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273
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1 -title: Constructing ZF Set Theory in Agda
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2
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3 --author: Shinji KONO
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4
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279
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5 --ZF in Agda
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6
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7 zf.agda axiom of ZF
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8 zfc.agda axiom of choice
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9 Ordinals.agda axiom of Ordinals
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10 ordinal.agda countable model of Ordinals
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11 OD.agda model of ZF based on Ordinal Definable Set with assumptions
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12 ODC.agda Law of exclude middle from axiom of choice assumptions
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13 LEMC.agda model of choice with assumption of the Law of exclude middle
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14 OPair.agda ordered pair on OD
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15
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16 BAlgbra.agda Boolean algebra on OD (not yet done)
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17 filter.agda Filter on OD (not yet done)
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18 cardinal.agda Caedinal number on OD (not yet done)
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19
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20 logic.agda some basics on logic
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21 nat.agda some basics on Nat
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22
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273
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23 --Programming Mathematics
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24
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25 Programming is processing data structure with λ terms.
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26
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27 We are going to handle Mathematics in intuitionistic logic with λ terms.
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28
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29 Mathematics is a functional programming which values are proofs.
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30
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31 Programming ZF Set Theory in Agda
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32
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33 --Target
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34
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35 Describe ZF axioms in Agda
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36 Construction a Model of ZF Set Theory in Agda
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37 Show necessary assumptions for the model
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38 Show validities of ZF axioms on the model
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39
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40 This shows consistency of Set Theory (with some assumptions),
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41 without circulating ZF Theory assumption.
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42
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43 <a href="https://github.com/shinji-kono/zf-in-agda">
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44 ZF in Agda https://github.com/shinji-kono/zf-in-agda
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45 </a>
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46
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47 --Why Set Theory
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48
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49 If we can formulate Set theory, it suppose to work on any mathematical theory.
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50
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51 Set Theory is a difficult point for beginners especially axiom of choice.
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52
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53 It has some amount of difficulty and self circulating discussion.
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54
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55 I'm planning to do it in my old age, but I'm enough age now.
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56
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336
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57 if you familier with Agda, you can skip to
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58 <a href="#set-theory">
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59 there
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60 </a>
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273
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61
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62 --Agda and Intuitionistic Logic
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63
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64 Curry Howard Isomorphism
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65
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66 Proposition : Proof ⇔ Type : Value
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67
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68 which means
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69
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70 constructing a typed lambda calculus which corresponds a logic
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71
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72 Typed lambda calculus which allows complex type as a value of a variable (System FC)
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73
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74 First class Type / Dependent Type
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75
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76 Agda is a such a programming language which has similar syntax of Haskell
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77
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78 Coq is specialized in proof assistance such as command and tactics .
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79
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80 --Introduction of Agda
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81
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82 A length of a list of type A.
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83
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84 length : {A : Set } → List A → Nat
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85 length [] = zero
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86 length (_ ∷ t) = suc ( length t )
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87
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88 Simple functional programming language. Type declaration is mandatory.
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89 A colon means type, an equal means value. Indentation based.
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90
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91 Set is a base type (which may have a level ).
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92
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93 {} means implicit variable which can be omitted if Agda infers its value.
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94
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95 --data ( Sum type )
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96
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97 A data type which as exclusive multiple constructors. A similar one as
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98 union in C or case class in Scala.
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99
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100 It has a similar syntax as Haskell but it has a slight difference.
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101
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102 data List (A : Set ) : Set where
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103 [] : List A
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104 _∷_ : A → List A → List A
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105
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106 _∷_ means infix operator. If use explicit _, it can be used in a normal function
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107 syntax.
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108
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109 Natural number can be defined as a usual way.
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110
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111 data Nat : Set where
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112 zero : Nat
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113 suc : Nat → Nat
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114
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115 -- A → B means "A implies B"
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116
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117 In Agda, a type can be a value of a variable, which is usually called dependent type.
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118 Type has a name Set in Agda.
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119
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120 ex3 : {A B : Set} → Set
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121 ex3 {A}{B} = A → B
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122
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123 ex3 is a type : A → B, which is a value of Set. It also means a formula : A implies B.
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124
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125 A type is a formula, the value is the proof
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126
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127 A value of A → B can be interpreted as an inference from the formula A to the formula B, which
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128 can be a function from a proof of A to a proof of B.
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129
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130 --introduction と elimination
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131
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132 For a logical operator, there are two types of inference, an introduction and an elimination.
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133
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134 intro creating symbol / constructor / introduction
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135 elim using symbolic / accessors / elimination
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136
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137 In Natural deduction, this can be written in proof schema.
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138
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139 A
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140 :
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141 B A A → B
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142 ------------- →intro ------------------ →elim
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143 A → B B
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144
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145 In Agda, this is a pair of type and value as follows. Introduction of → uses λ.
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146
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147 →intro : {A B : Set } → A → B → ( A → B )
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148 →intro _ b = λ x → b
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149
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150 →elim : {A B : Set } → A → ( A → B ) → B
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151 →elim a f = f a
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152
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153 Important
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154
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155 {A B : Set } → A → B → ( A → B )
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156
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157 is
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158
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159 {A B : Set } → ( A → ( B → ( A → B ) ))
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160
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161 This makes currying of function easy.
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162
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163 -- To prove A → B
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164
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165 Make a left type as an argument. (intros in Coq)
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166
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167 ex5 : {A B C : Set } → A → B → C → ?
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168 ex5 a b c = ?
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169
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170 ? is called a hole, which is unspecified part. Agda tell us which kind type is required for the Hole.
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171
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172 We are going to fill the holes, and if we have no warnings nor errors such as type conflict (Red),
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173 insufficient proof or instance (Yellow), Non-termination, the proof is completed.
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174
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175 -- A ∧ B
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176
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177 Well known conjunction's introduction and elimination is as follow.
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178
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179 A B A ∧ B A ∧ B
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180 ------------- ----------- proj1 ---------- proj2
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181 A ∧ B A B
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182
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183 We can introduce a corresponding structure in our functional programming language.
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184
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185 -- record
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186
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187 record _∧_ A B : Set
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188 field
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189 proj1 : A
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190 proj2 : B
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191
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192 _∧_ means infix operator. _∧_ A B can be written as A ∧ B (Haskell uses (∧) )
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193
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194 This a type which constructed from type A and type B. You may think this as an object
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195 or struct.
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196
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197 record { proj1 = x ; proj2 = y }
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198
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199 is a constructor of _∧_.
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200
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201 ex3 : {A B : Set} → A → B → ( A ∧ B )
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202 ex3 a b = record { proj1 = a ; proj2 = b }
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203
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204 ex1 : {A B : Set} → ( A ∧ B ) → A
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205 ex1 a∧b = proj1 a∧b
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206
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207 a∧b is a variable name. If we have no spaces in a string, it is a word even if we have symbols
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208 except parenthesis or colons. A symbol requires space separation such as a type defining colon.
