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Hierarchies of Sets in Classical and Constructive Set Theories Setting the Stage A Modified Hierarchy Applications Hierarchies of Sets in Classical and Constructive Set Theories Albert Ziegler School of Mathematics University of Leeds March 8, 2012 Albert Ziegler Hierarchies of Sets in Classical and Constructive Set Theories 1 Start with the empty set V0 = ∅. 2 Take the powerset of what you have so far (i.e. take all subsets). 3 Go to step 2. The Cumulative Hierarchy [ Vα = P(Vβ) β<α [ V = Vα α Setting the Stage The Cumulative Hierarchy A Modified Hierarchy CZF Applications The Cumulative Hierarchy and CZF Creating the Universe in Three Simple Steps Albert Ziegler Hierarchies of Sets in Classical and Constructive Set Theories 2 Take the powerset of what you have so far (i.e. take all subsets). 3 Go to step 2. The Cumulative Hierarchy [ Vα = P(Vβ) β<α [ V = Vα α Setting the Stage The Cumulative Hierarchy A Modified Hierarchy CZF Applications The Cumulative Hierarchy and CZF Creating the Universe in Three Simple Steps 1 Start with the empty set V0 = ∅. Albert Ziegler Hierarchies of Sets in Classical and Constructive Set Theories 3 Go to step 2. The Cumulative Hierarchy [ Vα = P(Vβ) β<α [ V = Vα α Setting the Stage The Cumulative Hierarchy A Modified Hierarchy CZF Applications The Cumulative Hierarchy and CZF Creating the Universe in Three Simple Steps 1 Start with the empty set V0 = ∅. 2 Take the powerset of what you have so far (i.e. take all subsets). Albert Ziegler Hierarchies of Sets in Classical and Constructive Set Theories The Cumulative Hierarchy [ Vα = P(Vβ) β<α [ V = Vα α Setting the Stage The Cumulative Hierarchy A Modified Hierarchy CZF Applications The Cumulative Hierarchy and CZF Creating the Universe in Three Simple Steps 1 Start with the empty set V0 = ∅. 2 Take the powerset of what you have so far (i.e. take all subsets). 3 Go to step 2. Albert Ziegler Hierarchies of Sets in Classical and Constructive Set Theories Setting the Stage The Cumulative Hierarchy A Modified Hierarchy CZF Applications The Cumulative Hierarchy and CZF Creating the Universe in Three Simple Steps 1 Start with the empty set V0 = ∅. 2 Take the powerset of what you have so far (i.e. take all subsets). 3 Go to step 2. The Cumulative Hierarchy [ Vα = P(Vβ) β<α [ V = Vα α Albert Ziegler Hierarchies of Sets in Classical and Constructive Set Theories But... But there is some ambiguity in how exactly to state the axioms: Classically equivalent formulations of the axioms can become constructively different. Setting the Stage The Cumulative Hierarchy A Modified Hierarchy CZF Applications The Cumulative Hierarchy and CZF Set Theory with Constructive Logic Classical Set Theory can serve as a framework for all classical mathematics The concept of set is just as compatible with constructivism Use set theory with constructive logic to serve as a framework for constructive mathematics For CZF, take same language and axioms as ZF Albert Ziegler Hierarchies of Sets in Classical and Constructive Set Theories Setting the Stage The Cumulative Hierarchy A Modified Hierarchy CZF Applications The Cumulative Hierarchy and CZF Set Theory with Constructive Logic Classical Set Theory can serve as a framework for all classical mathematics The concept of set is just as compatible with constructivism Use set theory with constructive logic to serve as a framework for constructive mathematics For CZF, take same language and axioms as ZF But... But there is some ambiguity in how exactly to state the axioms: Classically equivalent formulations of the axioms can become constructively different. Albert Ziegler Hierarchies of Sets in Classical and Constructive Set Theories CZF instead includes the axiom Fullness ∀A, B∃C∀R. ∀x ∈ A∃y ∈ B (x, y) ∈ R → ∃R0 ∈ C.R0 ⊆ R ∧ ∀x ∈ A∃y ∈ B(x, y) ∈ R0 Setting the Stage The Cumulative Hierarchy A Modified Hierarchy CZF Applications The Cumulative Hierarchy and CZF Powerset and Exponentiation The following are classically equivalent: Powerset ∀a∃b. b = P(a) := {x|x ⊆ a} Binary Exponentiation ∀a∃b. b = a2 := {f |f : a → 2} Albert Ziegler Hierarchies of Sets in Classical and Constructive Set Theories Setting the Stage The Cumulative Hierarchy A Modified Hierarchy CZF Applications The Cumulative Hierarchy and CZF Powerset and Exponentiation The following are classically equivalent: Powerset ∀a∃b. b = P(a) := {x|x ⊆ a} Binary Exponentiation ∀a∃b. b = a2 := {f |f : a → 2} CZF instead includes the axiom Fullness ∀A, B∃C∀R. ∀x ∈ A∃y ∈ B (x, y) ∈ R → ∃R0 ∈ C.R0 ⊆ R ∧ ∀x ∈ A∃y ∈ B(x, y) ∈ R0 Albert Ziegler Hierarchies of Sets in Classical and Constructive Set Theories Setting the Stage The Cumulative Hierarchy A Modified Hierarchy CZF Applications The Cumulative Hierarchy and CZF ZF and CZF ZF CZF Extensionality Extensionality Foundation ∈-Induction Pairing Pairing Union Axiom Union Axiom Infinity Infinity Separation Separation for ∆0-formulae Replacement Strong Collection Powerset Fullness Albert Ziegler Hierarchies