# HG changeset patch # User Shinji KONO # Date 1591529490 -32400 # Node ID 4fcac1eebc74af8ce383aeb9efd3230eecab5136 # Parent 6e1c60866788d7217a3ceb34a10cf64ad74f7664# Parent 5de8905a5a2b83d3a4a364fca1d881e450135cec axiom of choice clean up diff -r 6e1c60866788 -r 4fcac1eebc74 .hgtags --- a/.hgtags Thu Aug 29 16:18:37 2019 +0900 +++ b/.hgtags Sun Jun 07 20:31:30 2020 +0900 @@ -15,3 +15,9 @@ 1b1620e2053cfc340a4df0d63de65b9059b19b6f current 1b1620e2053cfc340a4df0d63de65b9059b19b6f current 2ea2a19f9cd638b29af51a47fa3dabdaea381d5c current +2ea2a19f9cd638b29af51a47fa3dabdaea381d5c current +29a85a427ed21beb9be068728f7c55a7070a0a9f current +29a85a427ed21beb9be068728f7c55a7070a0a9f current +d9d3654baee12c1b93a25a3994146bc001877b2b current +d9d3654baee12c1b93a25a3994146bc001877b2b current +313140ae5e3d1793f8b2dc9055159658d63874e4 current diff -r 6e1c60866788 -r 4fcac1eebc74 BAlgbra.agda --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/BAlgbra.agda Sun Jun 07 20:31:30 2020 +0900 @@ -0,0 +1,109 @@ +open import Level +open import Ordinals +module BAlgbra {n : Level } (O : Ordinals {n}) where + +open import zf +open import logic +import OD +import ODC + +open import Relation.Nullary +open import Relation.Binary +open import Data.Empty +open import Relation.Binary +open import Relation.Binary.Core +open import Relation.Binary.PropositionalEquality +open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) + +open inOrdinal O +open OD O +open OD.OD +open ODAxiom odAxiom + +open _∧_ +open _∨_ +open Bool + +_∩_ : ( A B : OD ) → OD +A ∩ B = record { def = λ x → def A x ∧ def B x } + +_∪_ : ( A B : OD ) → OD +A ∪ B = record { def = λ x → def A x ∨ def B x } + +_＼_ : ( A B : OD ) → OD +A ＼ B = record { def = λ x → def A x ∧ ( ¬ ( def B x ) ) } + +∪-Union : { A B : OD } → Union (A , B) ≡ ( A ∪ B ) +∪-Union {A} {B} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where + lemma1 : {x : Ordinal} → def (Union (A , B)) x → def (A ∪ B) x + lemma1 {x} lt = lemma3 lt where + lemma4 : {y : Ordinal} → def (A , B) y ∧ def (ord→od y) x → ¬ (¬ ( def A x ∨ def B x) ) + lemma4 {y} z with proj1 z + lemma4 {y} z | case1 refl = double-neg (case1 ( subst (λ k → def k x ) oiso (proj2 z)) ) + lemma4 {y} z | case2 refl = double-neg (case2 ( subst (λ k → def k x ) oiso (proj2 z)) ) + lemma3 : (((u : Ordinals.ord O) → ¬ def (A , B) u ∧ def (ord→od u) x) → ⊥) → def (A ∪ B) x + lemma3 not = ODC.double-neg-eilm O (FExists _ lemma4 not) -- choice + lemma2 : {x : Ordinal} → def (A ∪ B) x → def (Union (A , B)) x + lemma2 {x} (case1 A∋x) = subst (λ k → def (Union (A , B)) k) diso ( IsZF.union→ isZF (A , B) (ord→od x) A + (record { proj1 = case1 refl ; proj2 = subst (λ k → def A k) (sym diso) A∋x})) + lemma2 {x} (case2 B∋x) = subst (λ k → def (Union (A , B)) k) diso ( IsZF.union→ isZF (A , B) (ord→od x) B + (record { proj1 = case2 refl ; proj2 = subst (λ k → def B k) (sym diso) B∋x})) + +∩-Select : { A B : OD } → Select A ( λ x → ( A ∋ x ) ∧ ( B ∋ x ) ) ≡ ( A ∩ B ) +∩-Select {A} {B} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where + lemma1 : {x : Ordinal} → def (Select A (λ x₁ → (A ∋ x₁) ∧ (B ∋ x₁))) x → def (A ∩ B) x + lemma1 {x} lt = record { proj1 = proj1 lt ; proj2 = subst (λ k → def B k ) diso (proj2 (proj2 lt)) } + lemma2 : {x : Ordinal} → def (A ∩ B) x → def (Select A (λ x₁ → (A ∋ x₁) ∧ (B ∋ x₁))) x + lemma2 {x} lt = record { proj1 = proj1 lt ; proj2 = + record { proj1 = subst (λ k → def A k) (sym diso) (proj1 lt) ; proj2 = subst (λ k → def B k ) (sym diso) (proj2 lt) } } + +dist-ord : {p q r : OD } → p ∩ ( q ∪ r ) ≡ ( p ∩ q ) ∪ ( p ∩ r ) +dist-ord {p} {q} {r} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where + lemma1 : {x : Ordinal} → def (p ∩ (q ∪ r)) x → def ((p ∩ q) ∪ (p ∩ r)) x + lemma1 {x} lt with proj2 lt + lemma1 {x} lt | case1 q∋x = case1 ( record { proj1 = proj1 lt ; proj2 = q∋x } ) + lemma1 {x} lt | case2 r∋x = case2 ( record { proj1 = proj1 lt ; proj2 = r∋x } ) + lemma2 : {x : Ordinal} → def ((p ∩ q) ∪ (p ∩ r)) x → def (p ∩ (q ∪ r)) x + lemma2 {x} (case1 p∩q) = record { proj1 = proj1 p∩q ; proj2 = case1 (proj2 p∩q ) } + lemma2 {x} (case2 p∩r) = record { proj1 = proj1 p∩r ; proj2 = case2 (proj2 p∩r ) } + +dist-ord2 : {p q r : OD } → p ∪ ( q ∩ r ) ≡ ( p ∪ q ) ∩ ( p ∪ r ) +dist-ord2 {p} {q} {r} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where + lemma1 : {x : Ordinal} → def (p ∪ (q ∩ r)) x → def ((p ∪ q) ∩ (p ∪ r)) x + lemma1 {x} (case1 cp) = record { proj1 = case1 cp ; proj2 = case1 cp } + lemma1 {x} (case2 cqr) = record { proj1 = case2 (proj1 cqr) ; proj2 = case2 (proj2 cqr) } + lemma2 : {x : Ordinal} → def ((p ∪ q) ∩ (p ∪ r)) x → def (p ∪ (q ∩ r)) x + lemma2 {x} lt with proj1 lt | proj2 lt + lemma2 {x} lt | case1 cp | _ = case1 cp + lemma2 {x} lt | _ | case1 cp = case1 cp + lemma2 {x} lt | case2 cq | case2 cr = case2 ( record { proj1 = cq ; proj2 = cr } ) + +record IsBooleanAlgebra ( L : Set n) + ( b1 : L ) + ( b0 : L ) + ( -_ : L → L ) + ( _+_ : L → L → L ) + ( _*_ : L → L → L ) : Set (suc n) where + field + +-assoc : {a b c : L } → a + ( b + c ) ≡ (a + b) + c + *-assoc : {a b c : L } → a * ( b * c ) ≡ (a * b) * c + +-sym : {a b : L } → a + b ≡ b + a + -sym : {a b : L } → a * b ≡ b * a + -aab : {a b : L } → a + ( a * b ) ≡ a + *-aab : {a b : L } → a * ( a + b ) ≡ a + -dist : {a b c : L } → a + ( b * c ) ≡ ( a * b ) + ( a * c ) + *-dist : {a b c : L } → a * ( b + c ) ≡ ( a + b ) * ( a + c ) + a+0 : {a : L } → a + b0 ≡ a + a*1 : {a : L } → a * b1 ≡ a + a+-a1 : {a : L } → a + ( - a ) ≡ b1 + a*-a0 : {a : L } → a * ( - a ) ≡ b0 + +record BooleanAlgebra ( L : Set n) : Set (suc n) where + field + b1 : L + b0 : L + -_ : L → L + _++_ : L → L → L + _**_ : L → L → L + isBooleanAlgebra : IsBooleanAlgebra L b1 b0 -_ _++_ _**_ + diff -r 6e1c60866788 -r 4fcac1eebc74 LEMC.agda --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/LEMC.agda Sun Jun 07 20:31:30 2020 +0900 @@ -0,0 +1,131 @@ +open import Level +open import Ordinals +open import logic +open import Relation.Nullary +module LEMC {n : Level } (O : Ordinals {n} ) (p∨¬p : ( p : Set (suc n)) → p ∨ ( ¬ p )) where + +open import zf +open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) +open import Relation.Binary.PropositionalEquality +open import Data.Nat.Properties +open import Data.Empty +open import Relation.Binary +open import Relation.Binary.Core + +open import nat +import OD + +open inOrdinal O +open OD O +open OD.OD +open OD._==_ +open ODAxiom odAxiom + +open import zfc + +--- With assuption of OD is ordered, p ∨ ( ¬ p ) <=> axiom of choice +--- +record choiced ( X : OD) : Set (suc n) where + field + a-choice : OD + is-in : X ∋ a-choice + +open choiced + +OD→ZFC : ZFC +OD→ZFC = record { + ZFSet = OD + ; _∋_ = _∋_ + ; _≈_ = _==_ + ; ∅ = od∅ + ; Select = Select + ; isZFC = isZFC + } where + -- infixr 200 _∈_ + -- infixr 230 _∩_ _∪_ + isZFC : IsZFC (OD ) _∋_ _==_ od∅ Select + isZFC = record { + choice-func = λ A {X} not A∋X → a-choice (choice-func X not ); + choice = λ A {X} A∋X not → is-in (choice-func X not) + } where + choice-func : (X : OD ) → ¬ ( X == od∅ ) → choiced X + choice-func X not = have_to_find where + ψ : ( ox : Ordinal ) → Set (suc n) + ψ ox = (( x : Ordinal ) → x o< ox → ( ¬ def X x )) ∨ choiced X + lemma-ord : ( ox : Ordinal ) → ψ ox + lemma-ord ox = TransFinite {ψ} induction ox where + ∋-p : (A x : OD ) → Dec ( A ∋ x ) + ∋-p A x with p∨¬p (Lift (suc n) ( A ∋ x )) -- LEM + ∋-p A x | case1 (lift t) = yes t + ∋-p A x | case2 t = no (λ x → t (lift x )) + ∀-imply-or : {A : Ordinal → Set n } {B : Set (suc n) } + → ((x : Ordinal ) → A x ∨ B) → ((x : Ordinal ) → A x) ∨ B + ∀-imply-or {A} {B} ∀AB with p∨¬p (Lift ( suc n ) ((x : Ordinal ) → A x)) -- LEM + ∀-imply-or {A} {B} ∀AB | case1 (lift t) = case1 t + ∀-imply-or {A} {B} ∀AB | case2 x = case2 (lemma (λ not → x (lift not ))) where + lemma : ¬ ((x : Ordinal ) → A x) → B + lemma not with p∨¬p B + lemma not | case1 b = b + lemma not | case2 ¬b = ⊥-elim (not (λ x → dont-orb (∀AB x) ¬b )) + induction : (x : Ordinal) → ((y : Ordinal) → y o< x → ψ y) → ψ x + induction x prev with ∋-p X ( ord→od x) + ... | yes p = case2 ( record { a-choice = ord→od x ; is-in = p } ) + ... | no ¬p = lemma where + lemma1 : (y : Ordinal) → (y o< x → def X y → ⊥) ∨ choiced X + lemma1 y with ∋-p X (ord→od y) + lemma1 y | yes y : (x y : OD) → OD -< x , y > = (x , x ) , (x , y ) - -exg-pair : { x y : OD } → (x , y ) == ( y , x ) -exg-pair {x} {y} = record { eq→ = left ; eq← = right } where - left : {z : Ordinal} → def (x , y) z → def (y , x) z - left (case1 t) = case2 t - left (case2 t) = case1 t - right : {z : Ordinal} → def (y , x) z → def (x , y) z - right (case1 t) = case2 t - right (case2 t) = case1 t - -==-trans : { x y z : OD } → x == y → y == z → x == z -==-trans x=y y=z = record { eq→ = λ {m} t → eq→ y=z (eq→ x=y t) ; eq← = λ {m} t → eq← x=y (eq← y=z t) } - -==-sym : { x y : OD } → x == y → y == x -==-sym x=y = record { eq→ = λ {m} t → eq← x=y t ; eq← = λ {m} t → eq→ x=y t } - -ord≡→≡ : { x y : OD } → od→ord x ≡ od→ord y → x ≡ y -ord≡→≡ eq = subst₂ (λ j k → j ≡ k ) oiso oiso ( cong ( λ k → ord→od k ) eq ) - -od≡→≡ : { x y : Ordinal } → ord→od x ≡ ord→od y → x ≡ y -od≡→≡ eq = subst₂ (λ j k → j ≡ k ) diso diso ( cong ( λ k → od→ord k ) eq ) - -eq-prod : { x x' y y' : OD } → x ≡ x' → y ≡ y' → < x , y > ≡ < x' , y' > -eq-prod refl refl = refl - -prod-eq : { x x' y y' : OD } → < x , y > == < x' , y' > → (x ≡ x' ) ∧ ( y ≡ y' ) -prod-eq {x} {x'} {y} {y'} eq = record { proj1 = lemmax ; proj2 = lemmay } where - lemma0 : {x y z : OD } → ( x , x ) == ( z , y ) → x ≡ y - lemma0 {x} {y} eq with trio< (od→ord x) (od→ord y) - lemma0 {x} {y} eq | tri< a ¬b ¬c with eq← eq {od→ord y} (case2 refl) - lemma0 {x} {y} eq | tri< a ¬b ¬c | case1 s = ⊥-elim ( o<¬≡ (sym s) a ) - lemma0 {x} {y} eq | tri< a ¬b ¬c | case2 s = ⊥-elim ( o<¬≡ (sym s) a ) - lemma0 {x} {y} eq | tri≈ ¬a b ¬c = ord≡→≡ b - lemma0 {x} {y} eq | tri> ¬a ¬b c with eq← eq {od→ord y} (case2 refl) - lemma0 {x} {y} eq | tri> ¬a ¬b c | case1 s = ⊥-elim ( o<¬≡ s c ) - lemma0 {x} {y} eq | tri> ¬a ¬b c | case2 s = ⊥-elim ( o<¬≡ s c ) - lemma2 : {x y z : OD } → ( x , x ) == ( z , y ) → z ≡ y - lemma2 {x} {y} {z} eq = trans (sym (lemma0 lemma3 )) ( lemma0 eq ) where - lemma3 : ( x , x ) == ( y , z ) - lemma3 = ==-trans eq exg-pair - lemma1 : {x y : OD } → ( x , x ) == ( y , y ) → x ≡ y - lemma1 {x} {y} eq with eq← eq {od→ord y} (case2 refl) - lemma1 {x} {y} eq | case1 s = ord≡→≡ (sym s) - lemma1 {x} {y} eq | case2 s = ord≡→≡ (sym s) - lemma4 : {x y z : OD } → ( x , y ) == ( x , z ) → y ≡ z - lemma4 {x} {y} {z} eq with eq← eq {od→ord z} (case2 refl) - lemma4 {x} {y} {z} eq | case1 s with ord≡→≡ s -- x ≡ z - ... | refl with lemma2 (==-sym eq ) - ... | refl = refl - lemma4 {x} {y} {z} eq | case2 s = ord≡→≡ (sym s) -- y ≡ z - lemmax : x ≡ x' - lemmax with eq→ eq {od→ord (x , x)} (case1 refl) - lemmax | case1 s = lemma1 (ord→== s ) -- (x,x)≡(x',x') - lemmax | case2 s with lemma2 (ord→== s ) -- (x,x)≡(x',y') with x'≡y' - ... | refl = lemma1 (ord→== s ) - lemmay : y ≡ y' - lemmay with lemmax - ... | refl with lemma4 eq -- with (x,y)≡(x,y') - ... | eq1 = lemma4 (ord→== (cong (λ k → od→ord k ) eq1 )) - -ppp : { p : Set n } { a : OD } → record { def = λ x → p } ∋ a → p -ppp {p} {a} d = d - --- --- Axiom of choice in intutionistic logic implies the exclude middle --- https://plato.stanford.edu/entries/axiom-choice/#AxiChoLog --- -p∨¬p : ( p : Set n ) → p ∨ ( ¬ p ) -- assuming axiom of choice -p∨¬p p with is-o∅ ( od→ord ( record { def = λ x → p } )) -p∨¬p p | yes eq = case2 (¬p eq) where - ps = record { def = λ x → p } - lemma : ps == od∅ → p → ⊥ - lemma eq p0 = ¬x<0 {od→ord ps} (eq→ eq p0 ) - ¬p : (od→ord ps ≡ o∅) → p → ⊥ - ¬p eq = lemma ( subst₂ (λ j k → j == k ) oiso o∅≡od∅ ( o≡→== eq )) -p∨¬p p | no ¬p = case1 (ppp {p} {minimul ps (λ eq → ¬p (eqo∅ eq))} lemma) where - ps = record { def = λ x → p } - eqo∅ : ps == od∅ → od→ord ps ≡ o∅ - eqo∅ eq = subst (λ k → od→ord ps ≡ k) ord-od∅ ( cong (λ k → od→ord k ) (==→o≡ eq)) - lemma : ps ∋ minimul ps (λ eq → ¬p (eqo∅ eq)) - lemma = x∋minimul ps (λ eq → ¬p (eqo∅ eq)) - -decp : ( p : Set n ) → Dec p -- assuming axiom of choice -decp p with p∨¬p p -decp p | case1 x = yes x -decp p | case2 x = no x - -double-neg-eilm : {A : Set n} → ¬ ¬ A → A -- we don't have this in intutionistic logic -double-neg-eilm {A} notnot with decp A -- assuming axiom of choice -... | yes p = p -... | no ¬p = ⊥-elim ( notnot ¬p ) - -OrdP : ( x : Ordinal ) ( y : OD ) → Dec ( Ord x ∋ y ) -OrdP x y with trio< x (od→ord y) -OrdP x y | tri< a ¬b ¬c = no ¬c -OrdP x y | tri≈ ¬a refl ¬c = no ( o<¬≡ refl ) -OrdP x y | tri> ¬a ¬b c = yes c -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) -- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality n (suc n) @@ -285,33 +199,38 @@ ZFSubset A x = record { def = λ y → def A y ∧ def x y } -- roughly x = A → Set Def : (A : OD ) → OD -Def A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) ) +Def A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A x) ) ) +-- _⊆_ : ( A B : OD ) → ∀{ x : OD } → Set n +-- _⊆_ A B {x} = A ∋ x → B ∋ x -_⊆_ : ( A B : OD ) → ∀{ x : OD } → Set n -_⊆_ A B {x} = A ∋ x → B ∋ x +record _⊆_ ( A B : OD ) : Set (suc n) where + field + incl : { x : OD } → A ∋ x → B ∋ x + +open _⊆_ infixr 220 _⊆_ -subset-lemma : {A x y : OD } → ( x ∋ y → ZFSubset A x ∋ y ) ⇔ ( _⊆_ x A {y} ) -subset-lemma {A} {x} {y} = record { - proj1 = λ z lt → proj1 (z lt) - ; proj2 = λ x⊆A lt → record { proj1 = x⊆A lt ; proj2 = lt } +subset-lemma : {A x : OD } → ( {y : OD } → x ∋ y → ZFSubset A x ∋ y ) ⇔ ( x ⊆ A ) +subset-lemma {A} {x} = record { + proj1 = λ lt → record { incl = λ x∋z → proj1 (lt x∋z) } + ; proj2 = λ x⊆A lt → record { proj1 = incl x⊆A lt ; proj2 = lt } } --- Constructible Set on α --- Def X = record { def = λ y → ∀ (x : OD ) → y < X ∧ y < od→ord x } --- L (Φ 0) = Φ --- L (OSuc lv n) = { Def ( L n ) } --- L (Φ (Suc n)) = Union (L α) (α < Φ (Suc n) ) --- L : {n : Level} → (α : Ordinal ) → OD --- L record { lv = Zero ; ord = (Φ .0) } = od∅ --- L record { lv = lx ; ord = (OSuc lv ox) } = Def ( L ( record { lv = lx ; ord = ox } ) ) --- L record { lv = (Suc lx) ; ord = (Φ (Suc lx)) } = -- Union ( L α ) --- cseq ( Ord (od→ord (L (record { lv = lx ; ord = Φ lx })))) +open import Data.Unit --- L0 : {n : Level} → (α : Ordinal ) → L (osuc α) ∋ L α --- L1 : {n : Level} → (α β : Ordinal ) → α o< β → ∀ (x : OD ) → L α ∋ x → L β ∋ x +ε-induction : { ψ : OD → Set (suc n)} + → ( {x : OD } → ({ y : OD } → x ∋ y → ψ y ) → ψ x ) + → (x : OD ) → ψ x +ε-induction {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (osuc (od→ord x)) <-osuc ) where + induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (ord→od oy)) → ψ (ord→od ox) + induction ox prev = ind ( λ {y} lt → subst (λ k → ψ k ) oiso (prev (od→ord y) (o<-subst (c<→o< lt) refl diso ))) + ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (ord→od oy) + ε-induction-ord ox {oy} lt = TransFinite {λ oy → ψ (ord→od oy)} induction oy + +-- minimal-2 : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → (y : OD ) → ¬ ( def (minimal x ne) (od→ord y)) ∧ (def x (od→ord y) ) +-- minimal-2 x ne y = contra-position ( ε-induction (λ {z} ind → {!!