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209
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210 Defining record can be recursively, but we don't use the recursion here.
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211
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212 -- Mathematical structure
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213
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214 We have types of elements and the relationship in a mathematical structure.
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215
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216 logical relation has no ordering
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217 there is a natural ordering in arguments and a value in a function
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218
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219 So we have typical definition style of mathematical structure with records.
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220
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221 record IsOrdinals {n : Level} (ord : Set n)
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222 (_o<_ : ord → ord → Set n) : Set (suc (suc n)) where
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223 field
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224 Otrans : {x y z : ord } → x o< y → y o< z → x o< z
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225
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226 record Ordinals {n : Level} : Set (suc (suc n)) where
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227 field
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228 ord : Set n
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229 _o<_ : ord → ord → Set n
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230 isOrdinal : IsOrdinals ord _o<_
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231
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232 In IsOrdinals, axioms are written in flat way. In Ordinal, we may have
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233 inputs and outputs are put in the field including IsOrdinal.
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234
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235 Fields of Ordinal is existential objects in the mathematical structure.
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236
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237 -- A Model and a theory
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238
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239 Agda record is a type, so we can write it in the argument, but is it really exists?
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240
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241 If we have a value of the record, it simply exists, that is, we need to create all the existence
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242 in the record satisfies all the axioms (= field of IsOrdinal) should be valid.
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243
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244 type of record = theory
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245 value of record = model
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246
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247 We call the value of the record as a model. If mathematical structure has a
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248 model, it exists. Pretty Obvious.
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249
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250 -- postulate と module
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251
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252 Agda proofs are separated by modules, which are large records.
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253
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254 postulates are assumptions. We can assume a type without proofs.
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255
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256 postulate
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257 sup-o : ( Ordinal → Ordinal ) → Ordinal
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258 sup-o< : { ψ : Ordinal → Ordinal } → ∀ {x : Ordinal } → ψ x o< sup-o ψ
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259
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260 sup-o is an example of upper bound of a function and sup-o< assumes it actually
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261 satisfies all the value is less than upper bound.
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262
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263 Writing some type in a module argument is the same as postulating a type, but
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264 postulate can be written the middle of a proof.
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265
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266 postulate can be constructive.
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267
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336
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268 postulate can be inconsistent, which result everything has a proof. Actualy this assumption
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269 doesnot work for Ordinals, we discuss this later.
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273
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270
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271 -- A ∨ B
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272
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273 data _∨_ (A B : Set) : Set where
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274 case1 : A → A ∨ B
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275 case2 : B → A ∨ B
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276
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277 As Haskell, case1/case2 are patterns.
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278
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279 ex3 : {A B : Set} → ( A ∨ A ) → A
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280 ex3 = ?
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281
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282 In a case statement, Agda command C-C C-C generates possible cases in the head.
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283
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284 ex3 : {A B : Set} → ( A ∨ A ) → A
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285 ex3 (case1 x) = ?
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286 ex3 (case2 x) = ?
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287
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288 Proof schema of ∨ is omit due to the complexity.
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289
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290 -- Negation
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291
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292 ⊥
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293 ------------- ⊥-elim
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294 A
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295
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296 Anything can be derived from bottom, in this case a Set A. There is no introduction rule
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297 in ⊥, which can be implemented as data which has no constructor.
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298
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299 data ⊥ : Set where
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300
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301 ⊥-elim can be proved like this.
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302
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303 ⊥-elim : {A : Set } -> ⊥ -> A
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304 ⊥-elim ()
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305
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306 () means no match argument nor value.
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307
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308 A negation can be defined using ⊥ like this.
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309
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310 ¬_ : Set → Set
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311 ¬ A = A → ⊥
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312
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313 --Equality
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314
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315 All the value in Agda are terms. If we have the same normalized form, two terms are equal.
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316 If we have variables in the terms, we will perform an unification. unifiable terms are equal.
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317 We don't go further on the unification.
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318
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319 { x : A } x ≡ y f x y
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320 --------------- ≡-intro --------------------- ≡-elim
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321 x ≡ x f x x
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322
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323 equality _≡_ can be defined as a data.
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324
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325 data _≡_ {A : Set } : A → A → Set where
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326 refl : {x : A} → x ≡ x
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327
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328 The elimination of equality is a substitution in a term.
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329
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330 subst : {A : Set } → { x y : A } → ( f : A → Set ) → x ≡ y → f x → f y
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331 subst {A} {x} {y} f refl fx = fx
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332
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333 ex5 : {A : Set} {x y z : A } → x ≡ y → y ≡ z → x ≡ z
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334 ex5 {A} {x} {y} {z} x≡y y≡z = subst ( λ k → x ≡ k ) y≡z x≡y
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335
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336
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337 --Equivalence relation
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338
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339 refl' : {A : Set} {x : A } → x ≡ x
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340 refl' = ?
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341 sym : {A : Set} {x y : A } → x ≡ y → y ≡ x
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342 sym = ?
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343 trans : {A : Set} {x y z : A } → x ≡ y → y ≡ z → x ≡ z
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344 trans = ?
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345 cong : {A B : Set} {x y : A } { f : A → B } → x ≡ y → f x ≡ f y
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346 cong = ?
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347
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348 --Ordering
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349
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350 Relation is a predicate on two value which has a same type.
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351
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352 A → A → Set
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353
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354 Defining order is the definition of this type with predicate or a data.
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355
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356 data _≤_ : Rel ℕ 0ℓ where
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357 z≤n : ∀ {n} → zero ≤ n
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358 s≤s : ∀ {m n} (m≤n : m ≤ n) → suc m ≤ suc n
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359
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360
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361 --Quantifier
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362
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363 Handling quantifier in an intuitionistic logic requires special cares.
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364
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365 In the input of a function, there are no restriction on it, that is, it has
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366 a universal quantifier. (If we explicitly write ∀, Agda gives us a type inference on it)
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367
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368 There is no ∃ in agda, the one way is using negation like this.
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369
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370 ∃ (x : A ) → p x = ¬ ( ( x : A ) → ¬ ( p x ) )
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371
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372 On the another way, f : A can be used like this.
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373
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374 p f
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375
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376 If we use a function which can be defined globally which has stronger meaning the
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377 usage of ∃ x in a logical expression.
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378
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379
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380 --Can we do math in this way?
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381
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382 Yes, we can. Actually we have Principia Mathematica by Russell and Whitehead (with out computer support).
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383
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384 In some sense, this story is a reprinting of the work, (but Principia Mathematica has a different formulation than ZF).
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385
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386 define mathematical structure as a record
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387 program inferences as if we have proofs in variables
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388
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389 --Things which Agda cannot prove
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390
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391 The infamous Internal Parametricity is a limitation of Agda, it cannot prove so called Free Theorem, which
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392 leads uniqueness of a functor in Category Theory.
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393
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394 Functional extensionality cannot be proved.
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395 (∀ x → f x ≡ g x) → f ≡ g
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396
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397 Agda has no law of exclude middle.
