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23 </script>
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24 <title>Constructing ZF Set Theory in Agda </title>
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25 </head>
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26 <body>
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27 <div class="main" id="mmm">
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28 <h1>Constructing ZF Set Theory in Agda </h1>
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29 <a href="#" right="0px" onclick="javascript:showElement('menu')">
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30 <span>Menu</span>
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31 </a>
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32 <a href="#" left="0px" onclick="javascript:showElement('menu')">
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33 <span>Menu</span>
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34 </a>
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35
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36 <p>
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37
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38 <author> Shinji KONO</author>
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39
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40 <hr/>
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41 <h2><a name="content000">Programming Mathematics</a></h2>
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42 Programming is processing data structure with λ terms.
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43 <p>
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44 We are going to handle Mathematics in intuitionistic logic with λ terms.
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45 <p>
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46 Mathematics is a functional programming which values are proofs.
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47 <p>
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48 Programming ZF Set Theory in Agda
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49 <p>
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50
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51 <hr/>
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52 <h2><a name="content001">Target</a></h2>
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53
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54 <pre>
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55 Describe ZF axioms in Agda
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56 Construction a Model of ZF Set Theory in Agda
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57 Show necessary assumptions for the model
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58 Show validities of ZF axioms on the model
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59
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60 </pre>
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61 This shows consistency of Set Theory (with some assumptions),
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62 without circulating ZF Theory assumption.
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63 <p>
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64 <a href="https://github.com/shinji-kono/zf-in-agda">
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65 ZF in Agda https://github.com/shinji-kono/zf-in-agda
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66 </a>
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67 <p>
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68
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69 <hr/>
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70 <h2><a name="content002">Why Set Theory</a></h2>
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71 If we can formulate Set theory, it suppose to work on any mathematical theory.
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72 <p>
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73 Set Theory is a difficult point for beginners especially axiom of choice.
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74 <p>
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75 It has some amount of difficulty and self circulating discussion.
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76 <p>
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77 I'm planning to do it in my old age, but I'm enough age now.
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78 <p>
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79 This is done during from May to September.
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80 <p>
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81
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82 <hr/>
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83 <h2><a name="content003">Agda and Intuitionistic Logic </a></h2>
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84 Curry Howard Isomorphism
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85 <p>
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86
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87 <pre>
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88 Proposition : Proof ⇔ Type : Value
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89
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90 </pre>
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91 which means
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92 <p>
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93 constructing a typed lambda calculus which corresponds a logic
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94 <p>
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95 Typed lambda calculus which allows complex type as a value of a variable (System FC)
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96 <p>
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97 First class Type / Dependent Type
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98 <p>
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99 Agda is a such a programming language which has similar syntax of Haskell
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100 <p>
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101 Coq is specialized in proof assistance such as command and tactics .
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102 <p>
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103
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104 <hr/>
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105 <h2><a name="content004">Introduction of Agda </a></h2>
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106 A length of a list of type A.
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107 <p>
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108
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109 <pre>
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110 length : {A : Set } → List A → Nat
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111 length [] = zero
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112 length (_ ∷ t) = suc ( length t )
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113
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114 </pre>
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115 Simple functional programming language. Type declaration is mandatory.
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116 A colon means type, an equal means value. Indentation based.
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117 <p>
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118 Set is a base type (which may have a level ).
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119 <p>
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120 {} means implicit variable which can be omitted if Agda infers its value.
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121 <p>
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122
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123 <hr/>
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124 <h2><a name="content005">data ( Sum type )</a></h2>
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125 A data type which as exclusive multiple constructors. A similar one as
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126 union in C or case class in Scala.
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127 <p>
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128 It has a similar syntax as Haskell but it has a slight difference.
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129 <p>
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130
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131 <pre>
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132 data List (A : Set ) : Set where
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133 [] : List A
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134 _∷_ : A → List A → List A
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135
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136 </pre>
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137 _∷_ means infix operator. If use explicit _, it can be used in a normal function
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138 syntax.
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139 <p>
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140 Natural number can be defined as a usual way.
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141 <p>
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142
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143 <pre>
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144 data Nat : Set where
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145 zero : Nat
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146 suc : Nat → Nat
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147
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148 </pre>
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149
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150 <hr/>
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151 <h2><a name="content006"> A → B means "A implies B"</a></h2>
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152
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153 <p>
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154 In Agda, a type can be a value of a variable, which is usually called dependent type.
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155 Type has a name Set in Agda.
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156 <p>
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157
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158 <pre>
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159 ex3 : {A B : Set} → Set
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160 ex3 {A}{B} = A → B
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161
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162 </pre>
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163 ex3 is a type : A → B, which is a value of Set. It also means a formula : A implies B.