} ) x ) ( λ p → {!!} ) OD→ZF : ZF OD→ZF = record { @@ -327,11 +246,11 @@ ; infinite = infinite ; isZF = isZF } where - ZFSet = OD + ZFSet = OD -- is less than Ords because of maxod Select : (X : OD ) → ((x : OD ) → Set n ) → OD Select X ψ = record { def = λ x → ( def X x ∧ ψ ( ord→od x )) } Replace : OD → (OD → OD ) → OD - Replace X ψ = record { def = λ x → (x o< sup-o ( λ x → od→ord (ψ (ord→od x )))) ∧ def (in-codomain X ψ) x } + Replace X ψ = record { def = λ x → (x o< sup-o ( λ x → od→ord (ψ x))) ∧ def (in-codomain X ψ) x } _∩_ : ( A B : ZFSet ) → ZFSet A ∩ B = record { def = λ x → def A x ∧ def B x } Union : OD → OD @@ -340,8 +259,8 @@ A ∈ B = B ∋ A Power : OD → OD Power A = Replace (Def (Ord (od→ord A))) ( λ x → A ∩ x ) - ｛_｝ : ZFSet → ZFSet - ｛ x ｝ = ( x , x ) + -- ｛_｝ : ZFSet → ZFSet + -- ｛ x ｝ = ( x , x ) -- it works but we don't use data infinite-d : ( x : Ordinal ) → Set n where iφ : infinite-d o∅ @@ -355,7 +274,7 @@ -- infixr 230 _∩_ _∪_ isZF : IsZF (OD ) _∋_ _==_ od∅ _,_ Union Power Select Replace infinite isZF = record { - isEquivalence = record { refl = eq-refl ; sym = eq-sym; trans = eq-trans } + isEquivalence = record { refl = ==-refl ; sym = ==-sym; trans = ==-trans } ; pair→ = pair→ ; pair← = pair← ; union→ = union→ @@ -364,14 +283,14 @@ ; power→ = power→ ; power← = power← ; extensionality = λ {A} {B} {w} → extensionality {A} {B} {w} - -- ; ε-induction = {!!} + ; ε-induction = ε-induction ; infinity∅ = infinity∅ ; infinity = infinity ; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y} ; replacement← = replacement← ; replacement→ = replacement→ - ; choice-func = choice-func - ; choice = choice + -- ; choice-func = choice-func + -- ; choice = choice } where pair→ : ( x y t : ZFSet ) → (x , y) ∋ t → ( t == x ) ∨ ( t == y ) @@ -382,28 +301,24 @@ pair← x y t (case1 t==x) = case1 (cong (λ k → od→ord k ) (==→o≡ t==x)) pair← x y t (case2 t==y) = case2 (cong (λ k → od→ord k ) (==→o≡ t==y)) - -- pair0 : (A B : OD ) → ((A , B) ∋ A) ∧ ((A , B) ∋ B) - -- proj1 (pair A B ) = omax-x (od→ord A) (od→ord B) - -- proj2 (pair A B ) = omax-y (od→ord A) (od→ord B) - empty : (x : OD ) → ¬ (od∅ ∋ x) empty x = ¬x<0 - o<→c< : {x y : Ordinal } {z : OD }→ x o< y → _⊆_ (Ord x) (Ord y) {z} - o<→c< lt lt1 = ordtrans lt1 lt + o<→c< : {x y : Ordinal } → x o< y → (Ord x) ⊆ (Ord y) + o<→c< lt = record { incl = λ z → ordtrans z lt } - ⊆→o< : {x y : Ordinal } → (∀ (z : OD) → _⊆_ (Ord x) (Ord y) {z} ) → x o< osuc y + ⊆→o< : {x y : Ordinal } → (Ord x) ⊆ (Ord y) → x o< osuc y ⊆→o< {x} {y} lt with trio< x y ⊆→o< {x} {y} lt | tri< a ¬b ¬c = ordtrans a <-osuc ⊆→o< {x} {y} lt | tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc - ⊆→o< {x} {y} lt | tri> ¬a ¬b c with lt (ord→od y) (o<-subst c (sym diso) refl ) + ⊆→o< {x} {y} lt | tri> ¬a ¬b c with (incl lt) (o<-subst c (sym diso) refl ) ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt diso refl )) union→ : (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z union→ X z u xx not = ⊥-elim ( not (od→ord u) ( record { proj1 = proj1 xx ; proj2 = subst ( λ k → def k (od→ord z)) (sym oiso) (proj2 xx) } )) union← : (X z : OD) (X∋z : Union X ∋ z) → ¬ ( (u : OD ) → ¬ ((X ∋ u) ∧ (u ∋ z ))) - union← X z UX∋z = TransFiniteExists _ lemma UX∋z where + union← X z UX∋z = FExists _ lemma UX∋z where lemma : {y : Ordinal} → def X y ∧ def (ord→od y) (od→ord z) → ¬ ((u : OD) → ¬ (X ∋ u) ∧ (u ∋ z)) lemma {y} xx not = not (ord→od y) record { proj1 = subst ( λ k → def X k ) (sym diso) (proj1 xx ) ; proj2 = proj2 xx } @@ -434,7 +349,6 @@ --- First consider ordinals in OD --- --- ZFSubset A x = record { def = λ y → def A y ∧ def x y } subset of A - --- Power X = ord→od ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) ) Power X is a sup of all subset of A -- -- ∩-≡ : { a b : OD } → ({x : OD } → (a ∋ x → b ∋ x)) → a == ( b ∩ a ) @@ -443,8 +357,10 @@ proj1 = def-subst {_} {_} {b} {x} (inc (def-subst {_} {_} {a} {_} x axiom of choice - --- - record choiced ( X : OD) : Set (suc n) where - field - a-choice : OD - is-in : X ∋ a-choice - - choice-func' : (X : OD ) → (p∨¬p : ( p : Set (suc n)) → p ∨ ( ¬ p )) → ¬ ( X == od∅ ) → choiced X - choice-func' X p∨¬p not = have_to_find where - ψ : ( ox : Ordinal ) → Set (suc n) - ψ ox = (( x : Ordinal ) → x o< ox → ( ¬ def X x )) ∨ choiced X - lemma-ord : ( ox : Ordinal ) → ψ ox - lemma-ord ox = TransFinite {ψ} induction ox where - ∋-p : (A x : OD ) → Dec ( A ∋ x ) - ∋-p A x with p∨¬p (Lift (suc n) ( A ∋ x )) - ∋-p A x | case1 (lift t) = yes t - ∋-p A x | case2 t = no (λ x → t (lift x )) - ∀-imply-or : {A : Ordinal → Set n } {B : Set (suc n) } - → ((x : Ordinal ) → A x ∨ B) → ((x : Ordinal ) → A x) ∨ B - ∀-imply-or {A} {B} ∀AB with p∨¬p (Lift ( suc n ) ((x : Ordinal ) → A x)) - ∀-imply-or {A} {B} ∀AB | case1 (lift t) = case1 t - ∀-imply-or {A} {B} ∀AB | case2 x = case2 (lemma (λ not → x (lift not ))) where - lemma : ¬ ((x : Ordinal ) → A x) → B - lemma not with p∨¬p B - lemma not | case1 b = b - lemma not | case2 ¬b = ⊥-elim (not (λ x → dont-orb (∀AB x) ¬b )) - induction : (x : Ordinal) → ((y : Ordinal) → y o< x → ψ y) → ψ x - induction x prev with ∋-p X ( ord→od x) - ... | yes p = case2 ( record { a-choice = ord→od x ; is-in = p } ) - ... | no ¬p = lemma where - lemma1 : (y : Ordinal) → (y o< x → def X y → ⊥) ∨ choiced X - lemma1 y with ∋-p X (ord→od y) - lemma1 y | yes y ¬a ¬b c = yes c + +open import zfc + +OD→ZFC : ZFC +OD→ZFC = record { + ZFSet = OD + ; _∋_ = _∋_ + ; _≈_ = _==_ + ; ∅ = od∅ + ; Select = Select + ; isZFC = isZFC + } where + -- infixr 200 _∈_ + -- infixr 230 _∩_ _∪_ + isZFC : IsZFC (OD ) _∋_ _==_ od∅ Select + isZFC = record { + choice-func = choice-func ; + choice = choice + } where + -- Axiom of choice ( is equivalent to the existence of minimal in our case ) + -- ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ] + choice-func : (X : OD ) → {x : OD } → ¬ ( x == od∅ ) → ( X ∋ x ) → OD + choice-func X {x} not X∋x = minimal x not + choice : (X : OD ) → {A : OD } → ( X∋A : X ∋ A ) → (not : ¬ ( A == od∅ )) → A ∋ choice-func X not X∋A + choice X {A} X∋A not = x∋minimal A not + diff -r 6e1c60866788 -r 4fcac1eebc74 OPair.agda --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/OPair.agda Sun Jun 07 20:31:30 2020 +0900 @@ -0,0 +1,128 @@ +open import Level +open import Ordinals +module OPair {n : Level } (O : Ordinals {n}) where + +open import zf +open import logic +import OD + +open import Relation.Nullary +open import Relation.Binary +open import Data.Empty +open import Relation.Binary +open import Relation.Binary.Core +open import Relation.Binary.PropositionalEquality +open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) + +open inOrdinal O +open OD O +open OD.OD +open ODAxiom odAxiom + +open _∧_ +open _∨_ +open Bool + +open _==_ + +<_,_> : (x y : OD) → OD +< x , y > = (x , x ) , (x , y ) + +exg-pair : { x y : OD } → (x , y ) == ( y , x ) +exg-pair {x} {y} = record { eq→ = left ; eq← = right } where + left : {z : Ordinal} → def (x , y) z → def (y , x) z + left (case1 t) = case2 t + left (case2 t) = case1 t + right : {z : Ordinal} → def (y , x) z → def (x , y) z + right (case1 t) = case2 t + right (case2 t) = case1 t + +ord≡→≡ : { x y : OD } → od→ord x ≡ od→ord y → x ≡ y +ord≡→≡ eq = subst₂ (λ j k → j ≡ k ) oiso oiso ( cong ( λ k → ord→od k ) eq ) + +od≡→≡ : { x y : Ordinal } → ord→od x ≡ ord→od y → x ≡ y +od≡→≡ eq = subst₂ (λ j k → j ≡ k ) diso diso ( cong ( λ k → od→ord k ) eq ) + +eq-prod : { x x' y y' : OD } → x ≡ x' → y ≡ y' → < x , y > ≡ < x' , y' > +eq-prod refl refl = refl + +prod-eq : { x x' y y' : OD } → < x , y > == < x' , y' > → (x ≡ x' ) ∧ ( y ≡ y' ) +prod-eq {x} {x'} {y} {y'} eq = record { proj1 = lemmax ; proj2 = lemmay } where + lemma0 : {x y z : OD } → ( x , x ) == ( z , y ) → x ≡ y + lemma0 {x} {y} eq with trio< (od→ord x) (od→ord y) + lemma0 {x} {y} eq | tri< a ¬b ¬c with eq← eq {od→ord y} (case2 refl) + lemma0 {x} {y} eq | tri< a ¬b ¬c | case1 s = ⊥-elim ( o<¬≡ (sym s) a ) + lemma0 {x} {y} eq | tri< a ¬b ¬c | case2 s = ⊥-elim ( o<¬≡ (sym s) a ) + lemma0 {x} {y} eq | tri≈ ¬a b ¬c = ord≡→≡ b + lemma0 {x} {y} eq | tri> ¬a ¬b c with eq← eq {od→ord y} (case2 refl) + lemma0 {x} {y} eq | tri> ¬a ¬b c | case1 s = ⊥-elim ( o<¬≡ s c ) + lemma0 {x} {y} eq | tri> ¬a ¬b c | case2 s = ⊥-elim ( o<¬≡ s c ) + lemma2 : {x y z : OD } → ( x , x ) == ( z , y ) → z ≡ y + lemma2 {x} {y} {z} eq = trans (sym (lemma0 lemma3 )) ( lemma0 eq ) where + lemma3 : ( x , x ) == ( y , z ) + lemma3 = ==-trans eq exg-pair + lemma1 : {x y : OD } → ( x , x ) == ( y , y ) → x ≡ y + lemma1 {x} {y} eq with eq← eq {od→ord y} (case2 refl) + lemma1 {x} {y} eq | case1 s = ord≡→≡ (sym s) + lemma1 {x} {y} eq | case2 s = ord≡→≡ (sym s) + lemma4 : {x y z : OD } → ( x , y ) == ( x , z ) → y ≡ z + lemma4 {x} {y} {z} eq with eq← eq {od→ord z} (case2 refl) + lemma4 {x} {y} {z} eq | case1 s with ord≡→≡ s -- x ≡ z + ... | refl with lemma2 (==-sym eq ) + ... | refl = refl + lemma4 {x} {y} {z} eq | case2 s = ord≡→≡ (sym s) -- y ≡ z + lemmax : x ≡ x' + lemmax with eq→ eq {od→ord (x , x)} (case1 refl) + lemmax | case1 s = lemma1 (ord→== s ) -- (x,x)≡(x',x') + lemmax | case2 s with lemma2 (ord→== s ) -- (x,x)≡(x',y') with x'≡y' + ... | refl = lemma1 (ord→== s ) + lemmay : y ≡ y' + lemmay with lemmax + ... | refl with lemma4 eq -- with (x,y)≡(x,y') + ... | eq1 = lemma4 (ord→== (cong (λ k → od→ord k ) eq1 )) + +data ord-pair : (p : Ordinal) → Set n where + pair : (x y : Ordinal ) → ord-pair ( od→ord ( < ord→od x , ord→od y > ) ) + +ZFProduct : OD +ZFProduct = record { def = λ x → ord-pair x } + +-- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) +-- eq-pair : { x x' y y' : Ordinal } → x ≡ x' → y ≡ y' → pair x y ≅ pair x' y' +-- eq-pair refl refl = HE.refl + +pi1 : { p : Ordinal } → ord-pair p → Ordinal +pi1 ( pair x y) = x + +π1 : { p : OD } → ZFProduct ∋ p → OD +π1 lt = ord→od (pi1 lt ) + +pi2 : { p : Ordinal } → ord-pair p → Ordinal +pi2 ( pair x y ) = y + +π2 : { p : OD } → ZFProduct ∋ p → OD +π2 lt = ord→od (pi2 lt ) + +op-cons : { ox oy : Ordinal } → ZFProduct ∋ < ord→od ox , ord→od oy > +op-cons {ox} {oy} = pair ox oy + +p-cons : ( x y : OD ) → ZFProduct ∋ < x , y > +p-cons x y = def-subst {_} {_} {ZFProduct} {od→ord (< x , y >)} (pair (od→ord x) ( od→ord y )) refl ( + let open ≡-Reasoning in begin + od→ord < ord→od (od→ord x) , ord→od (od→ord y) > + ≡⟨ cong₂ (λ j k → od→ord < j , k >) oiso oiso ⟩ + od→ord < x , y > + ∎ ) + +op-iso : { op : Ordinal } → (q : ord-pair op ) → od→ord < ord→od (pi1 q) , ord→od (pi2 q) > ≡ op +op-iso (pair ox oy) = refl + +p-iso : { x : OD } → (p : ZFProduct ∋ x ) → < π1 p , π2 p > ≡ x +p-iso {x} p = ord≡→≡ (op-iso p) + +p-pi1 : { x y : OD } → (p : ZFProduct ∋ < x , y > ) → π1 p ≡ x +p-pi1 {x} {y} p = proj1 ( prod-eq ( ord→== (op-iso p) )) + +p-pi2 : { x y : OD } → (p : ZFProduct ∋ < x , y > ) → π2 p ≡ y +p-pi2 {x} {y} p = proj2 ( prod-eq ( ord→== (op-iso p))) + diff -r 6e1c60866788 -r 4fcac1eebc74 Ordinals.agda --- a/Ordinals.agda Thu Aug 29 16:18:37 2019 +0900 +++ b/Ordinals.agda Sun Jun 07 20:31:30 2020 +0900 @@ -193,9 +193,9 @@ } } - TransFiniteExists : {m l : Level} → ( ψ : Ordinal → Set m ) + FExists : {m l : Level} → ( ψ : Ordinal → Set m ) → {p : Set l} ( P : { y : Ordinal } → ψ y → ¬ p ) → (exists : ¬ (∀ y → ¬ ( ψ y ) )) → ¬ p - TransFiniteExists {m} {l} ψ {p} P = contra-position ( λ p y ψy → P {y} ψy p ) + FExists {m} {l} ψ {p} P = contra-position ( λ p y ψy → P {y} ψy p ) diff -r 6e1c60866788 -r 4fcac1eebc74 README.md --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/README.md Sun Jun 07 20:31:30 2020 +0900 @@ -0,0 +1,1 @@ +zf-in-agda.ind \ No newline at end of file diff -r 6e1c60866788 -r 4fcac1eebc74 cardinal.agda --- a/cardinal.agda Thu Aug 29 16:18:37 2019 +0900 +++ b/cardinal.agda Sun Jun 07 20:31:30 2020 +0900 @@ -5,6 +5,8 @@ open import zf open import logic import OD +import ODC +import OPair open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) open import Relation.Binary.PropositionalEquality open import Data.Nat.Properties @@ -16,6 +18,8 @@ open inOrdinal O open OD O open OD.OD +open OPair O +open ODAxiom odAxiom open _∧_ open _∨_ @@ -24,50 +28,10 @@ -- we have to work on Ordinal to keep OD Level n -- since we use p∨¬p which works only on Level n --- < x , y > = (x , x) , (x , y) - -data ord-pair : (p : Ordinal) → Set n where - pair : (x y : Ordinal ) → ord-pair ( od→ord ( < ord→od x , ord→od y > ) ) - -ZFProduct : OD -ZFProduct = record { def = λ x → ord-pair x } - --- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) --- eq-pair : { x x' y y' : Ordinal } → x ≡ x' → y ≡ y' → pair x y ≅ pair x' y' --- eq-pair refl refl = HE.refl - -pi1 : { p : Ordinal } → ord-pair p → Ordinal -pi1 ( pair x y) = x - -π1 : { p : OD } → ZFProduct ∋ p → Ordinal -π1 lt = pi1 lt - -pi2 : { p : Ordinal } → ord-pair p → Ordinal -pi2 ( pair x y ) = y - -π2 : { p : OD } → ZFProduct ∋ p → Ordinal -π2 lt = pi2 lt - -p-cons : ( x y : OD ) → ZFProduct ∋ < x , y > -p-cons x y = def-subst {_} {_} {ZFProduct} {od→ord (< x , y >)} (pair (od→ord x) ( od→ord y )) refl ( - let open ≡-Reasoning in begin - od→ord < ord→od (od→ord x) , ord→od (od→ord y) > - ≡⟨ cong₂ (λ j k → od→ord < j , k >) oiso oiso ⟩ - od→ord < x , y > - ∎ ) - - -p-iso1 : { ox oy : Ordinal } → ZFProduct ∋ < ord→od ox , ord→od oy > -p-iso1 {ox} {oy} = pair ox oy - -p-iso : { x : OD } → (p : ZFProduct ∋ x ) → < ord→od (π1 p) , ord→od (π2 p) > ≡ x -p-iso {x} p = ord≡→≡ (lemma p) where - lemma : { op : Ordinal } → (q : ord-pair op ) → od→ord < ord→od (pi1 q) , ord→od (pi2 q) > ≡ op - lemma (pair ox oy) = refl ∋-p : (A x : OD ) → Dec ( A ∋ x ) -∋-p A x with p∨¬p ( A ∋ x ) +∋-p A x with ODC.p∨¬p O ( A ∋ x ) ∋-p A x | case1 t = yes t ∋-p A x | case2 t = no t @@ -88,7 +52,6 @@ -- power→ : ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x) - func→od : (f : Ordinal → Ordinal ) → ( dom : OD ) → OD func→od f dom = Replace dom ( λ x → < x , ord→od (f (od→ord x)) > ) @@ -98,17 +61,17 @@ func→od∈Func-1 : Func dom cod ∋ func→od func-1 dom od→func : { dom cod : OD } → {f : Ordinal } → def (Func dom cod ) f → Func←cd {dom} {cod} {f} -od→func {dom} {cod} {f} lt = record { func-1 = λ x → sup-o ( λ y → lemma x y ) ; func→od∈Func-1 = record { proj1 = {!!