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398
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399 a ∨ ( ¬ a )
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400
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401 For example, (A → B) → ¬ B → ¬ A can be proved but, ( ¬ B → ¬ A ) → A → B cannot.
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402
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403 It also other problems such as termination, type inference or unification which we may overcome with
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404 efforts or devices or may not.
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405
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406 If we cannot prove something, we can safely postulate it unless it leads a contradiction.
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407
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408 --Classical story of ZF Set Theory
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409
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336
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410 <a name="set-theory">
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273
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411 Assuming ZF, constructing a model of ZF is a flow of classical Set Theory, which leads
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412 a relative consistency proof of the Set Theory.
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413 Ordinal number is used in the flow.
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414
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415 In Agda, first we defines Ordinal numbers (Ordinals), then introduce Ordinal Definable Set (OD).
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416 We need some non constructive assumptions in the construction. A model of Set theory is
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417 constructed based on these assumptions.
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418
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419 <center><img src="fig/set-theory.svg"></center>
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420
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421 --Ordinals
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422
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423 Ordinals are our intuition of infinite things, which has ∅ and orders on the things.
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424 It also has a successor osuc.
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425
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426 record Ordinals {n : Level} : Set (suc (suc n)) where
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427 field
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428 ord : Set n
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429 o∅ : ord
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430 osuc : ord → ord
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431 _o<_ : ord → ord → Set n
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432 isOrdinal : IsOrdinals ord o∅ osuc _o<_
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433
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434 It is different from natural numbers in way. The order of Ordinals is not defined in terms
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435 of successor. It is given from outside, which make it possible to have higher order infinity.
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436
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437 --Axiom of Ordinals
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438
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439 Properties of infinite things. We request a transfinite induction, which states that if
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440 some properties are satisfied below all possible ordinals, the properties are true on all
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441 ordinals.
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442
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443 Successor osuc has no ordinal between osuc and the base ordinal. There are some ordinals
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444 which is not a successor of any ordinals. It is called limit ordinal.
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445
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446 Any two ordinal can be compared, that is less, equal or more, that is total order.
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447
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448 record IsOrdinals {n : Level} (ord : Set n) (o∅ : ord )
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449 (osuc : ord → ord )
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450 (_o<_ : ord → ord → Set n) : Set (suc (suc n)) where
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451 field
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452 Otrans : {x y z : ord } → x o< y → y o< z → x o< z
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453 OTri : Trichotomous {n} _≡_ _o<_
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454 ¬x<0 : { x : ord } → ¬ ( x o< o∅ )
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455 <-osuc : { x : ord } → x o< osuc x
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456 osuc-≡< : { a x : ord } → x o< osuc a → (x ≡ a ) ∨ (x o< a)
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457 TransFinite : { ψ : ord → Set (suc n) }
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458 → ( (x : ord) → ( (y : ord ) → y o< x → ψ y ) → ψ x )
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459 → ∀ (x : ord) → ψ x
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460
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336
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461 --Concrete Ordinals or Countable Ordinals
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273
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462
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463 We can define a list like structure with level, which is a kind of two dimensional infinite array.
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464
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465 data OrdinalD {n : Level} : (lv : Nat) → Set n where
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466 Φ : (lv : Nat) → OrdinalD lv
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467 OSuc : (lv : Nat) → OrdinalD {n} lv → OrdinalD lv
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468
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469 The order of the OrdinalD can be defined in this way.
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470
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471 data _d<_ {n : Level} : {lx ly : Nat} → OrdinalD {n} lx → OrdinalD {n} ly → Set n where
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472 Φ< : {lx : Nat} → {x : OrdinalD {n} lx} → Φ lx d< OSuc lx x
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473 s< : {lx : Nat} → {x y : OrdinalD {n} lx} → x d< y → OSuc lx x d< OSuc lx y
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474
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475 This is a simple data structure, it has no abstract assumptions, and it is countable many data
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476 structure.
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477
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478 Φ 0
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479 OSuc 2 ( Osuc 2 ( Osuc 2 (Φ 2)))
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480 Osuc 0 (Φ 0) d< Φ 1
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481
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482 --Model of Ordinals
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483
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484 It is easy to show OrdinalD and its order satisfies the axioms of Ordinals.
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485
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486 So our Ordinals has a mode. This means axiom of Ordinals are consistent.
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487
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488 --Debugging axioms using Model
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489
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490 Whether axiom is correct or not can be checked by a validity on a mode.
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491
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492 If not, we may fix the axioms or the model, such as the definitions of the order.
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493
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494 We can also ask whether the inputs exist.
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495
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496 --Countable Ordinals can contains uncountable set?
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497
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498 Yes, the ordinals contains any level of infinite Set in the axioms.
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499
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500 If we handle real-number in the model, only countable number of real-number is used.
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501
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502 from the outside view point, it is countable
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503 from the internal view point, it is uncountable
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504
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505 The definition of countable/uncountable is the same, but the properties are different
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506 depending on the context.
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507
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508 We don't show the definition of cardinal number here.
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509
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510 --What is Set
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511
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512 The word Set in Agda is not a Set of ZF Set, but it is a type (why it is named Set?).
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513
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514 From naive point view, a set i a list, but in Agda, elements have all the same type.
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515 A set in ZF may contain other Sets in ZF, which not easy to implement it as a list.
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516
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517 Finite set may be written in finite series of ∨, but ...
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518
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519 --We don't ask the contents of Set. It can be anything.
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520
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521 From empty set φ, we can think a set contains a φ, and a pair of φ and the set, and so on,
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522 and all of them, and again we repeat this.
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523
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524 φ {φ} {φ,{φ}}, {φ,{φ},...}
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525
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526 It is called V.
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527
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528 This operation can be performed within a ZF Set theory. Classical Set Theory assumes
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529 ZF, so this kind of thing is allowed.
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530
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531 But in our case, we have no ZF theory, so we are going to use Ordinals.
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532
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336
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533 The idea is to use an ordinal as a pointer to a record which defines a Set.
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534 If the recored defines a series of Ordinals which is a pointer to the Set.
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535 This record looks like a Set.
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273
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536
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537 --Ordinal Definable Set
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538
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539 We can define a sbuset of Ordinals using predicates. What is a subset?
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540
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541 a predicate has an Ordinal argument
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542
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543 is an Ordinal Definable Set (OD). In Agda, OD is defined as follows.
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544
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545 record OD : Set (suc n ) where
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546 field
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547 def : (x : Ordinal ) → Set n
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548
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549 Ordinals itself is not a set in a ZF Set theory but a class. In OD,
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550
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336
|
551 data One : Set n where
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552 OneObj : One
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273
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553
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336
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554 record { def = λ x → One }
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555
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556 means it accepets all Ordinals, i.e. this is Ordinals itself, so ODs are larger than ZF Set.
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557 You can say OD is a class in ZF Set Theory term.