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164 <p>
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165
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166 <pre>
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167 A type is a formula, the value is the proof
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168
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169 </pre>
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170 A value of A → B can be interpreted as an inference from the formula A to the formula B, which
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171 can be a function from a proof of A to a proof of B.
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172 <p>
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173
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174 <hr/>
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175 <h2><a name="content007">introduction と elimination</a></h2>
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176 For a logical operator, there are two types of inference, an introduction and an elimination.
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177 <p>
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178
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179 <pre>
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180 intro creating symbol / constructor / introduction
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181 elim using symbolic / accessors / elimination
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182
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183 </pre>
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184 In Natural deduction, this can be written in proof schema.
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185 <p>
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186
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187 <pre>
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188 A
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189 :
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190 B A A → B
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191 ------------- →intro ------------------ →elim
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192 A → B B
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193
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194 </pre>
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195 In Agda, this is a pair of type and value as follows. Introduction of → uses λ.
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196 <p>
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197
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198 <pre>
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199 →intro : {A B : Set } → A → B → ( A → B )
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200 →intro _ b = λ x → b
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201 →elim : {A B : Set } → A → ( A → B ) → B
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202 →elim a f = f a
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203
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204 </pre>
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205 Important
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206 <p>
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207
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208 <pre>
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209 {A B : Set } → A → B → ( A → B )
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210
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211 </pre>
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212 is
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213 <p>
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214
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215 <pre>
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216 {A B : Set } → ( A → ( B → ( A → B ) ))
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217
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218 </pre>
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219 This makes currying of function easy.
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220 <p>
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221
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222 <hr/>
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223 <h2><a name="content008"> To prove A → B </a></h2>
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224 Make a left type as an argument. (intros in Coq)
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225 <p>
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226
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227 <pre>
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228 ex5 : {A B C : Set } → A → B → C → ?
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229 ex5 a b c = ?
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230
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231 </pre>
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232 ? is called a hole, which is unspecified part. Agda tell us which kind type is required for the Hole.
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233 <p>
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234 We are going to fill the holes, and if we have no warnings nor errors such as type conflict (Red),
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235 insufficient proof or instance (Yellow), Non-termination, the proof is completed.
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236 <p>
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237
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238 <hr/>
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239 <h2><a name="content009"> A ∧ B</a></h2>
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240 Well known conjunction's introduction and elimination is as follow.
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241 <p>
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242
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243 <pre>
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244 A B A ∧ B A ∧ B
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245 ------------- ----------- proj1 ---------- proj2
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246 A ∧ B A B
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247
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248 </pre>
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249 We can introduce a corresponding structure in our functional programming language.
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250 <p>
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251
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252 <hr/>
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253 <h2><a name="content010"> record</a></h2>
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254
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255 <pre>
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256 record _∧_ A B : Set
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257 field
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258 proj1 : A
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259 proj2 : B
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260
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261 </pre>
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262 _∧_ means infix operator. _∧_ A B can be written as A ∧ B (Haskell uses (∧) )
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263 <p>
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264 This a type which constructed from type A and type B. You may think this as an object
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265 or struct.
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266 <p>
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267
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268 <pre>
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269 record { proj1 = x ; proj2 = y }
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270
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271 </pre>
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272 is a constructor of _∧_.
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273 <p>
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274
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275 <pre>
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276 ex3 : {A B : Set} → A → B → ( A ∧ B )
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277 ex3 a b = record { proj1 = a ; proj2 = b }
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278 ex1 : {A B : Set} → ( A ∧ B ) → A
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279 ex1 a∧b = proj1 a∧b
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280
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281 </pre>
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282 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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283 except parenthesis or colons. A symbol requires space separation such as a type defining colon.
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284 <p>
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285 Defining record can be recursively, but we don't use the recursion here.
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286 <p>
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287
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288 <hr/>
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289 <h2><a name="content011"> Mathematical structure</a></h2>
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290 We have types of elements and the relationship in a mathematical structure.
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291 <p>
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292
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293 <pre>
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294 logical relation has no ordering
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295 there is a natural ordering in arguments and a value in a function
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296
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297 </pre>
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298 So we have typical definition style of mathematical structure with records.
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299 <p>
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300
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301 <pre>
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302 record IsOrdinals {n : Level} (ord : Set n)
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303 (_o<_ : ord → ord → Set n) : Set (suc (suc n)) where
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304 field
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305 Otrans : {x y z : ord } → x o< y → y o< z → x o< z
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306 record Ordinals {n : Level} : Set (suc (suc n)) where
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307 field
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308 ord : Set n
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309 _o<_ : ord → ord → Set n
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310 isOrdinal : IsOrdinals ord _o<_
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311
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312 </pre>
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313 In IsOrdinals, axioms are written in flat way. In Ordinal, we may have
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314 inputs and outputs are put in the field including IsOrdinal.