} ; proj2 = {!!} } } where +od→func {dom} {cod} {f} lt = record { func-1 = λ x → sup-o ( λ y → lemma x {!!} ) ; func→od∈Func-1 = record { proj1 = {!!} ; proj2 = {!!} } } where lemma : Ordinal → Ordinal → Ordinal lemma x y with IsZF.power→ isZF (dom ⊗ cod) (ord→od f) (subst (λ k → def (Power (dom ⊗ cod)) k ) (sym diso) lt ) | ∋-p (ord→od f) (ord→od y) lemma x y | p | no n = o∅ - lemma x y | p | yes f∋y = lemma2 (proj1 (double-neg-eilm ( p {ord→od y} f∋y ))) where -- p : {y : OD} → f ∋ y → ¬ ¬ (dom ⊗ cod ∋ y) + lemma x y | p | yes f∋y = lemma2 (proj1 (ODC.double-neg-eilm O ( p {ord→od y} f∋y ))) where -- p : {y : OD} → f ∋ y → ¬ ¬ (dom ⊗ cod ∋ y) lemma2 : {p : Ordinal} → ord-pair p → Ordinal - lemma2 (pair x1 y1) with decp ( x1 ≡ x) + lemma2 (pair x1 y1) with ODC.decp O ( x1 ≡ x) lemma2 (pair x1 y1) | yes p = y1 lemma2 (pair x1 y1) | no ¬p = o∅ fod : OD - fod = Replace dom ( λ x → < x , ord→od (sup-o ( λ y → lemma (od→ord x) y )) > ) + fod = Replace dom ( λ x → < x , ord→od (sup-o ( λ y → lemma (od→ord x) {!!} )) > ) open Func←cd @@ -139,7 +102,7 @@ open Onto -onto-restrict : {X Y Z : OD} → Onto X Y → ({x : OD} → _⊆_ Z Y {x}) → Onto X Z +onto-restrict : {X Y Z : OD} → Onto X Y → Z ⊆ Y → Onto X Z onto-restrict {X} {Y} {Z} onto Z⊆Y = record { xmap = xmap1 ; ymap = zmap @@ -167,15 +130,15 @@ cardinal : (X : OD ) → Cardinal X cardinal X = record { - cardinal = sup-o ( λ x → proj1 ( cardinal-p x) ) + cardinal = sup-o ( λ x → proj1 ( cardinal-p {!!}) ) ; conto = onto ; cmax = cmax } where cardinal-p : (x : Ordinal ) → ( Ordinal ∧ Dec (Onto X (Ord x) ) ) - cardinal-p x with p∨¬p ( Onto X (Ord x) ) + cardinal-p x with ODC.p∨¬p O ( Onto X (Ord x) ) cardinal-p x | case1 True = record { proj1 = x ; proj2 = yes True } cardinal-p x | case2 False = record { proj1 = o∅ ; proj2 = no False } - S = sup-o (λ x → proj1 (cardinal-p x)) + S = sup-o (λ x → proj1 (cardinal-p {!!})) lemma1 : (x : Ordinal) → ((y : Ordinal) → y o< x → Lift (suc n) (y o< (osuc S) → Onto X (Ord y))) → Lift (suc n) (x o< (osuc S) → Onto X (Ord x) ) lemma1 x prev with trio< x (osuc S) @@ -192,9 +155,9 @@ ... | lift t = t <-osuc cmax : (y : Ordinal) → S o< y → ¬ Onto X (Ord y) cmax y lt ontoy = o<> lt (o<-subst {_} {_} {y} {S} - (sup-o< {λ x → proj1 ( cardinal-p x)}{y} ) lemma refl ) where + (sup-o< {λ x → proj1 ( cardinal-p {!!})}{{!!}} ) lemma refl ) where lemma : proj1 (cardinal-p y) ≡ y - lemma with p∨¬p ( Onto X (Ord y) ) + lemma with ODC.p∨¬p O ( Onto X (Ord y) ) lemma | case1 x = refl lemma | case2 not = ⊥-elim ( not ontoy ) diff -r 6e1c60866788 -r 4fcac1eebc74 fig/ODandOrdinals.graffle Binary file fig/ODandOrdinals.graffle has changed diff -r 6e1c60866788 -r 4fcac1eebc74 fig/ODandOrdinals.svg --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/fig/ODandOrdinals.svg Sun Jun 07 20:31:30 2020 +0900 @@ -0,0 +1,192 @@ + + + + + + + + + + + + + + + + + + + + + Produced by OmniGraffle 7.12.1 + 2019-11-28 04:44:00 +0000 + + + Canvas 1 + + + Layer 1 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + Ordinal + + + + + Ordinal + Definable + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + Ordinal + + + + + Ordinal + Definable + + + + + + + + + + + + + + + OD contains Ordinals + + + + + OD has a name in Ordinals + + + + + all arrows have to be acyclic + + + + + diff -r 6e1c60866788 -r 4fcac1eebc74 fig/Sets.graffle Binary file fig/Sets.graffle has changed diff -r 6e1c60866788 -r 4fcac1eebc74 fig/Sets.svg --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/fig/Sets.svg Sun Jun 07 20:31:30 2020 +0900 @@ -0,0 +1,84 @@ + + + + + + + + + + + + + + + + Produced by OmniGraffle 7.12.1 + 2019-11-27 13:34:32 +0000 + + + Canvas 1 + + + Layer 1 + + + 様々な集合 + + + + + 順序数の公理を満たすもの。自然数の延長 + + + + + 一つ一つ増やす。無限回繰り返して、それを全部。 + また一つ一つ増やすを繰り返して得られるもの + + + + + 述語で定義されるものを集めるのを繰り返して作られるもの + + + + + 順序数上の方程式を満たす順序数の集合 + 階層化されていない + + + + + Ordinal + + + + + V + + + + + OD + + + + + L + + + + + Set + + + + + 型。集まりではなく、値の種類を区別する記号。’ + 自身が値になると、一つLevelの高い型を持つ + + + + + diff -r 6e1c60866788 -r 4fcac1eebc74 fig/axiom-dependency.graffle Binary file fig/axiom-dependency.graffle has changed diff -r 6e1c60866788 -r 4fcac1eebc74 fig/axiom-dependency.svg --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/fig/axiom-dependency.svg Sun Jun 07 20:31:30 2020 +0900 @@ -0,0 +1,87 @@ + + + + + + + + + + + + + + + + + + + + + Produced by OmniGraffle 7.12.1 + 2020-01-11 11:10:39 +0000 + + + Canvas 1 + + + Layer 1 + + + + + + Choice + + + + + LEM + + + + + Well ordering outside + + + + + ε-induction + + + + + Strong Regularity + + + + + + + + + + + Well ordering inside + + + + + + + + + + + + + + + + + + + + + + diff -r 6e1c60866788 -r 4fcac1eebc74 fig/axiom-type.graffle Binary file fig/axiom-type.graffle has changed diff -r 6e1c60866788 -r 4fcac1eebc74 fig/axiom-type.svg --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/fig/axiom-type.svg Sun Jun 07 20:31:30 2020 +0900 @@ -0,0 +1,149 @@ + + + + + + + + + + + + + + + + + + + + + Produced by OmniGraffle 7.12.1 + 2020-01-11 11:09:31 +0000 + + + Canvas 1 + + + Layer 1 + + + Axioms of Set theory in Agda + + + + + Pure logical + + + + + Negation form + + + + + Non construtive + assumptions + + + + + pair + + + + + union + + + + + empty + + + + + power + + + + + extensionarity + + + + + infinity + + + + + selection + + + + + replacement + + + + + choice + + + + + ε-induction + + + + + regularity + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff -r 6e1c60866788 -r 4fcac1eebc74 fig/ord-od-mapping.graffle Binary file fig/ord-od-mapping.graffle has changed diff -r 6e1c60866788 -r 4fcac1eebc74 fig/ord-od-mapping.svg --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/fig/ord-od-mapping.svg Sun Jun 07 20:31:30 2020 +0900 @@ -0,0 +1,184 @@ + + + + + + + + + + + + + + + + + + + + + Produced by OmniGraffle 7.12.1 + 2019-11-28 05:45:48 +0000 + + + Canvas 1 + + + Layer 1 + + + + + + + + + + + + + + + + + + + + + + + Ordinal + + + + + Ordinal + Definable + + + + + max + = all + + + + + empty + + + + + Ordinal + max != Ordinal + + + + + + + + + + + empty + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + non-order preserving + + + + + order preserving + + + + + + + + larger + + + + + + + + defined + by + + + + + + + + + + + + + + + + + + + + + + + + + + + + + Total order + + + + + Partial order + + + + + diff -r 6e1c60866788 -r 4fcac1eebc74 fig/set-theory.graffle Binary file fig/set-theory.graffle has changed diff -r 6e1c60866788 -r 4fcac1eebc74 fig/set-theory.svg --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/fig/set-theory.svg Sun Jun 07 20:31:30 2020 +0900 @@ -0,0 +1,93 @@ + + + + + + + + + + + + + + + + Produced by OmniGraffle 7.12.1 + 2020-01-11 11:07:37 +0000 + + + Canvas 1 + + + Layer 1 + + + + + + + + + + + + + + + + + + primitive set theory + + + + + First order + predicate logic + + + + + Ordinals + + + + + Model on ZF + + + + + Higher order logic + + + + + Ordinals + + + + + ZF axioms + + + + + Non constructive + assumptions + + + + + Model on OD + + + + + ZF axioms + + + + + diff -r 6e1c60866788 -r 4fcac1eebc74 filter.agda --- a/filter.agda Thu Aug 29 16:18:37 2019 +0900 +++ b/filter.agda Sun Jun 07 20:31:30 2020 +0900 @@ -17,71 +17,76 @@ open inOrdinal O open OD O open OD.OD +open ODAxiom odAxiom open _∧_ open _∨_ open Bool -record Filter ( P max : OD ) : Set (suc n) where +_∩_ : ( A B : OD ) → OD +A ∩ B = record { def = λ x → def A x ∧ def B x } + +_∪_ : ( A B : OD ) → OD +A ∪ B = record { def = λ x → def A x ∨ def B x } + +_＼_ : ( A B : OD ) → OD +A ＼ B = record { def = λ x → def A x ∧ ( ¬ ( def B x ) ) } + + +record Filter ( L : OD ) : Set (suc n) where field - _⊇_ : OD → OD → Set n - G : OD - G∋1 : G ∋ max - Gmax : { p : OD } → P ∋ p → p ⊇ max - Gless : { p q : OD } → G ∋ p → P ∋ q → p ⊇ q → G ∋ q - Gcompat : { p q : OD } → G ∋ p → G ∋ q → ¬ ( - ( r : OD ) → (( p ⊇ r ) ∧ ( p ⊇ r ))) + filter : OD + proper : ¬ ( filter ∋ od∅ ) + inL : filter ⊆ L + filter1 : { p q : OD } → q ⊆ L → filter ∋ p → p ⊆ q → filter ∋ q + filter2 : { p q : OD } → filter ∋ p → filter ∋ q → filter ∋ (p ∩ q) -dense : Set (suc n) -dense = { D P p : OD } → ({x : OD } → P ∋ p → ¬ ( ( q : OD ) → D ∋ q → od→ord p o< od→ord q )) +open Filter + +L⊆L : (L : OD) → L ⊆ L +L⊆L L = record { incl = λ {x} lt → lt } + +L-filter : {L : OD} → (P : Filter L ) → {p : OD} → filter P ∋ p → filter P ∋ L +L-filter {L} P {p} lt = filter1 P {p} {L} (L⊆L L) lt {!!} -record NatFilter ( P : Nat → Set n) : Set (suc n) where +prime-filter : {L : OD} → Filter L → ∀ {p q : OD } → Set n +prime-filter {L} P {p} {q} = filter P ∋ ( p ∪ q) → ( filter P ∋ p ) ∨ ( filter P ∋ q ) + +ultra-filter : {L : OD} → Filter L → ∀ {p : OD } → Set n +ultra-filter {L} P {p} = L ∋ p → ( filter P ∋ p ) ∨ ( filter P ∋ ( L ＼ p) ) + + +filter-lemma1 : {L : OD} → (P : Filter L) → ∀ {p q : OD } → ( ∀ (p : OD ) → ultra-filter {L} P {p} ) → prime-filter {L} P {p} {q} +filter-lemma1 {L} P {p} {q} u lt = {!!} + +filter-lemma2 : {L : OD} → (P : Filter L) → ( ∀ {p q : OD } → prime-filter {L} P {p} {q}) → ∀ (p : OD ) → ultra-filter {L} P {p} +filter-lemma2 {L} P prime p with prime {!!} +... | t = {!!} + +generated-filter : {L : OD} → Filter L → (p : OD ) → Filter ( record { def = λ x → def L x ∨ (x ≡ od→ord p) } ) +generated-filter {L} P p = record { + proper = {!!} ; + filter = {!!} ; inL = {!!} ; + filter1 = {!!} ; filter2 = {!!} + } + +record Dense (P : OD ) : Set (suc n) where field - GN : Nat → Set n - GN∋1 : GN 0 - GNmax : { p : Nat } → P p → 0 ≤ p - GNless : { p q : Nat } → GN p → P q → q ≤ p → GN q - Gr : ( p q : Nat ) → GN p → GN q → Nat - GNcompat : { p q : Nat } → (gp : GN p) → (gq : GN q ) → ( Gr p q gp gq ≤ p ) ∨ ( Gr p q gp gq ≤ q ) - -record NatDense {n : Level} ( P : Nat → Set n) : Set (suc n) where - field - Gmid : { p : Nat } → P p → Nat - GDense : { D : Nat → Set n } → {x p : Nat } → (pp : P p ) → D (Gmid {p} pp) → Gmid pp ≤ p - -open OD.OD + dense : OD + incl : dense ⊆ P + dense-f : OD → OD + dense-p : { p : OD} → P ∋ p → p ⊆ (dense-f p) -- H(ω,2) = Power ( Power ω ) = Def ( Def ω)) -Pred : ( Dom : OD ) → OD -Pred dom = record { - def = λ x → def dom x → {!!} - } +infinite = ZF.infinite OD→ZF -Hω2 : OD -Hω2 = record { def = λ x → {dom : Ordinal } → x ≡ od→ord ( Pred ( ord→od dom )) } +module in-countable-ordinal {n : Level} where + + import ordinal -Hω2Filter : Filter Hω2 od∅ -Hω2Filter = record { - _⊇_ = _⊇_ - ; G = {!!} - ; G∋1 = {!!} - ; Gmax = {!!} - ; Gless = {!!} - ; Gcompat = {!!} - } where - P = Hω2 - _⊇_ : OD → OD → Set n - _⊇_ = {!!} - G : OD - G = {!!} - G∋1 : G ∋ od∅ - G∋1 = {!!} - Gmax : { p : OD } → P ∋ p → p ⊇ od∅ - Gmax = {!!} - Gless : { p q : OD } → G ∋ p → P ∋ q → p ⊇ q → G ∋ q - Gless = {!!} - Gcompat : { p q : OD } → G ∋ p → G ∋ q → ¬ ( - ( r : OD ) → (( p ⊇ r ) ∧ ( p ⊇ r ))) - Gcompat = {!!} + -- open ordinal.C-Ordinal-with-choice + -- both Power and infinite is too ZF, it is better to use simpler one + Hω2 : Filter (Power (Power infinite)) + Hω2 = {!!} diff -r 6e1c60866788 -r 4fcac1eebc74 ordinal.agda --- a/ordinal.agda Thu Aug 29 16:18:37 2019 +0900 +++ b/ordinal.agda Sun Jun 07 20:31:30 2020 +0900 @@ -203,7 +203,6 @@ lemma y lt | case1 refl = proj1 ( TransFinite1 lx ox ) lemma y lt | case2 lt1 = proj2 ( TransFinite1 lx ox ) y lt1 - open import Ordinals C-Ordinal : {n : Level} → Ordinals {suc n} @@ -234,75 +233,3 @@ ψ (record { lv = lx ; ord = OSuc lx x₁ }) caseOSuc lx ox prev = lt (ordinal lx (OSuc lx ox)) prev -module C-Ordinal-with-choice {n : Level} where - - import OD - -- open inOrdinal C-Ordinal - open OD (C-Ordinal {n}) - open OD.OD - - -- - -- another form of regularity - -- - ε-induction : {m : Level} { ψ : OD → Set m} - → ( {x : OD } → ({ y : OD } → x ∋ y → ψ y ) → ψ x ) - → (x : OD ) → ψ x - ε-induction {m} {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (lv (osuc (od→ord x))) (ord (osuc (od→ord x))) <-osuc) where - ε-induction-ord : (lx : Nat) ( ox : OrdinalD {suc n} lx ) {ly : Nat} {oy : OrdinalD {suc n} ly } - → (ly < lx) ∨ (oy d< ox ) → ψ (ord→od (record { lv = ly ; ord = oy } ) ) - ε-induction-ord lx (OSuc lx ox) {ly} {oy} y ¬a ¬b c = -- lz(a) - subst (λ k → ψ k ) oiso (ε-induction-ord lx (Φ lx) {_} {ord (od→ord z)} (case1 (<-trans lz ¬a ¬b c with d<→lv lz=ly -- lz(b) - ... | eq = subst (λ k → ψ k ) oiso - (ε-induction-ord lx (Φ lx) {_} {ord (od→ord z)} (case1 (subst (λ k → k < lx ) (trans (sym lemma1)(sym eq) ) c ))) - lemma z lt | case2 lz=ly | tri≈ ¬a lx=ly ¬c with d<→lv lz=ly -- lz(c) - ... | eq = subst (λ k → ψ k ) oiso ( ε-induction-ord lx (dz oz=lx) {lv (od→ord z)} {ord (od→ord z)} (case2 (dz + + + + +Constructing ZF Set Theory in Agda + + +
+