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273
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558
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559
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336
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560 --OD is not ZF Set
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561
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562 If we have 1 to 1 mapping between an OD and an Ordinal, OD contains several ODs and OD looks like
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563 a Set. The idea is to use an ordinal as a pointer to OD.
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564 Unfortunately this scheme does not work well. As we saw OD includes all Ordinals,
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565 which is a maximum of OD, but Ordinals has no maximum at all. So we have a contradction like
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273
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566
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336
|
567 ¬OD-order : ( od→ord : OD → Ordinal )
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568 → ( ord→od : Ordinal → OD ) → ( { x y : OD } → def y ( od→ord x ) → od→ord x o< od→ord y) → ⊥
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569 ¬OD-order od→ord ord→od c<→o< = ?
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570
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571 Actualy we can prove this contrdction, so we need some restrctions on OD.
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572
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573 This is a kind of Russel paradox, that is if OD contains everything, what happens if it contains itself.
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574
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575 -- HOD : Hereditarily Ordinal Definable
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576
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577 What we need is a bounded OD, the containment is limited by an ordinal.
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273
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578
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336
|
579 record HOD : Set (suc n) where
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580 field
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581 od : OD
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582 odmax : Ordinal
|
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583 <odmax : {y : Ordinal} → def od y → y o< odmax
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584
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585 In classical Set Theory, HOD stands for Hereditarily Ordinal Definable, which means
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586
|
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|
587 HOD = { x | TC x ⊆ OD }
|
|
273
|
588
|
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336
|
589 TC x is all transitive closure of x, that is elements of x and following all elements of them are all OD. But
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590 what is x? In this case, x is an Set which we don't have yet. In our case, HOD is a bounded OD.
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591
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592 --1 to 1 mapping between an HOD and an Ordinal
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593
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594 HOD is a predicate on Ordinals and the solution is bounded by some ordinal. If we have a mapping
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595
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596 od→ord : HOD → Ordinal
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597 ord→od : Ordinal → HOD
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598 oiso : {x : HOD } → ord→od ( od→ord x ) ≡ x
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599 diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x
|
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|
600
|
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|
601 we can check an HOD is an element of the OD using def.
|
|
273
|
602
|
|
|
603 A ∋ x can be define as follows.
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604
|
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336
|
605 _∋_ : ( A x : HOD ) → Set n
|
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|
606 _∋_ A x = def (od A) ( od→ord x )
|
|
273
|
607
|
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|
608 In ψ : Ordinal → Set, if A is a record { def = λ x → ψ x } , then
|
|
|
609
|
|
|
610 A x = def A ( od→ord x ) = ψ (od→ord x)
|
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|
611
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612 They say the existing of the mappings can be proved in Classical Set Theory, but we
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|
613 simply assumes these non constructively.
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614
|
|
|
615 <center><img src="fig/ord-od-mapping.svg"></center>
|
|
|
616
|
|
|
617 --Order preserving in the mapping of OD and Ordinal
|
|
|
618
|
|
336
|
619 Ordinals have the order and HOD has a natural order based on inclusion ( def / ∋ ).
|
|
273
|
620
|
|
336
|
621 def (od y) ( od→ord x )
|
|
273
|
622
|
|
336
|
623 An elements of HOD should be defined before the HOD, that is, an ordinal corresponding an elements
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|
|
624 have to be smaller than the corresponding ordinal of the containing OD.
|
|
|
625 We also assumes subset is always smaller. This is necessary to make a limit of Power Set.
|
|
273
|
626
|
|
336
|
627 c<→o< : {x y : HOD } → def (od y) ( od→ord x ) → od→ord x o< od→ord y
|
|
|
628 ⊆→o≤ : {y z : HOD } → ({x : Ordinal} → def (od y) x → def (od z) x ) → od→ord y o< osuc (od→ord z)
|
|
273
|
629
|
|
336
|
630 If wa assumes reverse order preservation,
|
|
273
|
631
|
|
|
632 o<→c< : {n : Level} {x y : Ordinal } → x o< y → def (ord→od y) x
|
|
|
633
|
|
336
|
634 ∀ x ∋ ∅ becomes true, which manes all OD becomes Ordinals in the model.
|
|
273
|
635
|
|
|
636 <center><img src="fig/ODandOrdinals.svg"></center>
|
|
|
637
|
|
|
638 --Various Sets
|
|
|
639
|
|
|
640 In classical Set Theory, there is a hierarchy call L, which can be defined by a predicate.
|
|
|
641
|
|
|
642 Ordinal / things satisfies axiom of Ordinal / extension of natural number
|
|
|
643 V / hierarchical construction of Set from φ
|
|
|
644 L / hierarchical predicate definable construction of Set from φ
|
|
336
|
645 HOD / Hereditarily Ordinal Definable
|
|
273
|
646 OD / equational formula on Ordinals
|
|
|
647 Agda Set / Type / it also has a level
|
|
|
648
|
|
|
649
|
|
|
650 --Fixes on ZF to intuitionistic logic
|
|
|
651
|
|
|
652 We use ODs as Sets in ZF, and defines record ZF, that is, we have to define
|
|
|
653 ZF axioms in Agda.
|
|
|
654
|
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|
655 It may not valid in our model. We have to debug it.
|
|
|
656
|
|
|
657 Fixes are depends on axioms.
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|
|
658
|
|
|
659 <center><img src="fig/axiom-type.svg"></center>
|
|
|
660
|
|
|
661 <a href="fig/zf-record.html">
|
|
|
662 ZFのrecord
|
|
|
663 </a>
|
|
|
664
|
|
|
665 --Pure logical axioms
|
|
|
666
|
|
279
|
667 empty, pair, select, ε-induction??infinity
|
|
273
|
668
|
|
|
669 These are logical relations among OD.
|
|
|
670
|
|
|
671 empty : ∀( x : ZFSet ) → ¬ ( ∅ ∋ x )
|
|
|
672 pair→ : ( x y t : ZFSet ) → (x , y) ∋ t → ( t ≈ x ) ∨ ( t ≈ y )
|
|
|
673 pair← : ( x y t : ZFSet ) → ( t ≈ x ) ∨ ( t ≈ y ) → (x , y) ∋ t
|
|
|
674 selection : { ψ : ZFSet → Set m } → ∀ { X y : ZFSet } → ( ( y ∈ X ) ∧ ψ y ) ⇔ (y ∈ Select X ψ )
|
|
|
675 infinity∅ : ∅ ∈ infinite
|
|
|
676 infinity : ∀( x : ZFSet ) → x ∈ infinite → ( x ∪ ( x , x ) ) ∈ infinite
|
|
|
677 ε-induction : { ψ : OD → Set (suc n)}
|
|
|
678 → ( {x : OD } → ({ y : OD } → x ∋ y → ψ y ) → ψ x )
|
|
|
679 → (x : OD ) → ψ x
|
|
|
680
|
|
|
681 finitely can be define by Agda data.
