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431
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1 open import Level
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2 open import Ordinals
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3 open import logic
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4 open import Relation.Nullary
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5 module LEMC {n : Level } (O : Ordinals {n} ) (p∨¬p : ( p : Set n) → p ∨ ( ¬ p )) where
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6
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7 open import zf
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8 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ )
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9 open import Relation.Binary.PropositionalEquality
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10 open import Data.Nat.Properties
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11 open import Data.Empty
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12 open import Relation.Binary
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13 open import Relation.Binary.Core
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14
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15 open import nat
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16 import OD
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17
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18 open inOrdinal O
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19 open OD O
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20 open OD.OD
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21 open OD._==_
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22 open ODAxiom odAxiom
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23 import OrdUtil
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24 import ODUtil
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25 open Ordinals.Ordinals O
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26 open Ordinals.IsOrdinals isOrdinal
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27 open Ordinals.IsNext isNext
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28 open OrdUtil O
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29 open ODUtil O
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30
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31 open import zfc
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32
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33 open HOD
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34
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35 decp : ( p : Set n ) → Dec p -- assuming axiom of choice
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36 decp p with p∨¬p p
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37 decp p | case1 x = yes x
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38 decp p | case2 x = no x
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39
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40 ∋-p : (A x : HOD ) → Dec ( A ∋ x )
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41 ∋-p A x with p∨¬p ( A ∋ x) -- LEM
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42 ∋-p A x | case1 t = yes t
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43 ∋-p A x | case2 t = no (λ x → t x)
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44
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45 double-neg-eilm : {A : Set n} → ¬ ¬ A → A -- we don't have this in intutionistic logic
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46 double-neg-eilm {A} notnot with decp A -- assuming axiom of choice
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47 ... | yes p = p
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48 ... | no ¬p = ⊥-elim ( notnot ¬p )
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49
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50 power→⊆ : ( A t : HOD) → Power A ∋ t → t ⊆ A
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1096
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51 power→⊆ A t PA∋t t∋x = subst (λ k → odef A k ) &iso ( t1 (subst (λ k → odef t k ) (sym &iso) t∋x)) where
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52 t1 : {x : HOD } → t ∋ x → A ∋ x
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431
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53 t1 = power→ A t PA∋t
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54
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55 --- With assuption of HOD is ordered, p ∨ ( ¬ p ) <=> axiom of choice
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56 ---
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57 record choiced ( X : Ordinal ) : Set n where
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58 field
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59 a-choice : Ordinal
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60 is-in : odef (* X) a-choice
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61
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62 open choiced
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63
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64 -- ∋→d : ( a : HOD ) { x : HOD } → * (& a) ∋ x → X ∋ * (a-choice (choice-func X not))
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65 -- ∋→d a lt = subst₂ (λ j k → odef j k) *iso (sym &iso) lt
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66
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67 oo∋ : { a : HOD} { x : Ordinal } → odef (* (& a)) x → a ∋ * x
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68 oo∋ lt = subst₂ (λ j k → odef j k) *iso (sym &iso) lt
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69
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70 ∋oo : { a : HOD} { x : Ordinal } → a ∋ * x → odef (* (& a)) x
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71 ∋oo lt = subst₂ (λ j k → odef j k ) (sym *iso) &iso lt
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72
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73 OD→ZFC : ZFC
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74 OD→ZFC = record {
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75 ZFSet = HOD
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76 ; _∋_ = _∋_
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77 ; _≈_ = _=h=_
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78 ; ∅ = od∅
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79 ; Select = Select
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80 ; isZFC = isZFC
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81 } where
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82 -- infixr 200 _∈_
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83 -- infixr 230 _∩_ _∪_
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84 isZFC : IsZFC (HOD ) _∋_ _=h=_ od∅ Select
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85 isZFC = record {
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86 choice-func = λ A {X} not A∋X → * (a-choice (choice-func X not) );
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87 choice = λ A {X} A∋X not → oo∋ (is-in (choice-func X not))
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88 } where
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89 --
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90 -- the axiom choice from LEM and OD ordering
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91 --
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92 choice-func : (X : HOD ) → ¬ ( X =h= od∅ ) → choiced (& X)
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93 choice-func X not = have_to_find where
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94 ψ : ( ox : Ordinal ) → Set n
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95 ψ ox = (( x : Ordinal ) → x o< ox → ( ¬ odef X x )) ∨ choiced (& X)
