Overview of theories Second-order parts of higher-order theories Topological definitions A bit of reverse topology Higher-Order Reverse Topology James Hunter (hunter@math.wisc.edu) University of Wisconsin Logic Colloquium 2007 Wroc� law University James Hunter Higher-Order Reverse Topology
Overview of theories Second-order parts of higher-order theories Topological definitions A bit of reverse topology Outline 1 Overview of theories 2 Second-order parts of higher-order theories 3 Topological definitions 4 A bit of reverse topology James Hunter Higher-Order Reverse Topology
Overview of theories Second-order parts of higher-order theories Topological definitions A bit of reverse topology Review of second-order reverse math Traditional reverse math studies subsystems of second-order arithmetic . Language: Number (type-0) and set (type-1) variables; { 0 , + , · , etc. } ; “= 0 ” for numbers (but not sets); “ ∈ ” relates numbers and sets. Base theory, RCA 0 : Axioms for number-theoretic N ; induction schema for Σ 0 1 formulas; comprehension schema for ∆ 0 1 formulas. The second-order part of the minimal ω -model of RCA 0 consists of all computable (recursive) sets. The first-order part of the theory RCA 0 is Σ 0 1 -PA [4]. A stronger theory, ACA 0 : Axioms for RCA 0 ; comprehension schema for arithmetical (or “Π 0 ∞ ”) formulas. The second-order part of the minimal ω -model of ACA 0 consists of all arithmetical sets. The first-order part of the theory ACA 0 is PA [4]. James Hunter Higher-Order Reverse Topology
Overview of theories Second-order parts of higher-order theories Topological definitions A bit of reverse topology Finite types Definition The finite types are defined inductively: 0 is a type. If σ and τ are types then ( σ → τ ) is a type. 0 is the type of natural numbers; ( σ → τ ) is the type of a functional mapping type- σ elements to type- τ elements. Definition The standard types are defined inductively: 0 is a standard type. If n is a standard type, then n + 1 := ( n → 0) is a standard type. Example: reals are of type 1. James Hunter Higher-Order Reverse Topology
Overview of theories Second-order parts of higher-order theories Topological definitions A bit of reverse topology Higher-order reverse math The language of second-order arithmetic may be too restrictive. In a recent paper [3], Kohlenbach described a base theory in a more-flexible, higher-order language. Language: Variables of all finite types; { 0 , + , · , etc. } as before; “= 0 ” only for (type-0) numbers, as before; plus— Combinators Π ρ,τ , Σ σ,ρ,τ (for λ -abstraction ); Some symbol for application , not shown here; and Symbol R 0 , for primitive recursion . 0 (= E-PRA ω + QF-AC 1 , 0 ): Axioms for Base theory, RCA ω number-theoretic N , as before; induction schema for quantifier-free formulas; axioms defining R 0 , the Π ρ,τ ’s, and the Σ σ,ρ,τ ’s; extensionality axioms; and QF-AC 1 , 0 : ∀ x 1 ∃ n 0 (Φ( x , n )) → ∃ F (1 → 0) ∀ x 1 (Φ( x , F ( x )) , where Φ is a quantifier-free formula. James Hunter Higher-Order Reverse Topology
Overview of theories Second-order parts of higher-order theories Topological definitions A bit of reverse topology The axioms ( E 1 ) and ( E 2 ) Definition The axiom ( E 1 ) is the statement: ∃ E 1 2 � ∀ x 1 ( E 1 ( x ) = 0 1) ↔ ∃ n 0 ( x ( n ) � = 0 0) � . Definition The axiom ( E 2 ) is the statement: ∃ E 2 3 � ∀ X 2 ( E 2 ( X ) = 0 1) ↔ ∃ x 1 ( X ( x ) � = 0 0) � . Higher-order equality is defined inductively. E.g., x 1 = 1 y 1 ⇐ ⇒ ∀ n 0 ( x ( n ) = 0 y ( n )). Think of E 1 as a functional determining type-1 equality: x 1 = 1 y 1 ⇐ ⇒ E 1 ( λ n 0 . ( x ( n ) − y ( n ))) = 0 0 . James Hunter Higher-Order Reverse Topology
Overview of theories Second-order parts of higher-order theories Topological definitions A bit of reverse topology Conservation results Proposition (Kohlenbach [3]) RCA ω 0 is conservative over and implies RCA 0 . Proposition (H.) 1 RCA ω 0 + ( E 1 ) is conservative over and implies ACA 0 . 0 + QF-AC 0 , 1 is conservative over and implies Σ 1 2 RCA ω 1 -AC 0 . 3 RCA ω 0 + ( E 2 ) is conservative over and implies Π 1 ∞ -CA 0 . 4 Etc. The proof of the second proposition uses term models and is analogous to the proof of the first. James Hunter Higher-Order Reverse Topology