|
|
|
682
|
|
|
683 data infinite-d : ( x : Ordinal ) → Set n where
|
|
|
684 iφ : infinite-d o∅
|
|
|
685 isuc : {x : Ordinal } → infinite-d x →
|
|
|
686 infinite-d (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) ))
|
|
|
687
|
|
|
688 Union (x , ( x , x )) should be an direct successor of x, but we cannot prove it in our model.
|
|
|
689
|
|
|
690 --Axiom of Pair
|
|
|
691
|
|
|
692 In the Tanaka's book, axiom of pair is as follows.
|
|
|
693
|
|
|
694 ∀ x ∀ y ∃ z ∀ t ( z ∋ t ↔ t ≈ x ∨ t ≈ y)
|
|
|
695
|
|
|
696 We have fix ∃ z, a function (x , y) is defined, which is _,_ .
|
|
|
697
|
|
|
698 _,_ : ( A B : ZFSet ) → ZFSet
|
|
|
699
|
|
|
700 using this, we can define two directions in separates axioms, like this.
|
|
|
701
|
|
|
702 pair→ : ( x y t : ZFSet ) → (x , y) ∋ t → ( t ≈ x ) ∨ ( t ≈ y )
|
|
|
703 pair← : ( x y t : ZFSet ) → ( t ≈ x ) ∨ ( t ≈ y ) → (x , y) ∋ t
|
|
|
704
|
|
|
705 This is already written in Agda, so we use these as axioms. All inputs have ∀.
|
|
|
706
|
|
|
707 --pair in OD
|
|
|
708
|
|
|
709 OD is an equation on Ordinals, we can simply write axiom of pair in the OD.
|
|
|
710
|
|
336
|
711 _,_ : HOD → HOD → HOD
|
|
|
712 x , y = record { od = record { def = λ t → (t ≡ od→ord x ) ∨ ( t ≡ od→ord y ) } ; odmax = ? ; <odmax = ? }
|
|
|
713
|
|
|
714 It is easy to find out odmax from odmax of x and y.
|
|
273
|
715
|
|
|
716 ≡ is an equality of λ terms, but please not that this is equality on Ordinals.
|
|
|
717
|
|
|
718 --Validity of Axiom of Pair
|
|
|
719
|
|
|
720 Assuming ZFSet is OD, we are going to prove pair→ .
|
|
|
721
|
|
|
722 pair→ : ( x y t : OD ) → (x , y) ∋ t → ( t == x ) ∨ ( t == y )
|
|
|
723 pair→ x y t p = ?
|
|
|
724
|
|
|
725 In this program, type of p is ( x , y ) ∋ t , that is
|
|
|
726 def ( x , y ) that is, (t ≡ od→ord x ) ∨ ( t ≡ od→ord y ) .
|
|
|
727
|
|
|
728 Since _∨_ is a data, it can be developed as (C-c C-c : agda2-make-case ).
|
|
|
729
|
|
|
730 pair→ x y t (case1 t≡x ) = ?
|
|
|
731 pair→ x y t (case2 t≡y ) = ?
|
|
|
732
|
|
|
733 The type of the ? is ( t == x ) ∨ ( t == y ), again it is data _∨_ .
|
|
|
734
|
|
|
735 pair→ x y t (case1 t≡x ) = case1 ?
|
|
|
736 pair→ x y t (case2 t≡y ) = case2 ?
|
|
|
737
|
|
|
738 The ? in case1 is t == x, so we have to create this from t≡x, which is a name of a variable
|
|
|
739 which type is
|
|
|
740
|
|
|
741 t≡x : od→ord t ≡ od→ord x
|
|
|
742
|
|
|
743 which is shown by an Agda command (C-C C-E : agda2-show-context ).
|
|
|
744
|
|
|
745 But we haven't defined == yet.
|
|
|
746
|
|
|
747 --Equality of OD and Axiom of Extensionality
|
|
|
748
|
|
|
749 OD is defined by a predicates, if we compares normal form of the predicates, even if
|
|
|
750 it contains the same elements, it may be different, which is no good as an equality of
|
|
|
751 Sets.
|
|
|
752
|
|
|
753 Axiom of Extensionality requires sets having the same elements are handled in the same way
|
|
|
754 each other.
|
|
|
755
|
|
|
756 ∀ z ( z ∈ x ⇔ z ∈ y ) ⇒ ∀ w ( x ∈ w ⇔ y ∈ w )
|
|
|
757
|
|
|
758 We can write this axiom in Agda as follows.
|
|
|
759
|
|
|
760 extensionality : { A B w : ZFSet } → ( (z : ZFSet) → ( A ∋ z ) ⇔ (B ∋ z) ) → ( A ∈ w ⇔ B ∈ w )
|
|
|
761
|
|
|
762 So we use ( A ∋ z ) ⇔ (B ∋ z) as an equality (_==_) of our model. We have to show
|
|
|
763 A ∈ w ⇔ B ∈ w from A == B.
|
|
|
764
|
|
|
765 x == y can be defined in this way.
|
|
|
766
|
|
|
767 record _==_ ( a b : OD ) : Set n where
|
|
|
768 field
|
|
|
769 eq→ : ∀ { x : Ordinal } → def a x → def b x
|
|
|
770 eq← : ∀ { x : Ordinal } → def b x → def a x
|
|
|
771
|
|
336
|
772 Actually, (z : HOD) → (A ∋ z) ⇔ (B ∋ z) implies od A == od B.
|
|
273
|
773
|
|
336
|
774 extensionality0 : {A B : HOD } → ((z : HOD) → (A ∋ z) ⇔ (B ∋ z)) → od A == od B
|
|
273
|
775 eq→ (extensionality0 {A} {B} eq ) {x} d = ?
|
|
|
776 eq← (extensionality0 {A} {B} eq ) {x} d = ?
|
|
|
777
|
|
|
778 ? are def B x and def A x and these are generated from eq : (z : OD) → (A ∋ z) ⇔ (B ∋ z) .