Constructing ZF Set Theory in Agda

+ +Menu + + +Menu + + +

+ + Shinji KONO + +

+

ZF in Agda

+ +
+    zf.agda            axiom of ZF
+    zfc.agda           axiom of choice
+    Ordinals.agda      axiom of Ordinals
+    ordinal.agda       countable model of Ordinals
+    OD.agda            model of ZF based on Ordinal Definable Set with assumptions
+    ODC.agda           Law of exclude middle from axiom of choice assumptions
+    LEMC.agda          model of choice with assumption of the Law of exclude middle 
+    OPair.agda         ordered pair on OD
+    BAlgbra.agda       Boolean algebra on OD (not yet done)
+    filter.agda        Filter on OD (not yet done)
+    cardinal.agda      Caedinal number on OD (not yet done)
+    logic.agda         some basics on logic
+    nat.agda           some basics on Nat
+
+
+ +
+

Programming Mathematics

+ +

+Programming is processing data structure with λ terms. +

+We are going to handle Mathematics in intuitionistic logic with λ terms. +

+Mathematics is a functional programming which values are proofs. +

+Programming ZF Set Theory in Agda +

+ +

+

Target

+ +
+   Describe ZF axioms in Agda
+   Construction a Model of ZF Set Theory in Agda
+   Show necessary assumptions for the model
+   Show validities of ZF axioms on the model
+
+
+This shows consistency of Set Theory (with some assumptions), +without circulating ZF Theory assumption. +

+ +ZF in Agda https://github.com/shinji-kono/zf-in-agda + +

+ +

+

Why Set Theory

+If we can formulate Set theory, it suppose to work on any mathematical theory. +

+Set Theory is a difficult point for beginners especially axiom of choice. +

+It has some amount of difficulty and self circulating discussion. +

+I'm planning to do it in my old age, but I'm enough age now. +

+This is done during from May to September. +

+ +

+

Agda and Intuitionistic Logic

+Curry Howard Isomorphism +

+ +

+    Proposition : Proof ⇔ Type : Value
+
+
+which means +

+　　constructing a typed lambda calculus which corresponds a logic +

+Typed lambda calculus which allows complex type as a value of a variable (System FC) +

+　　First class Type / Dependent Type +

+Agda is a such a programming language which has similar syntax of Haskell +

+Coq is specialized in proof assistance such as command and tactics . +

+ +

+

Introduction of Agda

+A length of a list of type A. +

+ +

+    length : {A : Set } → List A → Nat
+    length [] = zero
+    length (_ ∷ t)  = suc ( length t )
+
+
+Simple functional programming language. Type declaration is mandatory. +A colon means type, an equal means value. Indentation based. +

+Set is a base type (which may have a level ). +

+{} means implicit variable which can be omitted if Agda infers its value. +

+ +

+

data ( Sum type )

+A data type which as exclusive multiple constructors. A similar one as +union in C or case class in Scala. +

+It has a similar syntax as Haskell but it has a slight difference. +

+ +

+   data List (A : Set ) : Set where
+        [] : List A
+        _∷_ : A → List A → List A
+
+
+_∷_ means infix operator. If use explicit _, it can be used in a normal function +syntax. +

+Natural number can be defined as a usual way. +

+ +

+   data Nat : Set where
+        zero : Nat
+        suc  : Nat → Nat
+
+
+ +
+

A → B means "A implies B"

+ +

+In Agda, a type can be a value of a variable, which is usually called dependent type. +Type has a name Set in Agda. +

+ +

+    ex3 : {A B : Set} → Set 
+    ex3 {A}{B}  =  A → B
+
+
+ex3 is a type : A → B, which is a value of Set. It also means a formula : A implies B. +

+ +

+    A type is a formula, the value is the proof
+
+
+A value of A → B can be interpreted as an inference from the formula A to the formula B, which +can be a function from a proof of A to a proof of B. +

+ +

+

introduction と elimination

+For a logical operator, there are two types of inference, an introduction and an elimination. +

+ +

+  intro creating symbol  / constructor / introduction
+  elim  using symbolic / accessors / elimination
+
+
+In Natural deduction, this can be written in proof schema. +

+ +

+       A                   
+       :
+       B                    A       A → B
+   ------------- →intro   ------------------ →elim
+      A → B                     B
+
+
+In Agda, this is a pair of type and value as follows. Introduction of → uses λ. +

+ +

+    →intro : {A B : Set } → A →  B → ( A → B )
+    →intro _ b = λ x → b
+    →elim : {A B : Set } → A → ( A → B ) → B
+    →elim a f = f a
+
+
+Important +

+ +

+    {A B : Set } → A →  B → ( A → B )
+
+
+is +

+ +

+    {A B : Set } → ( A →  ( B → ( A → B ) ))
+
+
+This makes currying of function easy. +

+ +

+

To prove A → B

+Make a left type as an argument. (intros in Coq) +

+ +

+    ex5 : {A B C : Set } → A → B → C  → ?
+    ex5 a b c = ?
+
+
+? is called a hole, which is unspecified part. Agda tell us which kind type is required for the Hole. +

+We are going to fill the holes, and if we have no warnings nor errors such as type conflict (Red), +insufficient proof or instance (Yellow), Non-termination, the proof is completed. +

+ +

+

A ∧ B

+Well known conjunction's introduction and elimination is as follow. +

+ +

+     A    B                 A ∧ B           A ∧ B 
+   -------------         ----------- proj1   ---------- proj2
+      A ∧ B                   A               B
+
+
+We can introduce a corresponding structure in our functional programming language. +

+ +

+

record

+ +
+   record _∧_ A B : Set
+     field
+         proj1 : A
+         proj2 : B
+
+
+_∧_　means infix operator. _∧_ A B can be written as A ∧ B (Haskell uses (∧) ) +

+This a type which constructed from type A and type B. You may think this as an object +or struct. +

+ +

+   record { proj1 = x ; proj2 = y }    
+
+
+is a constructor of _∧_. +

+ +

+    ex3 : {A B : Set} → A → B → ( A ∧ B )
+    ex3 a b = record { proj1 = a ; proj2 = b }
+    ex1 : {A B : Set} → ( A ∧ B ) → A
+    ex1 a∧b = proj1 a∧b
+
+
+a∧b is a variable name. If we have no spaces in a string, it is a word even if we have symbols +except parenthesis or colons. A symbol requires space separation such as a type defining colon. +

+Defining record can be recursively, but we don't use the recursion here. +

+ +

+

Mathematical structure

+We have types of elements and the relationship in a mathematical structure. +

+ +

+  logical relation has no ordering
+  there is a natural ordering in arguments and a value in a function
+
+
+So we have typical definition style of mathematical structure with records. +

+ +

+  record IsOrdinals {n : Level} (ord : Set n)  
+    (_o<_ : ord → ord → Set n) : Set (suc (suc n)) where
+   field
+     Otrans :  {x y z : ord }  → x o< y → y o< z → x o< z
+  record Ordinals {n : Level} : Set (suc (suc n)) where
+       field 
+         ord : Set n
+         _o<_ : ord → ord → Set n
+         isOrdinal : IsOrdinals ord _o<_
+
+
+In IsOrdinals, axioms are written in flat way. In Ordinal, we may have +inputs and outputs are put in the field including IsOrdinal. +

+Fields of Ordinal is existential objects in the mathematical structure. +

+ +

+

A Model and a theory

+Agda record is a type, so we can write it in the argument, but is it really exists? +

+If we have a value of the record, it simply exists, that is, we need to create all the existence +in the record satisfies all the axioms (= field of IsOrdinal) should be valid. +

+ +

+   type of record = theory
+   value of record = model
+
+
+We call the value of the record as a model. If mathematical structure has a +model, it exists. Pretty Obvious. +

+ +

+

postulate と module

+Agda proofs are separated by modules, which are large records. +

+postulates are assumptions. We can assume a type without proofs. +

+ +

+    postulate      
+      sup-o  :  ( Ordinal  → Ordinal ) →  Ordinal 
+      sup-o< :  { ψ : Ordinal  →  Ordinal } → ∀ {x : Ordinal } → ψ x  o<  sup-o ψ 
+
+
+sup-o is an example of upper bound of a function and sup-o< assumes it actually satisfies all the value is less than upper bound. +

+Writing some type in a module argument is the same as postulating a type, but +postulate can be written the middle of a proof. +

+postulate can be constructive. +

+postulate can be inconsistent, which result everything has a proof. +

+ +

+

A ∨ B

+ +
+    data _∨_ (A B : Set) : Set where
+      case1 : A → A ∨ B
+      case2 : B → A ∨ B
+
+
+As Haskell, case1/case2 are patterns. +

+ +

+    ex3 : {A B : Set} → ( A ∨ A ) → A 
+    ex3 = ?
+
+
+In a case statement, Agda command C-C C-C generates possible cases in the head. +

+ +

+    ex3 : {A B : Set} → ( A ∨ A ) → A 
+    ex3 (case1 x) = ?
+    ex3 (case2 x) = ?
+
+
+Proof schema of ∨ is omit due to the complexity. +

+ +

+

Negation

+ +
+       ⊥
+    ------------- ⊥-elim
+       A
+
+
+Anything can be derived from bottom, in this case a Set A. There is no introduction rule +in ⊥, which can be implemented as data which has no constructor. +

+ +

+    data ⊥ : Set where
+
+
+⊥-elim can be proved like this. +

+ +

+    ⊥-elim : {A : Set } -> ⊥ -> A
+    ⊥-elim ()
+
+
+() means no match argument nor value. +

+A negation can be defined using ⊥ like this. +

+ +

+    ¬_ : Set → Set
+    ¬ A = A → ⊥
+
+
+ +
+

Equality

+ +

+All the value in Agda are terms. If we have the same normalized form, two terms are equal. +If we have variables in the terms, we will perform an unification. unifiable terms are equal. +We don't go further on the unification. +

+ +

+     { x : A }                 x ≡ y    f x y
+   --------------- ≡-intro   --------------------- ≡-elim
+      x ≡ x                         f x x
+
+
+equality _≡_ can be defined as a data. +

+ +

+    data _≡_ {A : Set } : A → A → Set where
+       refl :  {x : A} → x ≡ x
+
+
+The elimination of equality is a substitution in a term. +

+ +

+    subst : {A : Set } → { x y : A } → ( f : A → Set ) → x ≡ y → f x → f y
+    subst {A} {x} {y} f refl fx = fx
+    ex5 :   {A : Set} {x y z : A } →  x ≡ y → y ≡ z → x ≡ z
+    ex5 {A} {x} {y} {z} x≡y y≡z = subst ( λ k → x ≡ k ) y≡z x≡y
+
+
+ +
+

Equivalence relation

+ +

+ +

+    refl' : {A : Set} {x : A } → x ≡ x
+    refl'  = ?
+    sym : {A : Set} {x y : A } → x ≡ y → y ≡ x
+    sym = ?
+    trans : {A : Set} {x y z : A } → x ≡ y → y ≡ z → x ≡ z
+    trans = ?
+    cong : {A B : Set} {x y : A } { f : A → B } →   x ≡ y → f x ≡ f y
+    cong = ?
+
+
+ +
+

Ordering

+ +

+Relation is a predicate on two value which has a same type. +

+ +

+   A → A → Set 
+
+
+Defining order is the definition of this type with predicate or a data. +

+ +

+    data _≤_ : Rel ℕ 0ℓ where
+      z≤n : ∀ {n}                 → zero  ≤ n
+      s≤s : ∀ {m n} (m≤n : m ≤ n) → suc m ≤ suc n
+
+
+ +
+

Quantifier

+ +

+Handling quantifier in an intuitionistic logic requires special cares. +

+In the input of a function, there are no restriction on it, that is, it has +a universal quantifier. (If we explicitly write ∀, Agda gives us a type inference on it) +

+There is no ∃ in agda, the one way is using negation like this. +

+　∃ (x : A ) → p x = ¬ ( ( x : A ) → ¬ ( p x ) ) +

+On the another way, f : A can be used like this. +

+ +

+  p f
+
+
+If we use a function which can be defined globally which has stronger meaning the +usage of ∃ x in a logical expression. +

+ +

+

Can we do math in this way?