of Sets in Classical and Constructive Set Theories Constructively, the collection of all subsets (the ”powerset”) is too unstructured to be accepted as a set. Consequently, the Vα are not sets but only classes. S The description of the universe as α Vα still holds true. But it loses much of its power. Setting the Stage The Cumulative Hierarchy A Modified Hierarchy CZF Applications The Cumulative Hierarchy and CZF The Cumulative Hierarchy in a Constructive Context [ [ Vα = P(Vβ), V = Vα β<α α Albert Ziegler Hierarchies of Sets in Classical and Constructive Set Theories Setting the Stage The Cumulative Hierarchy A Modified Hierarchy CZF Applications The Cumulative Hierarchy and CZF The Cumulative Hierarchy in a Constructive Context [ [ Vα = P(Vβ), V = Vα β<α α Constructively, the collection of all subsets (the ”powerset”) is too unstructured to be accepted as a set. Consequently, the Vα are not sets but only classes. S The description of the universe as α Vα still holds true. But it loses much of its power. Albert Ziegler Hierarchies of Sets in Classical and Constructive Set Theories | ∀x ∈Veβ. {0|x ∈ X} ∈ {0, 1} ∪ Veβ} be defined by recursion over the ordinals. Setting the Stage Definition A Modified Hierarchy Basic Properties Applications Central Properties A Modified Hierarchy for Constructive Purposes Definition Let for α ∈ On [ Veα = {X ⊆ Veβ β<α Albert Ziegler Hierarchies of Sets in Classical and Constructive Set Theories ∪ Veβ} be defined by recursion over the ordinals. Setting the Stage Definition A Modified Hierarchy Basic Properties Applications Central Properties A Modified Hierarchy for Constructive Purposes Definition Let for α ∈ On [ Veα = {X ⊆ Veβ | ∀x ∈Veβ. {0|x ∈ X} ∈ {0, 1} β<α Albert Ziegler Hierarchies of Sets in Classical and Constructive Set Theories Setting the Stage Definition A Modified Hierarchy Basic Properties Applications Central Properties A Modified Hierarchy for Constructive Purposes Definition Let for α ∈ On [ Veα = {X ⊆ Veβ | ∀x ∈Veβ. {0|x ∈ X} ∈ {0, 1} ∪ Veβ} β<α be defined by recursion over the ordinals. Albert Ziegler Hierarchies of Sets in Classical and Constructive Set Theories Setting the Stage Definition A Modified Hierarchy Basic Properties Applications Central Properties Exploring the Modified Hierarchy Finite Stages n For n finite, Ven is also finite and has 2 − 1 elements: Ve0 = ∅ Ve1 = {∅} Ve2 = {∅, {∅}} Ve3 = {∅, {∅}, {{∅}}, {∅, {∅}}} ... Between 0 and 1 For 0 ≤ α ≤ 1, and α < β: Veα = {0|0 ∈ α} = α ∈ Veβ Albert Ziegler Hierarchies of Sets in Classical and Constructive Set Theories S 2 V = α Veα. More precisely, there is a class function rke : V → On such that ∀a.a ∈ Ve rke (a) Setting the Stage Definition A Modified Hierarchy Basic Properties Applications Central Properties The Good Stuff Theorem 1 For all α, the class Veα is actually a set. Albert Ziegler Hierarchies of Sets in Classical and Constructive Set Theories Setting the Stage Definition A Modified Hierarchy Basic Properties Applications Central Properties The Good Stuff Theorem 1 For all α, the class Veα is actually a set. S 2 V = α Veα. More precisely, there is a class function rke : V → On such that ∀a.a ∈ Ve rke (a) Albert Ziegler Hierarchies of Sets in Classical and Constructive Set Theories Setting the Stage Quantifier Elimination for CZF A Modified Hierarchy The Structure of Large Sets Applications A Characterization of Mahlo Sets Quantifier Elimination As an immediate consequence of this, all unbounded quantification in CZF can be replaced by bounded quantification and quantification over the ordinals. Theorem There is a definitional extension of CZF and a primitive recursive mapping φ 7→ φ∗ of formulas, such that ∗ 1 All quantifiers in φ are either bounded by sets (∀x ∈ a, ∃x ∈ a) or range over the class of ordinals (∀α ∈ On, ∃α ∈ On) ∗ 2 φ and φ are provably equivalent. Albert Ziegler Hierarchies of Sets in Classical and Constructive Set Theories Inaccessible Sets A set I is called inaccessible iff (I, ∈) CZF2 Setting the Stage Quantifier Elimination for CZF A Modified Hierarchy The Structure of Large Sets Applications A Characterization of Mahlo Sets Large Cardinals and Large Sets Large cardinals have become a central topic in classical set theory The classical concept of cardinals does not fit well with constructive set theory Instead of lifting the properties of a large cardinal κ to a constructive setting, better lift the properties of the universe Vκ. Albert Ziegler Hierarchies of Sets in Classical and Constructive Set Theories Setting the Stage Quantifier Elimination for CZF A Modified Hierarchy The Structure of Large Sets Applications A Characterization of Mahlo Sets Large Cardinals and Large Sets Large cardinals have become a central topic in classical set theory The classical concept of cardinals does not fit well with constructive set theory Instead of lifting the properties of a large cardinal κ to a constructive setting, better lift the properties of the universe Vκ.
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