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|
779
|
|
|
780 Actual proof is rather complicated.
|
|
|
781
|
|
336
|
782 eq→ (extensionality0 {A} {B} eq ) {x} d = odef-iso {A} {B} (sym diso) (proj1 (eq (ord→od x))) d
|
|
|
783 eq← (extensionality0 {A} {B} eq ) {x} d = odef-iso {B} {A} (sym diso) (proj2 (eq (ord→od x))) d
|
|
273
|
784
|
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|
785 where
|
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|
786
|
|
336
|
787 odef-iso : {A B : HOD } {x y : Ordinal } → x ≡ y → (def A (od y) → def (od B) y) → def (od A) x → def (od B) x
|
|
|
788 odef-iso refl t = t
|
|
273
|
789
|
|
|
790 --Validity of Axiom of Extensionality
|
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|
791
|
|
336
|
792 If we can derive (w ∋ A) ⇔ (w ∋ B) from od A == od B, the axiom becomes valid, but it seems impossible,
|
|
273
|
793 so we assumes
|
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|
794
|
|
336
|
795 ==→o≡ : { x y : HOD } → (od x == od y) → x ≡ y
|
|
273
|
796
|
|
|
797 Using this, we have
|
|
|
798
|
|
336
|
799 extensionality : {A B w : HOD } → ((z : HOD ) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B)
|
|
273
|
800 proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d
|
|
336
|
801 proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d
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|
273
|
802
|
|
|
803 --Non constructive assumptions so far
|
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804
|
|
336
|
805 od→ord : HOD → Ordinal
|
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|
806 ord→od : Ordinal → HOD
|
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|
807 c<→o< : {x y : HOD } → def (od y) ( od→ord x ) → od→ord x o< od→ord y
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|
|
808 ⊆→o≤ : {y z : HOD } → ({x : Ordinal} → def (od y) x → def (od z) x ) → od→ord y o< osuc (od→ord z)
|
|
|
809 oiso : {x : HOD } → ord→od ( od→ord x ) ≡ x
|
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|
810 diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x
|
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|
811 ==→o≡ : {x y : HOD } → (od x == od y) → x ≡ y
|
|
|
812 sup-o : (A : HOD) → ( ( x : Ordinal ) → def (od A) x → Ordinal ) → Ordinal
|
|
|
813 sup-o< : (A : HOD) → { ψ : ( x : Ordinal ) → def (od A) x → Ordinal } → ∀ {x : Ordinal } → (lt : def (od A) x ) → ψ x lt o< sup-o A ψ
|
|
273
|
814
|
|
|
815
|
|
|
816 --Axiom which have negation form
|
|
|
817
|
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|
818 Union, Selection
|
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|
819
|
|
|
820 These axioms contains ∃ x as a logical relation, which can be described in ¬ ( ∀ x ( ¬ p )).
|
|
|
821
|
|
|
822 Axiom of replacement uses upper bound of function on Ordinals, which makes it non-constructive.
|
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|
823
|
|
|
824 Power Set axiom requires double negation,
|
|
|
825
|
|
|
826 power→ : ∀( A t : ZFSet ) → Power A ∋ t → ∀ {x} → t ∋ x → ¬ ¬ ( A ∋ x )
|
|
|
827 power← : ∀( A t : ZFSet ) → t ⊆_ A → Power A ∋ t
|
|
|
828
|
|
|
829 If we have an assumption of law of exclude middle, we can recover the original A ∋ x form.
|
|
|
830
|
|
|
831 --Union
|
|
|
832
|
|
|
833 The original form of the Axiom of Union is
|
|
|
834
|
|
|
835 ∀ x ∃ y ∀ z (z ∈ y ⇔ ∃ u ∈ x ∧ (z ∈ u))
|
|
|
836
|
|
|
837 Union requires the existence of b in a ⊇ ∃ b ∋ x . We will use negation form of ∃.
|
|
|
838
|
|
|
839 union→ : ( X z u : ZFSet ) → ( X ∋ u ) ∧ (u ∋ z ) → Union X ∋ z
|
|
|
840 union← : ( X z : ZFSet ) → (X∋z : Union X ∋ z ) → ¬ ( (u : ZFSet ) → ¬ ((X ∋ u) ∧ (u ∋ z )))
|
|
|
841
|
|
|
842 The definition of Union in OD is like this.
|
|
|
843
|
|
|
844 Union : OD → OD
|
|
|
845 Union U = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((def U u) ∧ (def (ord→od u) x))) }
|
|
|
846
|
|
|
847 Proof of validity is straight forward.
|
|
|
848
|
|
|
849 union→ : (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z
|
|
|
850 union→ X z u xx not = ⊥-elim ( not (od→ord u) ( record { proj1 = proj1 xx
|
|
|
851 ; proj2 = subst ( λ k → def k (od→ord z)) (sym oiso) (proj2 xx) } ))
|
|
|
852 union← : (X z : OD) (X∋z : Union X ∋ z) → ¬ ( (u : OD ) → ¬ ((X ∋ u) ∧ (u ∋ z )))
|
|
|
853 union← X z UX∋z = FExists _ lemma UX∋z where
|
|
|
854 lemma : {y : Ordinal} → def X y ∧ def (ord→od y) (od→ord z) → ¬ ((u : OD) → ¬ (X ∋ u) ∧ (u ∋ z))
|
|
|
855 lemma {y} xx not = not (ord→od y) record { proj1 = subst ( λ k → def X k ) (sym diso) (proj1 xx ) ; proj2 = proj2 xx }
|
|
|
856
|
|
|
857 where
|
|
|
858
|
|
|
859 FExists : {m l : Level} → ( ψ : Ordinal → Set m )
|
|
|
860 → {p : Set l} ( P : { y : Ordinal } → ψ y → ¬ p )
|
|
|
861 → (exists : ¬ (∀ y → ¬ ( ψ y ) ))
|
|
|
862 → ¬ p
|
|
|
863 FExists {m} {l} ψ {p} P = contra-position ( λ p y ψy → P {y} ψy p )
|
|
|
864
|
|
|
865 which checks existence using contra-position.
|
|
|
866
|
|
|
867 --Axiom of replacement
|
|
|
868
|
|
|
869 We can replace the elements of a set by a function and it becomes a set. From the book,
|
|
|
870
|
|
|
871 ∀ x ∀ y ∀ z ( ( ψ ( x , y ) ∧ ψ ( x , z ) ) → y = z ) → ∀ X ∃ A ∀ y ( y ∈ A ↔ ∃ x ∈ X ψ ( x , y ) )
|
|
|
872
|
|
|
873 The existential quantifier can be related by a function,
|
|
|
874
|
|
|
875 Replace : OD → (OD → OD ) → OD
|
|
|
876
|
|
|
877 The axioms becomes as follows.
|
|
|
878
|
|
|
879 replacement← : {ψ : ZFSet → ZFSet} → ∀ ( X x : ZFSet ) → x ∈ X → ψ x ∈ Replace X ψ
|
|
|
880 replacement→ : {ψ : ZFSet → ZFSet} → ∀ ( X x : ZFSet ) → ( lt : x ∈ Replace X ψ ) → ¬ ( ∀ (y : ZFSet) → ¬ ( x ≈ ψ y ) )
|
|
|
881
|
|
|
882 In the axiom, the existence of the original elements is necessary. In order to do that we use OD which has
|
|
|
883 negation form of existential quantifier in the definition.
|
|
|
884
|
|
|
885 in-codomain : (X : OD ) → ( ψ : OD → OD ) → OD
|
|
|
886 in-codomain X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( def X y ∧ ( x ≡ od→ord (ψ (ord→od y ))))) }
|
|
|
887
|
|
|
888 Besides this upper bounds is required.
|
|
|
889
|
|
|
890 Replace : OD → (OD → OD ) → OD
|
|
|
891 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 }
|
|
|
892
|
|
|
893 We omit the proof of the validity, but it is rather straight forward.
|
|
|
894
|
|
|
895 --Validity of Power Set Axiom
|
|
|
896
|
|
|
897 The original Power Set Axiom is this.
|
|
|
898
|
|
|
899 ∀ X ∃ A ∀ t ( t ∈ A ↔ t ⊆ X ) )
|
|
|
900
|
|
|
901 The existential quantifier is replaced by a function
|
|
|
902
|
|
|
903 Power : ( A : OD ) → OD
|
|
|
904
|
|
|
905 t ⊆ X is a record like this.