+Yes, we can. Actually we have Principia Mathematica by Russell and Whitehead (with out computer support). +

+In some sense, this story is a reprinting of the work, (but Principia Mathematica has a different formulation than ZF). +

+ +

+    define mathematical structure as a record
+    program inferences as if we have proofs in variables
+
+
+ +
+

Things which Agda cannot prove

+ +

+The infamous Internal Parametricity is a limitation of Agda, it cannot prove so called Free Theorem, which +leads uniqueness of a functor in Category Theory. +

+Functional extensionality cannot be proved. +

+  (∀ x → f x ≡ g x) → f ≡ g
+
+
+Agda has no law of exclude middle. +

+ +

+  a ∨ ( ¬ a )
+
+
+For example, (A → B) → ¬ B → ¬ A can be proved but, ( ¬ B → ¬ A ) → A → B cannot. +

+It also other problems such as termination, type inference or unification which we may overcome with +efforts or devices or may not. +

+If we cannot prove something, we can safely postulate it unless it leads a contradiction. +

+ 
+
+
+ +
+

Classical story of ZF Set Theory

+ +

+Assuming ZF, constructing a model of ZF is a flow of classical Set Theory, which leads +a relative consistency proof of the Set Theory. +Ordinal number is used in the flow. +

+In Agda, first we defines Ordinal numbers (Ordinals), then introduce Ordinal Definable Set (OD). +We need some non constructive assumptions in the construction. A model of Set theory is +constructed based on these assumptions. +

+ + +

+ +

+

Ordinals

+Ordinals are our intuition of infinite things, which has ∅ and orders on the things. +It also has a successor osuc. +

+ +

+    record Ordinals {n : Level} : Set (suc (suc n)) where
+       field
+         ord : Set n
+         o∅ : ord
+         osuc : ord → ord
+         _o<_ : ord → ord → Set n
+         isOrdinal : IsOrdinals ord o∅ osuc _o<_
+
+
+It is different from natural numbers in way. The order of Ordinals is not defined in terms +of successor. It is given from outside, which make it possible to have higher order infinity. +

+ +

+

Axiom of Ordinals

+Properties of infinite things. We request a transfinite induction, which states that if +some properties are satisfied below all possible ordinals, the properties are true on all +ordinals. +

+Successor osuc has no ordinal between osuc and the base ordinal. There are some ordinals +which is not a successor of any ordinals. It is called limit ordinal. +

+Any two ordinal can be compared, that is less, equal or more, that is total order. +

+ +

+  record IsOrdinals {n : Level} (ord : Set n)  (o∅ : ord ) 
+    (osuc : ord → ord )  
+    (_o<_ : ord → ord → Set n) : Set (suc (suc n)) where
+   field
+     Otrans :  {x y z : ord }  → x o< y → y o< z → x o< z
+     OTri : Trichotomous {n} _≡_  _o<_ 
+     ¬x<0 :   { x  : ord  } → ¬ ( x o< o∅  )
+     <-osuc :  { x : ord  } → x o< osuc x
+     osuc-≡< :  { a x : ord  } → x o< osuc a  →  (x ≡ a ) ∨ (x o< a)  
+     TransFinite : { ψ : ord  → Set (suc n) }
+          → ( (x : ord)  → ( (y : ord  ) → y o< x → ψ y ) → ψ x )
+          →  ∀ (x : ord)  → ψ x
+
+
+ +
+

Concrete Ordinals

+ +

+We can define a list like structure with level, which is a kind of two dimensional infinite array. +

+ +

+    data OrdinalD {n : Level} :  (lv : Nat) → Set n where
+       Φ : (lv : Nat) → OrdinalD  lv
+       OSuc : (lv : Nat) → OrdinalD {n} lv → OrdinalD lv
+
+
+The order of the OrdinalD can be defined in this way. +

+ +

+    data _d<_ {n : Level} :   {lx ly : Nat} → OrdinalD {n} lx  →  OrdinalD {n} ly  → Set n where
+       Φ<  : {lx : Nat} → {x : OrdinalD {n} lx}  →  Φ lx d< OSuc lx x
+       s<  : {lx : Nat} → {x y : OrdinalD {n} lx}  →  x d< y  → OSuc  lx x d< OSuc  lx y
+
+
+This is a simple data structure, it has no abstract assumptions, and it is countable many data +structure. +

+ +

+   Φ 0
+   OSuc 2 ( Osuc 2 ( Osuc 2 (Φ 2)))
+   Osuc 0 (Φ 0) d< Φ 1
+
+
+ +
+

Model of Ordinals

+ +

+It is easy to show OrdinalD and its order satisfies the axioms of Ordinals. +

+So our Ordinals has a mode. This means axiom of Ordinals are consistent. +

+ +

+

Debugging axioms using Model

+Whether axiom is correct or not can be checked by a validity on a mode. +

+If not, we may fix the axioms or the model, such as the definitions of the order. +

+We can also ask whether the inputs exist. +

+ +

+

Countable Ordinals can contains uncountable set?

+Yes, the ordinals contains any level of infinite Set in the axioms. +

+If we handle real-number in the model, only countable number of real-number is used. +

+ +

+    from the outside view point, it is countable
+    from the internal view point, it is uncountable
+
+
+The definition of countable/uncountable is the same, but the properties are different +depending on the context. +

+We don't show the definition of cardinal number here. +

+ +

+

What is Set

+The word Set in Agda is not a Set of ZF Set, but it is a type (why it is named Set?). +

+From naive point view, a set i a list, but in Agda, elements have all the same type. +A set in ZF may contain other Sets in ZF, which not easy to implement it as a list. +

+Finite set may be written in finite series of ∨, but ... +

+ +

+

We don't ask the contents of Set. It can be anything.

+From empty set φ, we can think a set contains a φ, and a pair of φ and the set, and so on, +and all of them, and again we repeat this. +

+ +

+   φ {φ} {φ,{φ}}, {φ,{φ},...}
+
+
+It is called V. +

+This operation can be performed within a ZF Set theory. Classical Set Theory assumes +ZF, so this kind of thing is allowed. +

+But in our case, we have no ZF theory, so we are going to use Ordinals. +

+ +

+

Ordinal Definable Set

+We can define a sbuset of Ordinals using predicates. What is a subset? +

+ +

+   a predicate has an Ordinal argument
+
+
+is an Ordinal Definable Set (OD). In Agda, OD is defined as follows. +

+ +

+    record OD : Set (suc n ) where
+      field
+        def : (x : Ordinal  ) → Set n
+
+
+Ordinals itself is not a set in a ZF Set theory but a class. In OD, +

+ +

+   record { def = λ x → true }
+
+
+means Ordinals itself, so ODs are larger than a notion of ZF Set Theory, +but we don't care about it. +

+ +

+

∋ in OD

+OD is a predicate on Ordinals and it does not contains OD directly. If we have a mapping +

+ +

+      od→ord : OD  → Ordinal 
+
+
+we may check an OD is an element of the OD using def. +

+A ∋ x can be define as follows. +

+ +

+    _∋_ : ( A x : OD  ) → Set n
+    _∋_  A x  = def A ( od→ord x )
+
+
+In ψ : Ordinal → Set, if A is a record { def = λ x → ψ x } , then +

+ +

+    A x = def A ( od→ord x ) = ψ (od→ord x)
+
+
+ +
+

1 to 1 mapping between an OD and an Ordinal

+ +

+ +

+  od→ord : OD  → Ordinal 
+  ord→od : Ordinal  → OD  
+  oiso   :  {x : OD }      → ord→od ( od→ord x ) ≡ x
+  diso   :  {x : Ordinal } → od→ord ( ord→od x ) ≡ x
+
+
+They say the existing of the mappings can be proved in Classical Set Theory, but we +simply assumes these non constructively. +

+We use postulate, it may contains additional restrictions which are not clear now. It look like the mapping should be a subset of Ordinals, but we don't explicitly +state it. +

+ + +

+ +

+

Order preserving in the mapping of OD and Ordinal

+Ordinals have the order and OD has a natural order based on inclusion ( def / ∋ ). +

+ +

+     def y ( od→ord x )
+
+
+An elements of OD should be defined before the OD, that is, an ordinal corresponding an elements +have to be smaller than the corresponding ordinal of the containing OD. +

+ +

+  c<→o<  :  {x y : OD  }   → def y ( od→ord x ) → od→ord x o< od→ord y
+
+
+This is also said to be provable in classical Set Theory, but we simply assumes it. +

+OD has an order based on the corresponding ordinal, but it may not have corresponding def / ∋relation. This means the reverse order preservation, +

+ +

+  o<→c<  : {n : Level} {x y : Ordinal  } → x o< y → def (ord→od y) x 
+
+
+is not valid. If we assumes it, ∀ x ∋ ∅ becomes true, which manes all OD becomes Ordinals in +the model. +

+ + +

+ +

+

ISO

+It also requires isomorphisms, +

+ +

+  oiso   :  {x : OD }      → ord→od ( od→ord x ) ≡ x
+  diso   :  {x : Ordinal } → od→ord ( ord→od x ) ≡ x
+
+
+ +
+

Various Sets

+ +

+In classical Set Theory, there is a hierarchy call L, which can be defined by a predicate. +

+ +

+    Ordinal / things satisfies axiom of Ordinal / extension of natural number 
+    V / hierarchical construction of Set from φ   
+    L / hierarchical predicate definable construction of Set from φ   
+    OD / equational formula on Ordinals 
+    Agda Set / Type / it also has a level
+
+
+ +
+

Fixes on ZF to intuitionistic logic

+ +

+We use ODs as Sets in ZF, and defines record ZF, that is, we have to define +ZF axioms in Agda. +

+It may not valid in our model. We have to debug it. +

+Fixes are depends on axioms. +

+ + +

+ +ZFのrecord +

+ +

+

Pure logical axioms

+ +
+   empty, pair, select, ε-induction??infinity
+
+
+These are logical relations among OD. +

+ +

+     empty :  ∀( x : ZFSet  ) → ¬ ( ∅ ∋ x )
+     pair→ : ( x y t : ZFSet  ) →  (x , y)  ∋ t  → ( t ≈ x ) ∨ ( t ≈ y ) 
+     pair← : ( x y t : ZFSet  ) →  ( t ≈ x ) ∨ ( t ≈ y )  →  (x , y)  ∋ t 
+     selection : { ψ : ZFSet → Set m } → ∀ { X y : ZFSet  } →  ( ( y ∈ X ) ∧ ψ y ) ⇔ (y ∈  Select X ψ ) 
+     infinity∅ :  ∅ ∈ infinite
+     infinity :  ∀( x : ZFSet  ) → x ∈ infinite →  ( x ∪ ( x , x ) ) ∈ infinite 
+     ε-induction : { ψ : OD  → Set (suc n)}
+       → ( {x : OD } → ({ y : OD } →  x ∋ y → ψ y ) → ψ x )
+       → (x : OD ) → ψ x
+
+
+finitely can be define by Agda data. +

+ +

+    data infinite-d  : ( x : Ordinal  ) → Set n where
+        iφ :  infinite-d o∅
+        isuc : {x : Ordinal  } →   infinite-d  x  →
+                infinite-d  (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) ))
+
+
+Union (x , ( x , x )) should be an direct successor of x, but we cannot prove it in our model. +

+ +

+

Axiom of Pair

+In the Tanaka's book, axiom of pair is as follows. +

+ +

+     ∀ x ∀ y ∃ z ∀ t ( z ∋ t ↔ t ≈ x ∨ t ≈ y)
+
+
+We have fix ∃ z, a function (x , y) is defined, which is _,_ . +

+ +

+     _,_ : ( A B : ZFSet  ) → ZFSet
+
+
+using this, we can define two directions in separates axioms, like this. +

+ +

+     pair→ : ( x y t : ZFSet  ) →  (x , y)  ∋ t  → ( t ≈ x ) ∨ ( t ≈ y ) 
+     pair← : ( x y t : ZFSet  ) →  ( t ≈ x ) ∨ ( t ≈ y )  →  (x , y)  ∋ t 
+
+
+This is already written in Agda, so we use these as axioms. All inputs have ∀. +

+ +

+

pair in OD

+OD is an equation on Ordinals, we can simply write axiom of pair in the OD. +

+ +

+    _,_ : OD  → OD  → OD 
+    x , y = record { def = λ t → (t ≡ od→ord x ) ∨ ( t ≡ od→ord y ) } 
+
+
+≡ is an equality of λ terms, but please not that this is equality on Ordinals. +

+ +

+

Validity of Axiom of Pair

+Assuming ZFSet is OD, we are going to prove pair→ . +

+ +

+    pair→ : ( x y t : OD  ) →  (x , y)  ∋ t  → ( t == x ) ∨ ( t == y ) 
+    pair→ x y t p = ?
+
+
+In this program, type of p is ( x , y ) ∋ t , that is def ( x , y ) that is, (t ≡ od→ord x ) ∨ ( t ≡ od→ord y ) . +

+Since _∨_ is a data, it can be developed as (C-c C-c : agda2-make-case ). +

+ +

+    pair→ x y t (case1 t≡x ) = ?
+    pair→ x y t (case2 t≡y ) = ?
+
+
+The type of the ? is ( t == x ) ∨ ( t == y ), again it is data _∨_ . +

+ +

+    pair→ x y t (case1 t≡x ) = case1 ?
+    pair→ x y t (case2 t≡y ) = case2 ?
+
+
+The ? in case1 is t == x, so we have to create this from t≡x, which is a name of a variable +which type is +

+ +

+    t≡x : od→ord t ≡ od→ord x
+
+
+which is shown by an Agda command (C-C C-E : agda2-show-context ). +

+But we haven't defined == yet. +

+ +

+

Equality of OD and Axiom of Extensionality

+OD is defined by a predicates, if we compares normal form of the predicates, even if +it contains the same elements, it may be different, which is no good as an equality of +Sets. +

+Axiom of Extensionality requires sets having the same elements are handled in the same way +each other. +

+ +

+    ∀ z ( z ∈ x ⇔ z ∈ y ) ⇒ ∀ w ( x ∈ w ⇔ y ∈ w )
+
+
+We can write this axiom in Agda as follows. +

+ +

+     extensionality :  { A B w : ZFSet  } → ( (z : ZFSet) → ( A ∋ z ) ⇔ (B ∋ z)  ) → ( A ∈ w ⇔ B ∈ w )
+
+
+So we use ( A ∋ z ) ⇔ (B ∋ z) as an equality (_==_) of our model. We have to show +A ∈ w ⇔ B ∈ w from A == B. +

+x == y can be defined in this way. +

+ +

+    record _==_  ( a b :  OD  ) : Set n where
+      field
+         eq→ : ∀ { x : Ordinal  } → def a x → def b x
+         eq← : ∀ { x : Ordinal  } → def b x → def a x
+
+
+Actually, (z : OD) → (A ∋ z) ⇔ (B ∋ z) implies A == B. +

+ +

+    extensionality0 : {A B : OD } → ((z : OD) → (A ∋ z) ⇔ (B ∋ z)) → A == B
+    eq→ (extensionality0 {A} {B} eq ) {x} d = ?
+    eq← (extensionality0 {A} {B} eq ) {x} d = ?
+
+
+? are def B x and def A x and these are generated from eq : (z : OD) → (A ∋ z) ⇔ (B ∋ z) . +

+Actual proof is rather complicated. +

+ +

+   eq→ (extensionality0 {A} {B} eq ) {x} d = def-iso  {A} {B} (sym diso) (proj1 (eq (ord→od x))) d
+   eq← (extensionality0 {A} {B} eq ) {x} d = def-iso  {B} {A} (sym diso) (proj2 (eq (ord→od x))) d
+
+
+where +

+ +

+   def-iso : {A B : OD } {x y : Ordinal } → x ≡ y  → (def A y → def B y)  → def A x → def B x
+   def-iso refl t = t
+
+
+ +
+

Validity of Axiom of Extensionality

+ +

+If we can derive (w ∋ A) ⇔ (w ∋ B) from A == B, the axiom becomes valid, but it seems impossible, so we assumes +

+ +

+  ==→o≡ : { x y : OD  } → (x == y) → x ≡ y
+
+
+Using this, we have +

+ +

+    extensionality : {A B w : OD  } → ((z : OD ) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B)
+    proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d
+    proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d
+
+
+This assumption means we may have an equivalence set on any predicates. +

+ +

+

Non constructive assumptions so far

+We have correspondence between OD and Ordinals and one directional order preserving. +

+We also have equality assumption. +

+ +

+  od→ord : OD  → Ordinal
+  ord→od : Ordinal  → OD
+  oiso   :  {x : OD }      → ord→od ( od→ord x ) ≡ x
+  diso   :  {x : Ordinal } → od→ord ( ord→od x ) ≡ x
+  c<→o<  :  {x y : OD  }   → def y ( od→ord x ) → od→ord x o< od→ord y
+  ==→o≡ : { x y : OD  } → (x == y) → x ≡ y
+
+
+ +
+

Axiom which have negation form

+ +

+ +

+   Union, Selection
+
+
+These axioms contains ∃ x as a logical relation, which can be described in ¬ ( ∀ x ( ¬ p )). +

+Axiom of replacement uses upper bound of function on Ordinals, which makes it non-constructive. +

+Power Set axiom requires double negation, +

+ +

+     power→ : ∀( A t : ZFSet  ) → Power A ∋ t → ∀ {x}  →  t ∋ x → ¬ ¬ ( A ∋ x ) 
+     power← : ∀( A t : ZFSet  ) → t ⊆_ A → Power A ∋ t 
+
+
+If we have an assumption of law of exclude middle, we can recover the original A ∋ x form. +

+ +

+

Union

+The original form of the Axiom of Union is +

+ +

+     ∀ x ∃ y ∀ z (z ∈ y ⇔ ∃ u ∈ x  ∧ (z ∈ u))
+
+
+Union requires the existence of b in a ⊇ ∃ b ∋ x . We will use negation form of ∃. +

+ +

+     union→ : ( X z u : ZFSet ) → ( X ∋ u ) ∧ (u ∋ z ) → Union X ∋ z
+     union← : ( X z : ZFSet ) → (X∋z : Union X ∋ z ) →  ¬  ( (u : ZFSet ) → ¬ ((X ∋  u) ∧ (u ∋ z )))
+
+
+The definition of Union in OD is like this. +

+ +

+    Union : OD  → OD   
+    Union U = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((def U u) ∧ (def (ord→od u) x)))  }
+
+
+Proof of validity is straight forward. +

+ +

+         union→ :  (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z
+         union→ X z u xx not = ⊥-elim ( not (od→ord u) ( record { proj1 = proj1 xx
+              ; proj2 = subst ( λ k → def k (od→ord z)) (sym oiso) (proj2 xx) } ))
+         union← :  (X z : OD) (X∋z : Union X ∋ z) →  ¬  ( (u : OD ) → ¬ ((X ∋  u) ∧ (u ∋ z )))
+         union← X z UX∋z =  FExists _ lemma UX∋z where
+              lemma : {y : Ordinal} → def X y ∧ def (ord→od y) (od→ord z) → ¬ ((u : OD) → ¬ (X ∋ u) ∧ (u ∋ z))
+              lemma {y} xx not = not (ord→od y) record { proj1 = subst ( λ k → def X k ) (sym diso) (proj1 xx ) ; proj2 = proj2 xx }
+
+
+where +

+ +

+        FExists : {m l : Level} → ( ψ : Ordinal  → Set m )
+          → {p : Set l} ( P : { y : Ordinal  } →  ψ y → ¬ p )
+          → (exists : ¬ (∀ y → ¬ ( ψ y ) ))
+          → ¬ p
+        FExists  {m} {l} ψ {p} P = contra-position ( λ p y ψy → P {y} ψy p )
+
+
+which checks existence using contra-position. +

+ +

+

Axiom of replacement

+We can replace the elements of a set by a function and it becomes a set. From the book, +

+ +

+     ∀ x ∀ y ∀ z ( ( ψ ( x , y ) ∧ ψ ( x , z ) ) → y = z ) → ∀ X ∃ A ∀ y ( y ∈ A ↔ ∃ x ∈ X ψ ( x , y ) )
+
+
+The existential quantifier can be related by a function, +