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96 lemma-ord : ( ox : Ordinal ) → ψ ox
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97 lemma-ord ox = TransFinite0 {ψ} induction ox where
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98 ∀-imply-or : {A : Ordinal → Set n } {B : Set n }
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99 → ((x : Ordinal ) → A x ∨ B) → ((x : Ordinal ) → A x) ∨ B
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100 ∀-imply-or {A} {B} ∀AB with p∨¬p ((x : Ordinal ) → A x) -- LEM
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101 ∀-imply-or {A} {B} ∀AB | case1 t = case1 t
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102 ∀-imply-or {A} {B} ∀AB | case2 x = case2 (lemma (λ not → x not )) where
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103 lemma : ¬ ((x : Ordinal ) → A x) → B
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104 lemma not with p∨¬p B
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105 lemma not | case1 b = b
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106 lemma not | case2 ¬b = ⊥-elim (not (λ x → dont-orb (∀AB x) ¬b ))
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107 induction : (x : Ordinal) → ((y : Ordinal) → y o< x → ψ y) → ψ x
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108 induction x prev with ∋-p X ( * x)
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109 ... | yes p = case2 ( record { a-choice = x ; is-in = ∋oo p } )
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110 ... | no ¬p = lemma where
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111 lemma1 : (y : Ordinal) → (y o< x → odef X y → ⊥) ∨ choiced (& X)
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112 lemma1 y with ∋-p X (* y)
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113 lemma1 y | yes y<X = case2 ( record { a-choice = y ; is-in = ∋oo y<X } )
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114 lemma1 y | no ¬y<X = case1 ( λ lt y<X → ¬y<X (d→∋ X y<X) )
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115 lemma : ((y : Ordinal) → y o< x → odef X y → ⊥) ∨ choiced (& X)
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116 lemma = ∀-imply-or lemma1
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117 odef→o< : {X : HOD } → {x : Ordinal } → odef X x → x o< & X
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118 odef→o< {X} {x} lt = o<-subst {_} {_} {x} {& X} ( c<→o< ( subst₂ (λ j k → odef j k ) (sym *iso) (sym &iso) lt )) &iso &iso
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119 have_to_find : choiced (& X)
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120 have_to_find = dont-or (lemma-ord (& X )) ¬¬X∋x where
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121 ¬¬X∋x : ¬ ((x : Ordinal) → x o< (& X) → odef X x → ⊥)
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122 ¬¬X∋x nn = not record {
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123 eq→ = λ {x} lt → ⊥-elim (nn x (odef→o< lt) lt)
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124 ; eq← = λ {x} lt → ⊥-elim ( ¬x<0 lt )
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125 }
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126
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127 --
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128 -- axiom regurality from ε-induction (using axiom of choice above)
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129 --
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130 -- from https://math.stackexchange.com/questions/2973777/is-it-possible-to-prove-regularity-with-transfinite-induction-only
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131 --
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132 record Minimal (x : HOD) : Set (suc n) where
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133 field
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134 min : HOD
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135 x∋min : x ∋ min
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136 min-empty : (y : HOD ) → ¬ ( min ∋ y) ∧ (x ∋ y)
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137 open Minimal
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138 open _∧_
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139 induction : {x : HOD} → ({y : HOD} → x ∋ y → (u : HOD ) → (u∋x : u ∋ y) → Minimal u )
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140 → (u : HOD ) → (u∋x : u ∋ x) → Minimal u
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141 induction {x} prev u u∋x with p∨¬p ((y : Ordinal ) → ¬ (odef x y) ∧ (odef u y))
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142 ... | case1 P =
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143 record { min = x
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144 ; x∋min = u∋x
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145 ; min-empty = λ y → P (& y)
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146 }
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147 ... | case2 NP = min2 where
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148 p : HOD
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149 p = record { od = record { def = λ y1 → odef x y1 ∧ odef u y1 } ; odmax = omin (odmax x) (odmax u) ; <odmax = lemma } where
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150 lemma : {y : Ordinal} → OD.def (od x) y ∧ OD.def (od u) y → y o< omin (odmax x) (odmax u)
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151 lemma {y} lt = min1 (<odmax x (proj1 lt)) (<odmax u (proj2 lt))
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152 np : ¬ (p =h= od∅)
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153 np p∅ = NP (λ y p∋y → ∅< {p} {_} (d→∋ p p∋y) p∅ )
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154 y1choice : choiced (& p)
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155 y1choice = choice-func p np
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156 y1 : HOD
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157 y1 = * (a-choice y1choice)
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158 y1prop : (x ∋ y1) ∧ (u ∋ y1)
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159 y1prop = oo∋ (is-in y1choice)
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160 min2 : Minimal u
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161 min2 = prev (proj1 y1prop) u (proj2 y1prop)
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162 Min2 : (x : HOD) → (u : HOD ) → (u∋x : u ∋ x) → Minimal u
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163 Min2 x u u∋x = (ε-induction {λ y → (u : HOD ) → (u∋x : u ∋ y) → Minimal u } induction x u u∋x )
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164 cx : {x : HOD} → ¬ (x =h= od∅ ) → choiced (& x )
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165 cx {x} nx = choice-func x nx
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166 minimal : (x : HOD ) → ¬ (x =h= od∅ ) → HOD
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167 minimal x ne = min (Min2 (* (a-choice (cx {x} ne) )) x ( oo∋ (is-in (cx ne))) )
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168 x∋minimal : (x : HOD ) → ( ne : ¬ (x =h= od∅ ) ) → odef x ( & ( minimal x ne ) )
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169 x∋minimal x ne = x∋min (Min2 (* (a-choice (cx {x} ne) )) x ( oo∋ (is-in (cx ne))) )
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170 minimal-1 : (x : HOD ) → ( ne : ¬ (x =h= od∅ ) ) → (y : HOD ) → ¬ ( odef (minimal x ne) (& y)) ∧ (odef x (& y) )
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171 minimal-1 x ne y = min-empty (Min2 (* (a-choice (cx ne) )) x ( oo∋ (is-in (cx ne)))) y
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172
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173
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174
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175
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