|
|
|
906
|
|
|
907 record _⊆_ ( A B : OD ) : Set (suc n) where
|
|
|
908 field
|
|
|
909 incl : { x : OD } → A ∋ x → B ∋ x
|
|
|
910
|
|
|
911 Axiom becomes likes this.
|
|
|
912
|
|
|
913 power→ : ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x)
|
|
|
914 power← : (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t
|
|
|
915
|
|
|
916 The validity of the axioms are slight complicated, we have to define set of all subset. We define
|
|
|
917 subset in a different form.
|
|
|
918
|
|
|
919 ZFSubset : (A x : OD ) → OD
|
|
|
920 ZFSubset A x = record { def = λ y → def A y ∧ def x y }
|
|
|
921
|
|
|
922 We can prove,
|
|
|
923
|
|
|
924 ( {y : OD } → x ∋ y → ZFSubset A x ∋ y ) ⇔ ( x ⊆ A )
|
|
|
925
|
|
|
926 We only have upper bound as an ordinal, but we have an obvious OD based on the order of Ordinals,
|
|
|
927 which is an Ordinals with our Model.
|
|
|
928
|
|
|
929 Ord : ( a : Ordinal ) → OD
|
|
|
930 Ord a = record { def = λ y → y o< a }
|
|
|
931
|
|
|
932 Def : (A : OD ) → OD
|
|
|
933 Def A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) )
|
|
|
934
|
|
|
935 This is slight larger than Power A, so we replace all elements x by A ∩ x (some of them may empty).
|
|
|
936
|
|
|
937 Power : OD → OD
|
|
|
938 Power A = Replace (Def (Ord (od→ord A))) ( λ x → A ∩ x )
|
|
|
939
|
|
|
940 Creating Power Set of Ordinals is rather easy, then we use replacement axiom on A ∩ x since we have this.
|
|
|
941
|
|
|
942 ∩-≡ : { a b : OD } → ({x : OD } → (a ∋ x → b ∋ x)) → a == ( b ∩ a )
|
|
|
943
|
|
|
944 In case of Ord a intro of Power Set axiom becomes valid.
|
|
|
945
|
|
|
946 ord-power← : (a : Ordinal ) (t : OD) → ({x : OD} → (t ∋ x → (Ord a) ∋ x)) → Def (Ord a) ∋ t
|
|
|
947
|
|
|
948 Using this, we can prove,
|
|
|
949
|
|
|
950 power→ : ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x)
|
|
|
951 power← : (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t
|
|
|
952
|
|
|
953
|
|
|
954 --Axiom of regularity, Axiom of choice, ε-induction
|
|
|
955
|
|
|
956 Axiom of regularity requires non self intersectable elements (which is called minimum), if we
|
|
|
957 replace it by a function, it becomes a choice function. It makes axiom of choice valid.
|
|
|
958
|
|
|
959 This means we cannot prove axiom regularity form our model, and if we postulate this, axiom of
|
|
|
960 choice also becomes valid.
|
|
|
961
|
|
|
962 minimal : (x : OD ) → ¬ (x == od∅ )→ OD
|
|
|
963 x∋minimal : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → def x ( od→ord ( minimal x ne ) )
|
|
|
964 minimal-1 : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → (y : OD ) → ¬ ( def (minimal x ne) (od→ord y)) ∧ (def x (od→ord y) )
|
|
|
965
|
|
|
966 We can avoid this using ε-induction (a predicate is valid on all set if the predicate is true on some element of set).
|
|
|
967 Assuming law of exclude middle, they say axiom of regularity will be proved, but we haven't check it yet.
|
|
|
968
|
|
|
969 ε-induction : { ψ : OD → Set (suc n)}
|
|
|
970 → ( {x : OD } → ({ y : OD } → x ∋ y → ψ y ) → ψ x )
|
|
|
971 → (x : OD ) → ψ x
|
|
|
972
|
|
|
973 In our model, we assumes the mapping between Ordinals and OD, this is actually the TransFinite induction in Ordinals.
|
|
|
974
|
|
|
975 The axiom of choice in the book is complicated using any pair in a set, so we use use a form in the Wikipedia.
|
|
|
976
|
|
|
977 ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ]
|
|
|
978
|
|
|
979 We can formulate like this.
|
|
|
980
|
|
|
981 choice-func : (X : ZFSet ) → {x : ZFSet } → ¬ ( x ≈ ∅ ) → ( X ∋ x ) → ZFSet
|
|
|
982 choice : (X : ZFSet ) → {A : ZFSet } → ( X∋A : X ∋ A ) → (not : ¬ ( A ≈ ∅ )) → A ∋ choice-func X not X∋A
|
|
|
983
|
|
|
984 It does not requires ∅ ∉ X .
|
|
|
985
|
|
|
986 --Axiom of choice and Law of Excluded Middle
|
|
|
987
|
|
|
988 In our model, since OD has a mapping to Ordinals, it has evident order, which means well ordering theorem is valid,
|
|
|
989 but it don't have correct form of the axiom yet. They say well ordering axiom is equivalent to the axiom of choice,
|
|
|
990 but it requires law of the exclude middle.
|
|
|
991
|
|
|
992 Actually, it is well known to prove law of the exclude middle from axiom of choice in intuitionistic logic, and we can
|
|
|
993 perform the proof in our mode. Using the definition like this, predicates and ODs are related and we can ask the
|
|
|
994 set is empty or not if we have an axiom of choice, so we have the law of the exclude middle p ∨ ( ¬ p ) .
|
|
|
995
|
|
|
996 ppp : { p : Set n } { a : OD } → record { def = λ x → p } ∋ a → p
|
|
|
997 ppp {p} {a} d = d
|
|
|
998
|
|
|
999 We can prove axiom of choice from law excluded middle since we have TransFinite induction. So Axiom of choice
|
|
|
1000 and Law of Excluded Middle is equivalent in our mode.