+ +

+     Replace : OD  → (OD  → OD  ) → OD
+
+
+The axioms becomes as follows. +

+ +

+     replacement← : {ψ : ZFSet → ZFSet} → ∀ ( X x : ZFSet  ) → x ∈ X → ψ x ∈  Replace X ψ
+     replacement→ : {ψ : ZFSet → ZFSet} → ∀ ( X x : ZFSet  ) →  ( lt : x ∈  Replace X ψ ) → ¬ ( ∀ (y : ZFSet)  →  ¬ ( x ≈ ψ y ) )
+
+
+In the axiom, the existence of the original elements is necessary. In order to do that we use OD which has +negation form of existential quantifier in the definition. +

+ +

+    in-codomain : (X : OD  ) → ( ψ : OD  → OD  ) → OD 
+    in-codomain  X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( def X y ∧  ( x ≡ od→ord (ψ (ord→od y )))))  }
+
+
+Besides this upper bounds is required. +

+ +

+    Replace : OD  → (OD  → OD  ) → OD 
+    Replace X ψ = record { def = λ x → (x o< sup-o  ( λ x → od→ord (ψ (ord→od x )))) ∧ def (in-codomain X ψ) x }
+
+
+We omit the proof of the validity, but it is rather straight forward. +

+ +

+

Validity of Power Set Axiom

+The original Power Set Axiom is this. +

+ +

+     ∀ X ∃ A ∀ t ( t ∈ A ↔ t ⊆ X ) )
+
+
+The existential quantifier is replaced by a function +

+ +

+    Power : ( A : OD  ) → OD
+
+
+t ⊆ X is a record like this. +

+ +

+    record _⊆_ ( A B : OD   ) : Set (suc n) where
+      field
+         incl : { x : OD } → A ∋ x →  B ∋ x
+
+
+Axiom becomes likes this. +

+ +

+    power→ :  ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x)
+    power← :  (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t
+
+
+The validity of the axioms are slight complicated, we have to define set of all subset. We define +subset in a different form. +

+ +

+    ZFSubset : (A x : OD  ) → OD 
+    ZFSubset A x =  record { def = λ y → def A y ∧  def x y }  
+
+
+We can prove, +

+ +

+    ( {y : OD } → x ∋ y → ZFSubset A x ∋  y ) ⇔  ( x ⊆ A  ) 
+
+
+We only have upper bound as an ordinal, but we have an obvious OD based on the order of Ordinals, +which is an Ordinals with our Model. +

+ +

+    Ord : ( a : Ordinal  ) → OD 
+    Ord  a = record { def = λ y → y o< a }  
+    Def :  (A :  OD ) → OD 
+    Def  A = Ord ( sup-o  ( λ x → od→ord ( ZFSubset A (ord→od x )) ) )   
+
+
+This is slight larger than Power A, so we replace all elements x by A ∩ x (some of them may empty). +

+ +

+    Power : OD  → OD 
+    Power A = Replace (Def (Ord (od→ord A))) ( λ x → A ∩ x )
+
+
+Creating Power Set of Ordinals is rather easy, then we use replacement axiom on A ∩ x since we have this. +

+ +

+     ∩-≡ :  { a b : OD  } → ({x : OD  } → (a ∋ x → b ∋ x)) → a == ( b ∩ a )
+
+
+In case of Ord a intro of Power Set axiom becomes valid. +

+ +

+    ord-power← : (a : Ordinal ) (t : OD) → ({x : OD} → (t ∋ x → (Ord a) ∋ x)) → Def (Ord a) ∋ t
+
+
+Using this, we can prove, +

+ +

+         power→ :  ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x)
+         power← :  (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t
+
+
+ +
+

Axiom of regularity, Axiom of choice, ε-induction

+ +

+Axiom of regularity requires non self intersectable elements (which is called minimum), if we +replace it by a function, it becomes a choice function. It makes axiom of choice valid. +

+This means we cannot prove axiom regularity form our model, and if we postulate this, axiom of +choice also becomes valid. +

+ +

+  minimal : (x : OD  ) → ¬ (x == od∅ )→ OD
+  x∋minimal : (x : OD  ) → ( ne : ¬ (x == od∅ ) ) → def x ( od→ord ( minimal x ne ) )
+  minimal-1 : (x : OD  ) → ( ne : ¬ (x == od∅ ) ) → (y : OD ) → ¬ ( def (minimal x ne) (od→ord y)) ∧ (def x (od→ord  y) )
+
+
+We can avoid this using ε-induction (a predicate is valid on all set if the predicate is true on some element of set). +Assuming law of exclude middle, they say axiom of regularity will be proved, but we haven't check it yet. +

+ +

+    ε-induction : { ψ : OD  → Set (suc n)}
+       → ( {x : OD } → ({ y : OD } →  x ∋ y → ψ y ) → ψ x )
+       → (x : OD ) → ψ x
+
+
+In our model, we assumes the mapping between Ordinals and OD, this is actually the TransFinite induction in Ordinals. +

+The axiom of choice in the book is complicated using any pair in a set, so we use use a form in the Wikipedia. +

+ +

+     ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ]
+
+
+We can formulate like this. +

+ +

+     choice-func : (X : ZFSet ) → {x : ZFSet } → ¬ ( x ≈ ∅ ) → ( X ∋ x ) → ZFSet
+     choice : (X : ZFSet  ) → {A : ZFSet } → ( X∋A : X ∋ A ) → (not : ¬ ( A ≈ ∅ )) → A ∋ choice-func X not X∋A
+
+
+It does not requires ∅ ∉ X . +

+ +

+

Axiom of choice and Law of Excluded Middle

+In our model, since OD has a mapping to Ordinals, it has evident order, which means well ordering theorem is valid, +but it don't have correct form of the axiom yet. They say well ordering axiom is equivalent to the axiom of choice, +but it requires law of the exclude middle. +

+Actually, it is well known to prove law of the exclude middle from axiom of choice in intuitionistic logic, and we can +perform the proof in our mode. Using the definition like this, predicates and ODs are related and we can ask the +set is empty or not if we have an axiom of choice, so we have the law of the exclude middle p ∨ ( ¬ p ) . +

+ +

+    ppp :  { p : Set n } { a : OD  } → record { def = λ x → p } ∋ a → p
+    ppp  {p} {a} d = d
+
+
+We can prove axiom of choice from law excluded middle since we have TransFinite induction. So Axiom of choice +and Law of Excluded Middle is equivalent in our mode. +

+ +

+

Relation-ship among ZF axiom

+ + +

+ +

+

Non constructive assumption in our model

+mapping between OD and Ordinals +

+ +

+  od→ord : OD  → Ordinal
+  ord→od : Ordinal  → OD
+  oiso   :  {x : OD }      → ord→od ( od→ord x ) ≡ x
+  diso   :  {x : Ordinal } → od→ord ( ord→od x ) ≡ x
+  c<→o<  :  {x y : OD  }   → def y ( od→ord x ) → od→ord x o< od→ord y
+
+
+Equivalence on OD +

+ +

+  ==→o≡ : { x y : OD  } → (x == y) → x ≡ y
+
+
+Upper bound +

+ +

+  sup-o  :  ( Ordinal  → Ordinal ) →  Ordinal
+  sup-o< :  { ψ : Ordinal  →  Ordinal } → ∀ {x : Ordinal } → ψ x  o<  sup-o ψ
+
+
+Axiom of choice and strong axiom of regularity. +

+ +

+  minimal : (x : OD  ) → ¬ (x == od∅ )→ OD
+  x∋minimal : (x : OD  ) → ( ne : ¬ (x == od∅ ) ) → def x ( od→ord ( minimal x ne ) )
+  minimal-1 : (x : OD  ) → ( ne : ¬ (x == od∅ ) ) → (y : OD ) → ¬ ( def (minimal x ne) (od→ord y)) ∧ (def x (od→ord  y) )
+
+
+ +
+

So it this correct?

+ +

+Our axiom are syntactically the same in the text book, but negations are slightly different. +

+If we assumes excluded middle, these are exactly same. +

+Even if we assumes excluded middle, intuitionistic logic itself remains consistent, but we cannot prove it. +

+Except the upper bound, axioms are simple logical relation. +

+Proof of existence of mapping between OD and Ordinals are not obvious. We don't know we prove it or not. +

+Existence of the Upper bounds is a pure assumption, if we have not limit on Ordinals, it may contradicts, +but we don't have explicit upper limit on Ordinals. +

+Several inference on our model or our axioms are basically parallel to the set theory text book, so it looks like correct. +

+ +

+

How to use Agda Set Theory

+Assuming record ZF, classical set theory can be developed. If necessary, axiom of choice can be +postulated or assuming law of excluded middle. +

+Instead, simply assumes non constructive assumption, various theory can be developed. We haven't check +these assumptions are proved in record ZF, so we are not sure, these development is a result of ZF Set theory. +

+ZF record itself is not necessary, for example, topology theory without ZF can be possible. +

+ +

+

Topos and Set Theory

+Topos is a mathematical structure in Category Theory, which is a Cartesian closed category which has a +sub-object classifier. +

+Topos itself is model of intuitionistic logic. +

+Transitive Sets are objects of Cartesian closed category. +It is possible to introduce Power Set Functor on it +

+We can use replacement A ∩ x for each element in Transitive Set, +in the similar way of our power set axiom. I +

+A model of ZF Set theory can be constructed on top of the Topos which is shown in Oisus. +

+Our Agda model is a proof theoretic version of it. +

+ +

+

Cardinal number and Continuum hypothesis

+Axiom of choice is required to define cardinal number +

+definition of cardinal number is not yet done +

+definition of filter is not yet done +

+we may have a model without axiom of choice or without continuum hypothesis +

+Possible representation of continuum hypothesis is this. +

+ +

+     continuum-hyphotheis : (a : Ordinal) → Power (Ord a) ⊆ Ord (osuc a)
+
+
+ +
+

Filter

+ +

+filter is a dual of ideal on boolean algebra or lattice. Existence on natural number +is depends on axiom of choice. +

+ +

+    record Filter  ( L : OD  ) : Set (suc n) where
+       field
+           filter : OD
+           proper : ¬ ( filter ∋ od∅ )
+           inL :  filter ⊆ L 
+           filter1 : { p q : OD } →  q ⊆ L  → filter ∋ p →  p ⊆ q  → filter ∋ q
+           filter2 : { p q : OD } → filter ∋ p →  filter ∋ q  → filter ∋ (p ∩ q)
+
+
+We may construct a model of non standard analysis or set theory. +

+This may be simpler than classical forcing theory ( not yet done). +

+ +

+

Programming Mathematics

+Mathematics is a functional programming in Agda where proof is a value of a variable. The mathematical +structure are +

+ +

+   record and data
+
+
+Proof is check by type consistency not by the computation, but it may include some normalization. +

+Type inference and termination is not so clear in multi recursions. +

+Defining Agda record is a good way to understand mathematical theory, for examples, +

+ +

+    Category theory ( Yoneda lemma, Floyd Adjunction functor theorem, Applicative functor )
+    Automaton ( Subset construction、Language containment)
+
+
+are proved in Agda. +

+ +

+

link

+Summer school of foundation of mathematics (in Japanese)
https://www.sci.shizuoka.ac.jp/~math/yorioka/ss2019/ +

+Foundation of axiomatic set theory (in Japanese)
https://www.sci.shizuoka.ac.jp/~math/yorioka/ss2019/sakai0.pdf + +

+Agda +
https://agda.readthedocs.io/en/v2.6.0.1/ +

+ZF-in-Agda source +
https://github.com/shinji-kono/zf-in-agda.git + +

+Category theory in Agda source +
https://github.com/shinji-kono/category-exercise-in-agda + +

