectivity Axiom Consequently, for every y ∈ A there is z ∈ A such that Conclusion ∃ p ∈ N + . (( p < ∞ ) ∈ U ( y ) ∧ z = x p ) . (1) By Recursion Theorem there is y = z satisfying (1). For such y , p is not equal to ∞ because p = ∞ implies y = x ∞ and ⊥ = ( p < ∞ ) ∈ U ( y ) = U ( x ∞ ) = {⊤} , contradiction. By Markov Principle, p ∈ N so we have x p = y and ⊤ = ( p < ∞ ) ∈ U ( x p ) , as required.

First Steps in ω -Chain Complete Posets Synthetic Computability ◮ An ω -chain complete poset ( ω -cpo) is a poset in which Andrej Bauer enumerable chains have suprema. Introduction Constructive Math Basic Computability Theory Theorems for Free Enumerability Axiom Markov Principle Injectivity Axiom Conclusion

First Steps in ω -Chain Complete Posets Synthetic Computability ◮ An ω -chain complete poset ( ω -cpo) is a poset in which Andrej Bauer enumerable chains have suprema. Introduction ◮ A base for an ω -cpo ( A , ≤ ) is an enumerable subset Constructive Math S ⊆ A such that: Basic Computability ◮ For all x ∈ S , y ∈ A , ( x ≤ y ) ∈ Σ . Theory ◮ Every x ∈ A is the supremum of a chain in S . Theorems for Free Enumerability Axiom Markov Principle Injectivity Axiom Conclusion

First Steps in The Topology of ω -cpos Synthetic Computability Andrej Bauer Theorem Introduction 1. The open subsets of an ω -cpo are upward closed and Constructive Math inaccessible by chains. Basic Computability Theory 2. If an ω -cpo A has a base S, then every open is a union of � � � Theorems for Free � x ≤ y basic opens sets ↑ x = y ∈ A with x ∈ S. Enumerability Axiom Markov Principle Injectivity Axiom Conclusion Proof. If x ≤ y and x ∈ U ∈ Σ A , define a : N + → A by � a p = if k < p then x else y k ∈ N Then a ∞ = x ∈ U and by Berger’s Lemma there is k ∈ N such that y = a k ∈ U , too.

First Steps in The Injectivity Axiom Synthetic Computability A subset A ⊆ B is a subspace if every U ∈ Σ A is the Andrej Bauer restriction of some V ∈ Σ B . Introduction Constructive Math Basic Computability Theory Theorems for Free Enumerability Axiom Markov Principle Injectivity Axiom Conclusion

First Steps in The Injectivity Axiom Synthetic Computability A subset A ⊆ B is a subspace if every U ∈ Σ A is the Andrej Bauer restriction of some V ∈ Σ B . Introduction Constructive Math Axiom (Injectivity) Basic Computability A classical subset of N is a subspace of N . Theory Theorems for Free Enumerability Axiom Markov Principle Injectivity Axiom Conclusion

First Steps in The Injectivity Axiom Synthetic Computability A subset A ⊆ B is a subspace if every U ∈ Σ A is the Andrej Bauer restriction of some V ∈ Σ B . Introduction Constructive Math Axiom (Injectivity) Basic Computability A classical subset of N is a subspace of N . Theory Theorems for Free Enumerability Axiom In other words, Σ is injective with respect to classical Markov Principle Injectivity Axiom subsets of N . Conclusion

First Steps in Kreisel-Lacombe-Shoenfield Theorem Synthetic Computability Andrej Bauer Theorem (Kreisel-Lacombe-Shoenfield-Ceitin) Introduction Every map from a complete separable metric space to a metric Constructive Math space is ǫδ -continuous. Basic Computability Theory Theorems for Free Enumerability Axiom Markov Principle Injectivity Axiom Conclusion

First Steps in Kreisel-Lacombe-Shoenfield Theorem Synthetic Computability Andrej Bauer Theorem (Kreisel-Lacombe-Shoenfield-Ceitin) Introduction Every map from a complete separable metric space to a metric Constructive Math space is ǫδ -continuous. Basic Computability Theory Proof idea. Theorems for Free Enumerability Axiom Markov Principle Suppose f : M → L is such a function. Write B ( x , r ) for the Injectivity Axiom open ball with radius r and centered at x . Conclusion The proof uses Berger’s Lemma and the observation that ∀ t ∈ B ( x , r ) . f ( t ) ∈ B ( y , q ) is the negation of ∃ t ∈ B ( x , r ) . d ( f ( t ) , y ) > q , which is semidecidable.

First Steps in Where to go from here? Synthetic Computability ◮ Computable Analysis: Andrej Bauer ◮ 2 N is homeomorphic to N N , Introduction ◮ R is locally non-compact, in the sense that every Constructive Math interval contains a sequence without accumulation Basic point, Computability Theory ◮ R has measure zero: it can be covered by a sequence Theorems for Free of open intervals whose total length is bounded by Enumerability Axiom Markov Principle ǫ > 0. Injectivity Axiom Conclusion