|
|
|
1001
|
|
|
1002 --Relation-ship among ZF axiom
|
|
|
1003
|
|
|
1004 <center><img src="fig/axiom-dependency.svg"></center>
|
|
|
1005
|
|
|
1006 --Non constructive assumption in our model
|
|
|
1007
|
|
|
1008 mapping between OD and Ordinals
|
|
|
1009
|
|
|
1010 od→ord : OD → Ordinal
|
|
|
1011 ord→od : Ordinal → OD
|
|
|
1012 oiso : {x : OD } → ord→od ( od→ord x ) ≡ x
|
|
|
1013 diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x
|
|
|
1014 c<→o< : {x y : OD } → def y ( od→ord x ) → od→ord x o< od→ord y
|
|
|
1015
|
|
|
1016 Equivalence on OD
|
|
|
1017
|
|
|
1018 ==→o≡ : { x y : OD } → (x == y) → x ≡ y
|
|
|
1019
|
|
|
1020 Upper bound
|
|
|
1021
|
|
|
1022 sup-o : ( Ordinal → Ordinal ) → Ordinal
|
|
|
1023 sup-o< : { ψ : Ordinal → Ordinal } → ∀ {x : Ordinal } → ψ x o< sup-o ψ
|
|
|
1024
|
|
|
1025 Axiom of choice and strong axiom of regularity.
|
|
|
1026
|
|
|
1027 minimal : (x : OD ) → ¬ (x == od∅ )→ OD
|
|
|
1028 x∋minimal : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → def x ( od→ord ( minimal x ne ) )
|
|
|
1029 minimal-1 : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → (y : OD ) → ¬ ( def (minimal x ne) (od→ord y)) ∧ (def x (od→ord y) )
|
|
|
1030
|
|
|
1031 --So it this correct?
|
|
|
1032
|
|
|
1033 Our axiom are syntactically the same in the text book, but negations are slightly different.
|
|
|
1034
|
|
|
1035 If we assumes excluded middle, these are exactly same.
|
|
|
1036
|
|
|
1037 Even if we assumes excluded middle, intuitionistic logic itself remains consistent, but we cannot prove it.
|
|
|
1038
|
|
|
1039 Except the upper bound, axioms are simple logical relation.
|
|
|
1040
|
|
|
1041 Proof of existence of mapping between OD and Ordinals are not obvious. We don't know we prove it or not.
|
|
|
1042
|
|
|
1043 Existence of the Upper bounds is a pure assumption, if we have not limit on Ordinals, it may contradicts,
|
|
|
1044 but we don't have explicit upper limit on Ordinals.
|
|
|
1045
|
|
|
1046 Several inference on our model or our axioms are basically parallel to the set theory text book, so it looks like correct.
|
|
|
1047
|
|
|
1048 --How to use Agda Set Theory
|
|
|
1049
|
|
|
1050 Assuming record ZF, classical set theory can be developed. If necessary, axiom of choice can be
|
|
|
1051 postulated or assuming law of excluded middle.
|
|
|
1052
|
|
|
1053 Instead, simply assumes non constructive assumption, various theory can be developed. We haven't check
|
|
|
1054 these assumptions are proved in record ZF, so we are not sure, these development is a result of ZF Set theory.
|
|
|
1055
|
|
|
1056 ZF record itself is not necessary, for example, topology theory without ZF can be possible.
|
|
|
1057
|
|
|
1058 --Topos and Set Theory
|
|
|
1059
|
|
|
1060 Topos is a mathematical structure in Category Theory, which is a Cartesian closed category which has a
|
|
|
1061 sub-object classifier.
|
|
|
1062
|
|
|
1063 Topos itself is model of intuitionistic logic.
|
|
|
1064
|
|
|
1065 Transitive Sets are objects of Cartesian closed category.
|
|
|
1066 It is possible to introduce Power Set Functor on it
|
|
|
1067
|
|
|
1068 We can use replacement A ∩ x for each element in Transitive Set,
|
|
|
1069 in the similar way of our power set axiom. I
|
|
|
1070
|
|
|
1071 A model of ZF Set theory can be constructed on top of the Topos which is shown in Oisus.
|
|
|
1072
|
|
|
1073 Our Agda model is a proof theoretic version of it.
|
|
|
1074
|
|
|
1075 --Cardinal number and Continuum hypothesis
|
|
|
1076
|
|
|
1077 Axiom of choice is required to define cardinal number
|
|
|
1078
|
|
|
1079 definition of cardinal number is not yet done
|
|
|
1080
|
|
|
1081 definition of filter is not yet done
|
|
|
1082
|
|
|
1083 we may have a model without axiom of choice or without continuum hypothesis
|
|
|
1084
|
|
|
1085 Possible representation of continuum hypothesis is this.
|
|
|
1086
|
|
|
1087 continuum-hyphotheis : (a : Ordinal) → Power (Ord a) ⊆ Ord (osuc a)
|
|
|
1088
|
|
|
1089 --Filter
|
|
|
1090
|
|
|
1091 filter is a dual of ideal on boolean algebra or lattice. Existence on natural number
|
|
|
1092 is depends on axiom of choice.
|
|
|
1093
|
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1094 record Filter ( L : OD ) : Set (suc n) where
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1095 field
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1096 filter : OD
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1097 proper : ¬ ( filter ∋ od∅ )
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1098 inL : filter ⊆ L
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1099 filter1 : { p q : OD } → q ⊆ L → filter ∋ p → p ⊆ q → filter ∋ q
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1100 filter2 : { p q : OD } → filter ∋ p → filter ∋ q → filter ∋ (p ∩ q)
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1101
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1102 We may construct a model of non standard analysis or set theory.
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1103
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1104 This may be simpler than classical forcing theory ( not yet done).
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1105
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1106 --Programming Mathematics
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1107
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1108 Mathematics is a functional programming in Agda where proof is a value of a variable. The mathematical
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1109 structure are
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1110
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1111 record and data
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1112
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1113 Proof is check by type consistency not by the computation, but it may include some normalization.
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1114
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1115 Type inference and termination is not so clear in multi recursions.
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1116
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1117 Defining Agda record is a good way to understand mathematical theory, for examples,
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1118
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1119 Category theory ( Yoneda lemma, Floyd Adjunction functor theorem, Applicative functor )
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1120 Automaton ( Subset construction、Language containment)
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1121
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1122 are proved in Agda.
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1123
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1124 --link
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1125
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1126 Summer school of foundation of mathematics (in Japanese)
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1127 <br> <a href="https://www.sci.shizuoka.ac.jp/~math/yorioka/ss2019/">
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1128 https://www.sci.shizuoka.ac.jp/~math/yorioka/ss2019/
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1129 </a>
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1130
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1131 Foundation of axiomatic set theory (in Japanese)
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1132 <br> <a href="https://www.sci.shizuoka.ac.jp/~math/yorioka/ss2019/sakai0.pdf">
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1133 https://www.sci.shizuoka.ac.jp/~math/yorioka/ss2019/sakai0.pdf
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1134 </a>
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1135
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1136 Agda
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1137 <br> <a href="https://agda.readthedocs.io/en/v2.6.0.1/">
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1138 https://agda.readthedocs.io/en/v2.6.0.1/
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1139 </a>
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1140
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1141 ZF-in-Agda source
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1142 <br> <a href="https://github.com/shinji-kono/zf-in-agda.git">
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1143 https://github.com/shinji-kono/zf-in-agda.git
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1144 </a>
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1145
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1146 Category theory in Agda source
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1147 <br> <a href="https://github.com/shinji-kono/category-exercise-in-agda">
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1148 https://github.com/shinji-kono/category-exercise-in-agda
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1149 </a>
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1150
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1151
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1152
|