+

+ + +
Shinji KONO / Sat May 9 16:41:10 2020 + diff -r 6e1c60866788 -r 4fcac1eebc74 zf-in-agda.ind --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/zf-in-agda.ind Sun Jun 07 20:31:30 2020 +0900 @@ -0,0 +1,1127 @@ +-title: Constructing ZF Set Theory in Agda + +--author: Shinji KONO + +--ZF in Agda + + zf.agda axiom of ZF + zfc.agda axiom of choice + Ordinals.agda axiom of Ordinals + ordinal.agda countable model of Ordinals + OD.agda model of ZF based on Ordinal Definable Set with assumptions + ODC.agda Law of exclude middle from axiom of choice assumptions + LEMC.agda model of choice with assumption of the Law of exclude middle + OPair.agda ordered pair on OD + + BAlgbra.agda Boolean algebra on OD (not yet done) + filter.agda Filter on OD (not yet done) + cardinal.agda Caedinal number on OD (not yet done) + + logic.agda some basics on logic + nat.agda some basics on Nat + +--Programming Mathematics + +Programming is processing data structure with λ terms. + +We are going to handle Mathematics in intuitionistic logic with λ terms. + +Mathematics is a functional programming which values are proofs. + +Programming ZF Set Theory in Agda + +--Target + + Describe ZF axioms in Agda + Construction a Model of ZF Set Theory in Agda + Show necessary assumptions for the model + Show validities of ZF axioms on the model + +This shows consistency of Set Theory (with some assumptions), +without circulating ZF Theory assumption. + + +ZF in Agda https://github.com/shinji-kono/zf-in-agda + + +--Why Set Theory + +If we can formulate Set theory, it suppose to work on any mathematical theory. + +Set Theory is a difficult point for beginners especially axiom of choice. + +It has some amount of difficulty and self circulating discussion. + +I'm planning to do it in my old age, but I'm enough age now. + +This is done during from May to September. + +--Agda and Intuitionistic Logic + +Curry Howard Isomorphism + + Proposition : Proof ⇔ Type : Value + +which means + +　　constructing a typed lambda calculus which corresponds a logic + +Typed lambda calculus which allows complex type as a value of a variable (System FC) + +　　First class Type / Dependent Type + +Agda is a such a programming language which has similar syntax of Haskell + +Coq is specialized in proof assistance such as command and tactics . + +--Introduction of Agda + +A length of a list of type A. + + length : {A : Set } → List A → Nat + length [] = zero + length (_ ∷ t) = suc ( length t ) + +Simple functional programming language. Type declaration is mandatory. +A colon means type, an equal means value. Indentation based. + +Set is a base type (which may have a level ). + +{} means implicit variable which can be omitted if Agda infers its value. + +--data ( Sum type ) + +A data type which as exclusive multiple constructors. A similar one as +union in C or case class in Scala. + +It has a similar syntax as Haskell but it has a slight difference. + + data List (A : Set ) : Set where + [] : List A + _∷_ : A → List A → List A + +_∷_ means infix operator. If use explicit _, it can be used in a normal function +syntax. + +Natural number can be defined as a usual way. + + data Nat : Set where + zero : Nat + suc : Nat → Nat + +-- A → B means "A implies B" + +In Agda, a type can be a value of a variable, which is usually called dependent type. +Type has a name Set in Agda. + + ex3 : {A B : Set} → Set + ex3 {A}{B} = A → B + +ex3 is a type : A → B, which is a value of Set. It also means a formula : A implies B. + + A type is a formula, the value is the proof + +A value of A → B can be interpreted as an inference from the formula A to the formula B, which +can be a function from a proof of A to a proof of B. + +--introduction と elimination + +For a logical operator, there are two types of inference, an introduction and an elimination. + + intro creating symbol / constructor / introduction + elim using symbolic / accessors / elimination + +In Natural deduction, this can be written in proof schema. + + A + : + B A A → B + ------------- →intro ------------------ →elim + A → B B + +In Agda, this is a pair of type and value as follows. Introduction of → uses λ. + + →intro : {A B : Set } → A → B → ( A → B ) + →intro _ b = λ x → b + + →elim : {A B : Set } → A → ( A → B ) → B + →elim a f = f a + +Important + + {A B : Set } → A → B → ( A → B ) + +is + + {A B : Set } → ( A → ( B → ( A → B ) )) + +This makes currying of function easy. + +-- To prove A → B + +Make a left type as an argument. (intros in Coq) + + ex5 : {A B C : Set } → A → B → C → ? + ex5 a b c = ? + +? is called a hole, which is unspecified part. Agda tell us which kind type is required for the Hole. + +We are going to fill the holes, and if we have no warnings nor errors such as type conflict (Red), +insufficient proof or instance (Yellow), Non-termination, the proof is completed. + +-- A ∧ B + +Well known conjunction's introduction and elimination is as follow. + + A B A ∧ B A ∧ B + ------------- ----------- proj1 ---------- proj2 + A ∧ B A B + +We can introduce a corresponding structure in our functional programming language. + +-- record + + record _∧_ A B : Set + field + proj1 : A + proj2 : B + +_∧_　means infix operator. _∧_ A B can be written as A ∧ B (Haskell uses (∧) ) + +This a type which constructed from type A and type B. You may think this as an object +or struct. + + record { proj1 = x ; proj2 = y } + +is a constructor of _∧_. + + ex3 : {A B : Set} → A → B → ( A ∧ B ) + ex3 a b = record { proj1 = a ; proj2 = b } + + ex1 : {A B : Set} → ( A ∧ B ) → A + ex1 a∧b = proj1 a∧b + +a∧b is a variable name. If we have no spaces in a string, it is a word even if we have symbols +except parenthesis or colons. A symbol requires space separation such as a type defining colon. + +Defining record can be recursively, but we don't use the recursion here. + +-- Mathematical structure + +We have types of elements and the relationship in a mathematical structure. + + logical relation has no ordering + there is a natural ordering in arguments and a value in a function + +So we have typical definition style of mathematical structure with records. + + record IsOrdinals {n : Level} (ord : Set n) + (_o<_ : ord → ord → Set n) : Set (suc (suc n)) where + field + Otrans : {x y z : ord } → x o< y → y o< z → x o< z + + record Ordinals {n : Level} : Set (suc (suc n)) where + field + ord : Set n + _o<_ : ord → ord → Set n + isOrdinal : IsOrdinals ord _o<_ + +In IsOrdinals, axioms are written in flat way. In Ordinal, we may have +inputs and outputs are put in the field including IsOrdinal. + +Fields of Ordinal is existential objects in the mathematical structure. + +-- A Model and a theory + +Agda record is a type, so we can write it in the argument, but is it really exists? + +If we have a value of the record, it simply exists, that is, we need to create all the existence +in the record satisfies all the axioms (= field of IsOrdinal) should be valid. + + type of record = theory + value of record = model + +We call the value of the record as a model. If mathematical structure has a +model, it exists. Pretty Obvious. + +-- postulate と module + +Agda proofs are separated by modules, which are large records. + +postulates are assumptions. We can assume a type without proofs. + + postulate + sup-o : ( Ordinal → Ordinal ) → Ordinal + sup-o< : { ψ : Ordinal → Ordinal } → ∀ {x : Ordinal } → ψ x o< sup-o ψ + +sup-o is an example of upper bound of a function and sup-o< assumes it actually +satisfies all the value is less than upper bound. + +Writing some type in a module argument is the same as postulating a type, but +postulate can be written the middle of a proof. + +postulate can be constructive. + +postulate can be inconsistent, which result everything has a proof. + +-- A ∨ B + + data _∨_ (A B : Set) : Set where + case1 : A → A ∨ B + case2 : B → A ∨ B + +As Haskell, case1/case2 are patterns. + + ex3 : {A B : Set} → ( A ∨ A ) → A + ex3 = ? + +In a case statement, Agda command C-C C-C generates possible cases in the head. + + ex3 : {A B : Set} → ( A ∨ A ) → A + ex3 (case1 x) = ? + ex3 (case2 x) = ? + +Proof schema of ∨ is omit due to the complexity. + +-- Negation + + ⊥ + ------------- ⊥-elim + A + +Anything can be derived from bottom, in this case a Set A. There is no introduction rule +in ⊥, which can be implemented as data which has no constructor. + + data ⊥ : Set where + +⊥-elim can be proved like this. + + ⊥-elim : {A : Set } -> ⊥ -> A + ⊥-elim () + +() means no match argument nor value. + +A negation can be defined using ⊥ like this. + + ¬_ : Set → Set + ¬ A = A → ⊥ + +--Equality + +All the value in Agda are terms. If we have the same normalized form, two terms are equal. +If we have variables in the terms, we will perform an unification. unifiable terms are equal. +We don't go further on the unification. + + { x : A } x ≡ y f x y + --------------- ≡-intro --------------------- ≡-elim + x ≡ x f x x + +equality _≡_ can be defined as a data. + + data _≡_ {A : Set } : A → A → Set where + refl : {x : A} → x ≡ x + +The elimination of equality is a substitution in a term. + + subst : {A : Set } → { x y : A } → ( f : A → Set ) → x ≡ y → f x → f y + subst {A} {x} {y} f refl fx = fx + + ex5 : {A : Set} {x y z : A } → x ≡ y → y ≡ z → x ≡ z + ex5 {A} {x} {y} {z} x≡y y≡z = subst ( λ k → x ≡ k ) y≡z x≡y + + +--Equivalence relation + + refl' : {A : Set} {x : A } → x ≡ x + refl' = ? + sym : {A : Set} {x y : A } → x ≡ y → y ≡ x + sym = ? + trans : {A : Set} {x y z : A } → x ≡ y → y ≡ z → x ≡ z + trans = ? + cong : {A B : Set} {x y : A } { f : A → B } → x ≡ y → f x ≡ f y + cong = ? + +--Ordering + +Relation is a predicate on two value which has a same type. + + A → A → Set + +Defining order is the definition of this type with predicate or a data. + + data _≤_ : Rel ℕ 0ℓ where + z≤n : ∀ {n} → zero ≤ n + s≤s : ∀ {m n} (m≤n : m ≤ n) → suc m ≤ suc n + + +--Quantifier + +Handling quantifier in an intuitionistic logic requires special cares. + +In the input of a function, there are no restriction on it, that is, it has +a universal quantifier. (If we explicitly write ∀, Agda gives us a type inference on it) + +There is no ∃ in agda, the one way is using negation like this. + +　∃ (x : A ) → p x = ¬ ( ( x : A ) → ¬ ( p x ) ) + +On the another way, f : A can be used like this. + + p f + +If we use a function which can be defined globally which has stronger meaning the +usage of ∃ x in a logical expression. + + +--Can we do math in this way? + +Yes, we can. Actually we have Principia Mathematica by Russell and Whitehead (with out computer support). + +In some sense, this story is a reprinting of the work, (but Principia Mathematica has a different formulation than ZF). + + define mathematical structure as a record + program inferences as if we have proofs in variables + +--Things which Agda cannot prove + +The infamous Internal Parametricity is a limitation of Agda, it cannot prove so called Free Theorem, which +leads uniqueness of a functor in Category Theory. + +Functional extensionality cannot be proved. + (∀ x → f x ≡ g x) → f ≡ g + +Agda has no law of exclude middle. + + a ∨ ( ¬ a ) + +For example, (A → B) → ¬ B → ¬ A can be proved but, ( ¬ B → ¬ A ) → A → B cannot. + +It also other problems such as termination, type inference or unification which we may overcome with +efforts or devices or may not. + +If we cannot prove something, we can safely postulate it unless it leads a contradiction. + +--Classical story of ZF Set Theory + +Assuming ZF, constructing a model of ZF is a flow of classical Set Theory, which leads +a relative consistency proof of the Set Theory. +Ordinal number is used in the flow. + +In Agda, first we defines Ordinal numbers (Ordinals), then introduce Ordinal Definable Set (OD). +We need some non constructive assumptions in the construction. A model of Set theory is +constructed based on these assumptions. + +
+ +--Ordinals + +Ordinals are our intuition of infinite things, which has ∅ and orders on the things. +It also has a successor osuc. + + record Ordinals {n : Level} : Set (suc (suc n)) where + field + ord : Set n + o∅ : ord + osuc : ord → ord + _o<_ : ord → ord → Set n + isOrdinal : IsOrdinals ord o∅ osuc _o<_ + +It is different from natural numbers in way. The order of Ordinals is not defined in terms +of successor. It is given from outside, which make it possible to have higher order infinity. + +--Axiom of Ordinals + +Properties of infinite things. We request a transfinite induction, which states that if +some properties are satisfied below all possible ordinals, the properties are true on all +ordinals. + +Successor osuc has no ordinal between osuc and the base ordinal. There are some ordinals +which is not a successor of any ordinals. It is called limit ordinal. + +Any two ordinal can be compared, that is less, equal or more, that is total order. + + record IsOrdinals {n : Level} (ord : Set n) (o∅ : ord ) + (osuc : ord → ord ) + (_o<_ : ord → ord → Set n) : Set (suc (suc n)) where + field + Otrans : {x y z : ord } → x o< y → y o< z → x o< z + OTri : Trichotomous {n} _≡_ _o<_ + ¬x<0 : { x : ord } → ¬ ( x o< o∅ ) + <-osuc : { x : ord } → x o< osuc x + osuc-≡< : { a x : ord } → x o< osuc a → (x ≡ a ) ∨ (x o< a) + TransFinite : { ψ : ord → Set (suc n) } + → ( (x : ord) → ( (y : ord ) → y o< x → ψ y ) → ψ x ) + → ∀ (x : ord) → ψ x + +--Concrete Ordinals + +We can define a list like structure with level, which is a kind of two dimensional infinite array. + + data OrdinalD {n : Level} : (lv : Nat) → Set n where + Φ : (lv : Nat) → OrdinalD lv + OSuc : (lv : Nat) → OrdinalD {n} lv → OrdinalD lv + +The order of the OrdinalD can be defined in this way. + + data _d<_ {n : Level} : {lx ly : Nat} → OrdinalD {n} lx → OrdinalD {n} ly → Set n where + Φ< : {lx : Nat} → {x : OrdinalD {n} lx} → Φ lx d< OSuc lx x + s< : {lx : Nat} → {x y : OrdinalD {n} lx} → x d< y → OSuc lx x d< OSuc lx y + +This is a simple data structure, it has no abstract assumptions, and it is countable many data +structure. + + Φ 0 + OSuc 2 ( Osuc 2 ( Osuc 2 (Φ 2))) + Osuc 0 (Φ 0) d< Φ 1 + +--Model of Ordinals + +It is easy to show OrdinalD and its order satisfies the axioms of Ordinals. + +So our Ordinals has a mode. This means axiom of Ordinals are consistent. + +--Debugging axioms using Model + +Whether axiom is correct or not can be checked by a validity on a mode. + +If not, we may fix the axioms or the model, such as the definitions of the order. + +We can also ask whether the inputs exist. + +--Countable Ordinals can contains uncountable set? + +Yes, the ordinals contains any level of infinite Set in the axioms. + +If we handle real-number in the model, only countable number of real-number is used. + + from the outside view point, it is countable + from the internal view point, it is uncountable + +The definition of countable/uncountable is the same, but the properties are different +depending on the context. + +We don't show the definition of cardinal number here. + +--What is Set + +The word Set in Agda is not a Set of ZF Set, but it is a type (why it is named Set?). + +From naive point view, a set i a list, but in Agda, elements have all the same type. +A set in ZF may contain other Sets in ZF, which not easy to implement it as a list. + +Finite set may be written in finite series of ∨, but ... + +--We don't ask the contents of Set. It can be anything. + +From empty set φ, we can think a set contains a φ, and a pair of φ and the set, and so on, +and all of them, and again we repeat this. + + φ {φ} {φ,{φ}}, {φ,{φ},...} + +It is called V. + +This operation can be performed within a ZF Set theory. Classical Set Theory assumes +ZF, so this kind of thing is allowed. + +But in our case, we have no ZF theory, so we are going to use Ordinals. + + +--Ordinal Definable Set + +We can define a sbuset of Ordinals using predicates. What is a subset? + + a predicate has an Ordinal argument + +is an Ordinal Definable Set (OD). In Agda, OD is defined as follows. + + record OD : Set (suc n ) where + field + def : (x : Ordinal ) → Set n + +Ordinals itself is not a set in a ZF Set theory but a class. In OD, + + record { def = λ x → true } + +means Ordinals itself, so ODs are larger than a notion of ZF Set Theory, +but we don't care about it. + + +--∋ in OD + +OD is a predicate on Ordinals and it does not contains OD directly. If we have a mapping + + od→ord : OD → Ordinal + +we may check an OD is an element of the OD using def. + +A ∋ x can be define as follows. + + _∋_ : ( A x : OD ) → Set n + _∋_ A x = def A ( od→ord x ) + +In ψ : Ordinal → Set, if A is a record { def = λ x → ψ x } , then + + A x = def A ( od→ord x ) = ψ (od→ord x) + +--1 to 1 mapping between an OD and an Ordinal + + od→ord : OD → Ordinal + ord→od : Ordinal → OD + oiso : {x : OD } → ord→od ( od→ord x ) ≡ x + diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x + +They say the existing of the mappings can be proved in Classical Set Theory, but we +simply assumes these non constructively. + +We use postulate, it may contains additional restrictions which are not clear now. +It look like the mapping should be a subset of Ordinals, but we don't explicitly +state it. + +
+ +--Order preserving in the mapping of OD and Ordinal + +Ordinals have the order and OD has a natural order based on inclusion ( def / ∋ ). + + def y ( od→ord x ) + +An elements of OD should be defined before the OD, that is, an ordinal corresponding an elements +have to be smaller than the corresponding ordinal of the containing OD. + + c<→o< : {x y : OD } → def y ( od→ord x ) → od→ord x o< od→ord y + +This is also said to be provable in classical Set Theory, but we simply assumes it. + +OD has an order based on the corresponding ordinal, but it may not have corresponding def / ∋ +relation. This means the reverse order preservation, + + o<→c< : {n : Level} {x y : Ordinal } → x o< y → def (ord→od y) x + +is not valid. If we assumes it, ∀ x ∋ ∅ becomes true, which manes all OD becomes Ordinals in +the model. + +
+ +--ISO + +It also requires isomorphisms, + + oiso : {x : OD } → ord→od ( od→ord x ) ≡ x + diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x + + +--Various Sets + +In classical Set Theory, there is a hierarchy call L, which can be defined by a predicate. + + Ordinal / things satisfies axiom of Ordinal / extension of natural number + V / hierarchical construction of Set from φ + L / hierarchical predicate definable construction of Set from φ + OD / equational formula on Ordinals + Agda Set / Type / it also has a level + + +--Fixes on ZF to intuitionistic logic + +We use ODs as Sets in ZF, and defines record ZF, that is, we have to define +ZF axioms in Agda. + +It may not valid in our model. We have to debug it. + +Fixes are depends on axioms. + +