Overview of theories Second-order parts of higher-order theories Topological definitions A bit of reverse topology Sets, families Definition 1 A real is a (type-1) function. 2 A set is a (type-2) functional X such that ∀ x 1 ( X ( x ) = 0 0 ∨ X ( x ) = 0 1) . 3 A family is a (type-3) functional F such that ∀ X 2 ( F ( x ) = 0 0 ∨ F ( X ) = 0 1) . (We write “x ∈ X” as shorthand for “X ( x ) = 0 1 .”) We consider only topologies on the reals. James Hunter Higher-Order Reverse Topology
Overview of theories Second-order parts of higher-order theories Topological definitions A bit of reverse topology Topologies Definition A family F is a topology iff: 1 ∅ := ( λ x 1 . 0) ∈ F ; N N := ( λ x 1 . 1) ∈ F ; 2 3 if X ∈ F and Y ∈ F then X ∩ Y := ( λ x . min( X ( x ) , Y ( x ))) ∈ F ; and 4 if G ⊆ 2 F and � G := { x : ∃ X 2 ∈ G ( x ∈ X ) } exists, then � G ∈ F . Examples: The indiscrete topology is {∅ , N N } , and the discrete topology is ( λ X 2 . 1). James Hunter Higher-Order Reverse Topology
Overview of theories Second-order parts of higher-order theories Topological definitions A bit of reverse topology Simple equivalences Proposition (H. and folklore) Over RCA ω 0 , we have the following equivalences: 1 ( E 2 ) ⇐ ⇒ there is a topology for a connected space (i.e., only ∅ and N N are clopen). 2 ( E 2 ) ⇐ ⇒ there is a topology with a dense, nowhere-dense set. 3 ( E 2 ) ⇐ ⇒ there is a topology generated by a countable enumeration for a basis. A consequence of (3) is that any topological statement examined in second-order reverse math follows, in higher-order reverse math, from the existence of such a formal topology. So second-order reverse topology does not carry over nicely to higher-order theories. James Hunter Higher-Order Reverse Topology
Overview of theories Second-order parts of higher-order theories Topological definitions A bit of reverse topology More simple equivalences Proposition (H. and folklore) Over RCA ω 0 + ( E 1 ) , we have the following equivalences: 1 ( E 2 ) ⇐ ⇒ there is a topology with a countable dense set. 2 ( E 2 ) ⇐ ⇒ there is a topology for a space that is the countable union of nowhere-dense sets (i.e., is of first category). James Hunter Higher-Order Reverse Topology
Overview of theories Second-order parts of higher-order theories Topological definitions A bit of reverse topology Topology in RCA ω 0 + ( E 1 ) Proposition (H.) If T is a topology existing in a minimal term model of RCA ω 0 + ( E 1 ) then T is equivalent to T × P ( N N \ X ) , where X = { x 0 , x 1 , . . . } is a countable set and T is a topology on X. (In other words T is essentially just a topology on N .) James Hunter Higher-Order Reverse Topology
Overview of theories Second-order parts of higher-order theories Topological definitions A bit of reverse topology Open questions Over RCA ω 0 + ( E 2 ): QF-AC 1 , 2 = ⇒ “every T 2 space has a witnessing functional” ⇒ QF-AC 1 , 1 . = “Every T 2 space has a witnessing functional” = ⇒ : every compact T 2 space is T 4 . every compact T 2 space has a basis of size ≤ 2 ℵ 0 . The principle ( E 3 ) = ⇒ that every compact, T 2 space is T 4 . Open question: what about reversals? James Hunter Higher-Order Reverse Topology
Overview of theories Second-order parts of higher-order theories Topological definitions A bit of reverse topology Bibliography [1] Avigad, Jeremy, and Solomon Feferman. “G¨ odel’s Functional (‘Dialectica’) Interpretation.” Handbook of Proof Theory (Samuel R. Buss, ed.). (Elsevier, 1998.) [2] Jech, Thomas J. The Axiom of Choice. (North-Holland, 1973.) [3] Kohlenbach, Ulrich. “Higher Order Reverse Mathematics.” Reverse Mathematics 2001 (Stephen Simpson, ed.). (A K Peters, 2005.) [4] Simpson, Stephen G. Subsystems of Second Order Arithmetic . (Springer, 1999.) James Hunter Higher-Order Reverse Topology
Recommend
More recommend