+ + +ZFのrecord + + +--Pure logical axioms + + empty, pair, select, ε-induction??infinity + +These are logical relations among OD. + + empty : ∀( x : ZFSet ) → ¬ ( ∅ ∋ x ) + pair→ : ( x y t : ZFSet ) → (x , y) ∋ t → ( t ≈ x ) ∨ ( t ≈ y ) + pair← : ( x y t : ZFSet ) → ( t ≈ x ) ∨ ( t ≈ y ) → (x , y) ∋ t + selection : { ψ : ZFSet → Set m } → ∀ { X y : ZFSet } → ( ( y ∈ X ) ∧ ψ y ) ⇔ (y ∈ Select X ψ ) + infinity∅ : ∅ ∈ infinite + infinity : ∀( x : ZFSet ) → x ∈ infinite → ( x ∪ ( x , x ) ) ∈ infinite + ε-induction : { ψ : OD → Set (suc n)} + → ( {x : OD } → ({ y : OD } → x ∋ y → ψ y ) → ψ x ) + → (x : OD ) → ψ x + +finitely can be define by Agda data. + + data infinite-d : ( x : Ordinal ) → Set n where + iφ : infinite-d o∅ + isuc : {x : Ordinal } → infinite-d x → + infinite-d (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) )) + +Union (x , ( x , x )) should be an direct successor of x, but we cannot prove it in our model. + +--Axiom of Pair + +In the Tanaka's book, axiom of pair is as follows. + + ∀ x ∀ y ∃ z ∀ t ( z ∋ t ↔ t ≈ x ∨ t ≈ y) + +We have fix ∃ z, a function (x , y) is defined, which is _,_ . + + _,_ : ( A B : ZFSet ) → ZFSet + +using this, we can define two directions in separates axioms, like this. + + pair→ : ( x y t : ZFSet ) → (x , y) ∋ t → ( t ≈ x ) ∨ ( t ≈ y ) + pair← : ( x y t : ZFSet ) → ( t ≈ x ) ∨ ( t ≈ y ) → (x , y) ∋ t + +This is already written in Agda, so we use these as axioms. All inputs have ∀. + +--pair in OD + +OD is an equation on Ordinals, we can simply write axiom of pair in the OD. + + _,_ : OD → OD → OD + x , y = record { def = λ t → (t ≡ od→ord x ) ∨ ( t ≡ od→ord y ) } + +≡ is an equality of λ terms, but please not that this is equality on Ordinals. + +--Validity of Axiom of Pair + +Assuming ZFSet is OD, we are going to prove pair→ . + + pair→ : ( x y t : OD ) → (x , y) ∋ t → ( t == x ) ∨ ( t == y ) + pair→ x y t p = ? + +In this program, type of p is ( x , y ) ∋ t , that is +def ( x , y ) that is, (t ≡ od→ord x ) ∨ ( t ≡ od→ord y ) . + +Since _∨_ is a data, it can be developed as (C-c C-c : agda2-make-case ). + + pair→ x y t (case1 t≡x ) = ? + pair→ x y t (case2 t≡y ) = ? + +The type of the ? is ( t == x ) ∨ ( t == y ), again it is data _∨_ . + + pair→ x y t (case1 t≡x ) = case1 ? + pair→ x y t (case2 t≡y ) = case2 ? + +The ? in case1 is t == x, so we have to create this from t≡x, which is a name of a variable +which type is + + t≡x : od→ord t ≡ od→ord x + +which is shown by an Agda command (C-C C-E : agda2-show-context ). + +But we haven't defined == yet. + +--Equality of OD and Axiom of Extensionality + +OD is defined by a predicates, if we compares normal form of the predicates, even if +it contains the same elements, it may be different, which is no good as an equality of +Sets. + +Axiom of Extensionality requires sets having the same elements are handled in the same way +each other. + + ∀ z ( z ∈ x ⇔ z ∈ y ) ⇒ ∀ w ( x ∈ w ⇔ y ∈ w ) + +We can write this axiom in Agda as follows. + + extensionality : { A B w : ZFSet } → ( (z : ZFSet) → ( A ∋ z ) ⇔ (B ∋ z) ) → ( A ∈ w ⇔ B ∈ w ) + +So we use ( A ∋ z ) ⇔ (B ∋ z) as an equality (_==_) of our model. We have to show +A ∈ w ⇔ B ∈ w from A == B. + +x == y can be defined in this way. + + record _==_ ( a b : OD ) : Set n where + field + eq→ : ∀ { x : Ordinal } → def a x → def b x + eq← : ∀ { x : Ordinal } → def b x → def a x + +Actually, (z : OD) → (A ∋ z) ⇔ (B ∋ z) implies A == B. + + extensionality0 : {A B : OD } → ((z : OD) → (A ∋ z) ⇔ (B ∋ z)) → A == B + eq→ (extensionality0 {A} {B} eq ) {x} d = ? + eq← (extensionality0 {A} {B} eq ) {x} d = ? + +? are def B x and def A x and these are generated from eq : (z : OD) → (A ∋ z) ⇔ (B ∋ z) . + +Actual proof is rather complicated. + + eq→ (extensionality0 {A} {B} eq ) {x} d = def-iso {A} {B} (sym diso) (proj1 (eq (ord→od x))) d + eq← (extensionality0 {A} {B} eq ) {x} d = def-iso {B} {A} (sym diso) (proj2 (eq (ord→od x))) d + +where + + def-iso : {A B : OD } {x y : Ordinal } → x ≡ y → (def A y → def B y) → def A x → def B x + def-iso refl t = t + +--Validity of Axiom of Extensionality + +If we can derive (w ∋ A) ⇔ (w ∋ B) from A == B, the axiom becomes valid, but it seems impossible, +so we assumes + + ==→o≡ : { x y : OD } → (x == y) → x ≡ y + +Using this, we have + + extensionality : {A B w : OD } → ((z : OD ) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B) + proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d + proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d + +This assumption means we may have an equivalence set on any predicates. + +--Non constructive assumptions so far + +We have correspondence between OD and Ordinals and one directional order preserving. + +We also have equality assumption. + + od→ord : OD → Ordinal + ord→od : Ordinal → OD + oiso : {x : OD } → ord→od ( od→ord x ) ≡ x + diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x + c<→o< : {x y : OD } → def y ( od→ord x ) → od→ord x o< od→ord y + ==→o≡ : { x y : OD } → (x == y) → x ≡ y + + +--Axiom which have negation form + + Union, Selection + +These axioms contains ∃ x as a logical relation, which can be described in ¬ ( ∀ x ( ¬ p )). + +Axiom of replacement uses upper bound of function on Ordinals, which makes it non-constructive. + +Power Set axiom requires double negation, + + power→ : ∀( A t : ZFSet ) → Power A ∋ t → ∀ {x} → t ∋ x → ¬ ¬ ( A ∋ x ) + power← : ∀( A t : ZFSet ) → t ⊆_ A → Power A ∋ t + +If we have an assumption of law of exclude middle, we can recover the original A ∋ x form. + +--Union + +The original form of the Axiom of Union is + + ∀ x ∃ y ∀ z (z ∈ y ⇔ ∃ u ∈ x ∧ (z ∈ u)) + +Union requires the existence of b in a ⊇ ∃ b ∋ x . We will use negation form of ∃. + + union→ : ( X z u : ZFSet ) → ( X ∋ u ) ∧ (u ∋ z ) → Union X ∋ z + union← : ( X z : ZFSet ) → (X∋z : Union X ∋ z ) → ¬ ( (u : ZFSet ) → ¬ ((X ∋ u) ∧ (u ∋ z ))) + +The definition of Union in OD is like this. + + Union : OD → OD + Union U = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((def U u) ∧ (def (ord→od u) x))) } + +Proof of validity is straight forward. + + union→ : (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z + union→ X z u xx not = ⊥-elim ( not (od→ord u) ( record { proj1 = proj1 xx + ; proj2 = subst ( λ k → def k (od→ord z)) (sym oiso) (proj2 xx) } )) + union← : (X z : OD) (X∋z : Union X ∋ z) → ¬ ( (u : OD ) → ¬ ((X ∋ u) ∧ (u ∋ z ))) + union← X z UX∋z = FExists _ lemma UX∋z where + lemma : {y : Ordinal} → def X y ∧ def (ord→od y) (od→ord z) → ¬ ((u : OD) → ¬ (X ∋ u) ∧ (u ∋ z)) + lemma {y} xx not = not (ord→od y) record { proj1 = subst ( λ k → def X k ) (sym diso) (proj1 xx ) ; proj2 = proj2 xx } + +where + + FExists : {m l : Level} → ( ψ : Ordinal → Set m ) + → {p : Set l} ( P : { y : Ordinal } → ψ y → ¬ p ) + → (exists : ¬ (∀ y → ¬ ( ψ y ) )) + → ¬ p + FExists {m} {l} ψ {p} P = contra-position ( λ p y ψy → P {y} ψy p ) + +which checks existence using contra-position. + +--Axiom of replacement + +We can replace the elements of a set by a function and it becomes a set. From the book, + + ∀ x ∀ y ∀ z ( ( ψ ( x , y ) ∧ ψ ( x , z ) ) → y = z ) → ∀ X ∃ A ∀ y ( y ∈ A ↔ ∃ x ∈ X ψ ( x , y ) ) + +The existential quantifier can be related by a function, + + Replace : OD → (OD → OD ) → OD + +The axioms becomes as follows. + + replacement← : {ψ : ZFSet → ZFSet} → ∀ ( X x : ZFSet ) → x ∈ X → ψ x ∈ Replace X ψ + replacement→ : {ψ : ZFSet → ZFSet} → ∀ ( X x : ZFSet ) → ( lt : x ∈ Replace X ψ ) → ¬ ( ∀ (y : ZFSet) → ¬ ( x ≈ ψ y ) ) + +In the axiom, the existence of the original elements is necessary. In order to do that we use OD which has +negation form of existential quantifier in the definition. + + in-codomain : (X : OD ) → ( ψ : OD → OD ) → OD + in-codomain X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( def X y ∧ ( x ≡ od→ord (ψ (ord→od y ))))) } + +Besides this upper bounds is required. + + Replace : OD → (OD → OD ) → OD + Replace X ψ = record { def = λ x → (x o< sup-o ( λ x → od→ord (ψ (ord→od x )))) ∧ def (in-codomain X ψ) x } + +We omit the proof of the validity, but it is rather straight forward. + +--Validity of Power Set Axiom + +The original Power Set Axiom is this. + + ∀ X ∃ A ∀ t ( t ∈ A ↔ t ⊆ X ) ) + +The existential quantifier is replaced by a function + + Power : ( A : OD ) → OD + +t ⊆ X is a record like this. + + record _⊆_ ( A B : OD ) : Set (suc n) where + field + incl : { x : OD } → A ∋ x → B ∋ x + +Axiom becomes likes this. + + power→ : ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x) + power← : (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t + +The validity of the axioms are slight complicated, we have to define set of all subset. We define +subset in a different form. + + ZFSubset : (A x : OD ) → OD + ZFSubset A x = record { def = λ y → def A y ∧ def x y } + +We can prove, + + ( {y : OD } → x ∋ y → ZFSubset A x ∋ y ) ⇔ ( x ⊆ A ) + +We only have upper bound as an ordinal, but we have an obvious OD based on the order of Ordinals, +which is an Ordinals with our Model. + + Ord : ( a : Ordinal ) → OD + Ord a = record { def = λ y → y o< a } + + Def : (A : OD ) → OD + Def A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) ) + +This is slight larger than Power A, so we replace all elements x by A ∩ x (some of them may empty). + + Power : OD → OD + Power A = Replace (Def (Ord (od→ord A))) ( λ x → A ∩ x ) + +Creating Power Set of Ordinals is rather easy, then we use replacement axiom on A ∩ x since we have this. + + ∩-≡ : { a b : OD } → ({x : OD } → (a ∋ x → b ∋ x)) → a == ( b ∩ a ) + +In case of Ord a intro of Power Set axiom becomes valid. + + ord-power← : (a : Ordinal ) (t : OD) → ({x : OD} → (t ∋ x → (Ord a) ∋ x)) → Def (Ord a) ∋ t + +Using this, we can prove, + + power→ : ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x) + power← : (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t + + +--Axiom of regularity, Axiom of choice, ε-induction + +Axiom of regularity requires non self intersectable elements (which is called minimum), if we +replace it by a function, it becomes a choice function. It makes axiom of choice valid. + +This means we cannot prove axiom regularity form our model, and if we postulate this, axiom of +choice also becomes valid. + + minimal : (x : OD ) → ¬ (x == od∅ )→ OD + x∋minimal : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → def x ( od→ord ( minimal x ne ) ) + minimal-1 : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → (y : OD ) → ¬ ( def (minimal x ne) (od→ord y)) ∧ (def x (od→ord y) ) + +We can avoid this using ε-induction (a predicate is valid on all set if the predicate is true on some element of set). +Assuming law of exclude middle, they say axiom of regularity will be proved, but we haven't check it yet. + + ε-induction : { ψ : OD → Set (suc n)} + → ( {x : OD } → ({ y : OD } → x ∋ y → ψ y ) → ψ x ) + → (x : OD ) → ψ x + +In our model, we assumes the mapping between Ordinals and OD, this is actually the TransFinite induction in Ordinals. + +The axiom of choice in the book is complicated using any pair in a set, so we use use a form in the Wikipedia. + + ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ] + +We can formulate like this. + + choice-func : (X : ZFSet ) → {x : ZFSet } → ¬ ( x ≈ ∅ ) → ( X ∋ x ) → ZFSet + choice : (X : ZFSet ) → {A : ZFSet } → ( X∋A : X ∋ A ) → (not : ¬ ( A ≈ ∅ )) → A ∋ choice-func X not X∋A + +It does not requires ∅ ∉ X . + +--Axiom of choice and Law of Excluded Middle + +In our model, since OD has a mapping to Ordinals, it has evident order, which means well ordering theorem is valid, +but it don't have correct form of the axiom yet. They say well ordering axiom is equivalent to the axiom of choice, +but it requires law of the exclude middle. + +Actually, it is well known to prove law of the exclude middle from axiom of choice in intuitionistic logic, and we can +perform the proof in our mode. Using the definition like this, predicates and ODs are related and we can ask the +set is empty or not if we have an axiom of choice, so we have the law of the exclude middle p ∨ ( ¬ p ) . + + ppp : { p : Set n } { a : OD } → record { def = λ x → p } ∋ a → p + ppp {p} {a} d = d + +We can prove axiom of choice from law excluded middle since we have TransFinite induction. So Axiom of choice +and Law of Excluded Middle is equivalent in our mode. + +--Relation-ship among ZF axiom + +
+ +--Non constructive assumption in our model + +mapping between OD and Ordinals + + od→ord : OD → Ordinal + ord→od : Ordinal → OD + oiso : {x : OD } → ord→od ( od→ord x ) ≡ x + diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x + c<→o< : {x y : OD } → def y ( od→ord x ) → od→ord x o< od→ord y + +Equivalence on OD + + ==→o≡ : { x y : OD } → (x == y) → x ≡ y + +Upper bound + + sup-o : ( Ordinal → Ordinal ) → Ordinal + sup-o< : { ψ : Ordinal → Ordinal } → ∀ {x : Ordinal } → ψ x o< sup-o ψ + +Axiom of choice and strong axiom of regularity. + + minimal : (x : OD ) → ¬ (x == od∅ )→ OD + x∋minimal : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → def x ( od→ord ( minimal x ne ) ) + minimal-1 : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → (y : OD ) → ¬ ( def (minimal x ne) (od→ord y)) ∧ (def x (od→ord y) ) + +--So it this correct? + +Our axiom are syntactically the same in the text book, but negations are slightly different. + +If we assumes excluded middle, these are exactly same. + +Even if we assumes excluded middle, intuitionistic logic itself remains consistent, but we cannot prove it. + +Except the upper bound, axioms are simple logical relation. + +Proof of existence of mapping between OD and Ordinals are not obvious. We don't know we prove it or not. + +Existence of the Upper bounds is a pure assumption, if we have not limit on Ordinals, it may contradicts, +but we don't have explicit upper limit on Ordinals. + +Several inference on our model or our axioms are basically parallel to the set theory text book, so it looks like correct. + +--How to use Agda Set Theory + +Assuming record ZF, classical set theory can be developed. If necessary, axiom of choice can be +postulated or assuming law of excluded middle. + +Instead, simply assumes non constructive assumption, various theory can be developed. We haven't check +these assumptions are proved in record ZF, so we are not sure, these development is a result of ZF Set theory. + +ZF record itself is not necessary, for example, topology theory without ZF can be possible. + +--Topos and Set Theory + +Topos is a mathematical structure in Category Theory, which is a Cartesian closed category which has a +sub-object classifier. + +Topos itself is model of intuitionistic logic. + +Transitive Sets are objects of Cartesian closed category. +It is possible to introduce Power Set Functor on it + +We can use replacement A ∩ x for each element in Transitive Set, +in the similar way of our power set axiom. I + +A model of ZF Set theory can be constructed on top of the Topos which is shown in Oisus. + +Our Agda model is a proof theoretic version of it. + +--Cardinal number and Continuum hypothesis + +Axiom of choice is required to define cardinal number + +definition of cardinal number is not yet done + +definition of filter is not yet done + +we may have a model without axiom of choice or without continuum hypothesis + +Possible representation of continuum hypothesis is this. + + continuum-hyphotheis : (a : Ordinal) → Power (Ord a) ⊆ Ord (osuc a) + +--Filter + +filter is a dual of ideal on boolean algebra or lattice. Existence on natural number +is depends on axiom of choice. + + record Filter ( L : OD ) : Set (suc n) where + field + filter : OD + proper : ¬ ( filter ∋ od∅ ) + inL : filter ⊆ L + filter1 : { p q : OD } → q ⊆ L → filter ∋ p → p ⊆ q → filter ∋ q + filter2 : { p q : OD } → filter ∋ p → filter ∋ q → filter ∋ (p ∩ q) + +We may construct a model of non standard analysis or set theory. + +This may be simpler than classical forcing theory ( not yet done). + +--Programming Mathematics + +Mathematics is a functional programming in Agda where proof is a value of a variable. The mathematical +structure are + + record and data + +Proof is check by type consistency not by the computation, but it may include some normalization. + +Type inference and termination is not so clear in multi recursions. + +Defining Agda record is a good way to understand mathematical theory, for examples, + + Category theory ( Yoneda lemma, Floyd Adjunction functor theorem, Applicative functor ) + Automaton ( Subset construction、Language containment) + +are proved in Agda. + +--link + +Summer school of foundation of mathematics (in Japanese) +
+https://www.sci.shizuoka.ac.jp/~math/yorioka/ss2019/ + + +Foundation of axiomatic set theory (in Japanese) +
+https://www.sci.shizuoka.ac.jp/~math/yorioka/ss2019/sakai0.pdf + + +Agda +
+https://agda.readthedocs.io/en/v2.6.0.1/ + + +ZF-in-Agda source +
+https://github.com/shinji-kono/zf-in-agda.git + + +Category theory in Agda source +
+https://github.com/shinji-kono/category-exercise-in-agda + + + + diff -r 6e1c60866788 -r 4fcac1eebc74 zf.agda --- a/zf.agda Thu Aug 29 16:18:37 2019 +0900 +++ b/zf.agda Sun Jun 07 20:31:30 2020 +0900 @@ -19,7 +19,7 @@ (Select : (X : ZFSet ) → ( ψ : (x : ZFSet ) → Set m ) → ZFSet ) (Replace : ZFSet → ( ZFSet → ZFSet ) → ZFSet ) (infinite : ZFSet) - : Set (suc (n ⊔ m)) where + : Set (suc (n ⊔ suc m)) where field isEquivalence : IsEquivalence {n} {m} {ZFSet} _≈_ -- ∀ x ∀ y ∃ z ∀ t ( z ∋ t → t ≈ x ∨ t ≈ y) @@ -35,7 +35,7 @@ _∩_ : ( A B : ZFSet ) → ZFSet A ∩ B = Select A ( λ x → ( A ∋ x ) ∧ ( B ∋ x ) ) _∪_ : ( A B : ZFSet ) → ZFSet - A ∪ B = Union (A , B) -- Select A ( λ x → ( A ∋ x ) ∨ ( B ∋ x ) ) is easier + A ∪ B = Union (A , B) ｛_｝ : ZFSet → ZFSet ｛ x ｝ = ( x , x ) infixr 200 _∈_ @@ -48,14 +48,10 @@ power← : ∀( A t : ZFSet ) → ( ∀ {x} → _⊆_ t A {x}) → Power A ∋ t -- extensionality : ∀ z ( z ∈ x ⇔ z ∈ y ) ⇒ ∀ w ( x ∈ w ⇔ y ∈ w ) extensionality : { A B w : ZFSet } → ( (z : ZFSet) → ( A ∋ z ) ⇔ (B ∋ z) ) → ( A ∈ w ⇔ B ∈ w ) - -- This form of regurality forces choice function - -- regularity : ∀ x ( x ≠ ∅ → ∃ y ∈ x ( y ∩ x = ∅ ) ) - -- minimul : (x : ZFSet ) → ¬ (x ≈ ∅) → ZFSet - -- regularity : ∀( x : ZFSet ) → (not : ¬ (x ≈ ∅)) → ( minimul x not ∈ x ∧ ( minimul x not ∩ x ≈ ∅ ) ) - -- another form of regularity - -- ε-induction : { ψ : ZFSet → Set m} - -- → ( {x : ZFSet } → ({ y : ZFSet } → x ∋ y → ψ y ) → ψ x ) - -- → (x : ZFSet ) → ψ x + -- regularity without minimum + ε-induction : { ψ : ZFSet → Set (suc m)} + → ( {x : ZFSet } → ({ y : ZFSet } → x ∋ y → ψ y ) → ψ x ) + → (x : ZFSet ) → ψ x -- infinity : ∃ A ( ∅ ∈ A ∧ ∀ x ∈ A ( x ∪ { x } ∈ A ) ) infinity∅ : ∅ ∈ infinite infinity : ∀( x : ZFSet ) → x ∈ infinite → ( x ∪ ｛ x ｝) ∈ infinite @@ -63,11 +59,8 @@ -- replacement : ∀ x ∀ y ∀ z ( ( ψ ( x , y ) ∧ ψ ( x , z ) ) → y = z ) → ∀ X ∃ A ∀ y ( y ∈ A ↔ ∃ x ∈ X ψ ( x , y ) ) replacement← : {ψ : ZFSet → ZFSet} → ∀ ( X x : ZFSet ) → x ∈ X → ψ x ∈ Replace X ψ replacement→ : {ψ : ZFSet → ZFSet} → ∀ ( X x : ZFSet ) → ( lt : x ∈ Replace X ψ ) → ¬ ( ∀ (y : ZFSet) → ¬ ( x ≈ ψ y ) ) - -- ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ] - choice-func : (X : ZFSet ) → {x : ZFSet } → ¬ ( x ≈ ∅ ) → ( X ∋ x ) → ZFSet - choice : (X : ZFSet ) → {A : ZFSet } → ( X∋A : X ∋ A ) → (not : ¬ ( A ≈ ∅ )) → A ∋ choice-func X not X∋A -record ZF {n m : Level } : Set (suc (n ⊔ m)) where +record ZF {n m : Level } : Set (suc (n ⊔ suc m)) where infixr 210 _,_ infixl 200 _∋_ infixr 220 _≈_ diff -r 6e1c60866788 -r 4fcac1eebc74 zfc.agda --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/zfc.agda Sun Jun 07 20:31:30 2020 +0900 @@ -0,0 +1,34 @@ +module zfc where + +open import Level +open import Relation.Binary +open import Relation.Nullary +open import logic + +record IsZFC {n m : Level } + (ZFSet : Set n) + (_∋_ : ( A x : ZFSet ) → Set m) + (_≈_ : Rel ZFSet m) + (∅ : ZFSet) + (Select : (X : ZFSet ) → ( ψ : (x : ZFSet ) → Set m ) → ZFSet ) + : Set (suc (n ⊔ suc m)) where + field + -- ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ] + choice-func : (X : ZFSet ) → {x : ZFSet } → ¬ ( x ≈ ∅ ) → ( X ∋ x ) → ZFSet + choice : (X : ZFSet ) → {A : ZFSet } → ( X∋A : X ∋ A ) → (not : ¬ ( A ≈ ∅ )) → A ∋ choice-func X not X∋A + infixr 200 _∈_ + infixr 230 _∩_ + _∈_ : ( A B : ZFSet ) → Set m + A ∈ B = B ∋ A + _∩_ : ( A B : ZFSet ) → ZFSet + A ∩ B = Select A ( λ x → ( A ∋ x ) ∧ ( B ∋ x ) ) + +record ZFC {n m : Level } : Set (suc (n ⊔ suc m)) where + field + ZFSet : Set n + _∋_ : ( A x : ZFSet ) → Set m + _≈_ : ( A B : ZFSet ) → Set m + ∅ : ZFSet + Select : (X : ZFSet ) → ( ψ : (x : ZFSet ) → Set m ) → ZFSet + isZFC : IsZFC ZFSet _∋_ _≈_ ∅ Select +