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315 <p>
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316 Fields of Ordinal is existential objects in the mathematical structure.
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317 <p>
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318
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319 <hr/>
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320 <h2><a name="content012"> A Model and a theory</a></h2>
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321 Agda record is a type, so we can write it in the argument, but is it really exists?
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322 <p>
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323 If we have a value of the record, it simply exists, that is, we need to create all the existence
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324 in the record satisfies all the axioms (= field of IsOrdinal) should be valid.
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325 <p>
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326
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327 <pre>
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328 type of record = theory
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329 value of record = model
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330
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331 </pre>
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332 We call the value of the record as a model. If mathematical structure has a
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333 model, it exists. Pretty Obvious.
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334 <p>
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335
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336 <hr/>
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337 <h2><a name="content013"> postulate と module</a></h2>
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338 Agda proofs are separated by modules, which are large records.
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339 <p>
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340 postulates are assumptions. We can assume a type without proofs.
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341 <p>
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342
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343 <pre>
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344 postulate
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345 sup-o : ( Ordinal → Ordinal ) → Ordinal
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346 sup-o< : { ψ : Ordinal → Ordinal } → ∀ {x : Ordinal } → ψ x o< sup-o ψ
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347
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348 </pre>
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349 sup-o is an example of upper bound of a function and sup-o< assumes it actually satisfies all the value is less than upper bound.
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350 <p>
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351 Writing some type in a module argument is the same as postulating a type, but
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352 postulate can be written the middle of a proof.
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353 <p>
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354 postulate can be constructive.
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355 <p>
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356 postulate can be inconsistent, which result everything has a proof.
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357 <p>
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358
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359 <hr/>
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360 <h2><a name="content014"> A ∨ B</a></h2>
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361
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362 <pre>
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363 data _∨_ (A B : Set) : Set where
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364 case1 : A → A ∨ B
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365 case2 : B → A ∨ B
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366
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367 </pre>
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368 As Haskell, case1/case2 are patterns.
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369 <p>
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370
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371 <pre>
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372 ex3 : {A B : Set} → ( A ∨ A ) → A
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373 ex3 = ?
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374
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375 </pre>
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376 In a case statement, Agda command C-C C-C generates possible cases in the head.
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377 <p>
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378
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379 <pre>
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380 ex3 : {A B : Set} → ( A ∨ A ) → A
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381 ex3 (case1 x) = ?
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382 ex3 (case2 x) = ?
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383
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384 </pre>
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385 Proof schema of ∨ is omit due to the complexity.
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386 <p>
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387
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388 <hr/>
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389 <h2><a name="content015"> Negation</a></h2>
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390
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391 <pre>
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392 ⊥
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393 ------------- ⊥-elim
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394 A
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395
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396 </pre>
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397 Anything can be derived from bottom, in this case a Set A. There is no introduction rule
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398 in ⊥, which can be implemented as data which has no constructor.
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399 <p>
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400
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401 <pre>
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402 data ⊥ : Set where
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403
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404 </pre>
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405 ⊥-elim can be proved like this.
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406 <p>
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407
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408 <pre>
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409 ⊥-elim : {A : Set } -> ⊥ -> A
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410 ⊥-elim ()
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411
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412 </pre>
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413 () means no match argument nor value.
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414 <p>
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415 A negation can be defined using ⊥ like this.
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416 <p>
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417
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418 <pre>
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419 ¬_ : Set → Set
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420 ¬ A = A → ⊥
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421
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422 </pre>
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423
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424 <hr/>
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425 <h2><a name="content016">Equality </a></h2>
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426
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427 <p>
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428 All the value in Agda are terms. If we have the same normalized form, two terms are equal.
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429 If we have variables in the terms, we will perform an unification. unifiable terms are equal.
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430 We don't go further on the unification.
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431 <p>
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432
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433 <pre>
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434 { x : A } x ≡ y f x y
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435 --------------- ≡-intro --------------------- ≡-elim
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436 x ≡ x f x x
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437
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438 </pre>
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439 equality _≡_ can be defined as a data.
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440 <p>
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441
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442 <pre>
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443 data _≡_ {A : Set } : A → A → Set where
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444 refl : {x : A} → x ≡ x
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445
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446 </pre>
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447 The elimination of equality is a substitution in a term.
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448 <p>
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449
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450 <pre>
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451 subst : {A : Set } → { x y : A } → ( f : A → Set ) → x ≡ y → f x → f y
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452 subst {A} {x} {y} f refl fx = fx
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453 ex5 : {A : Set} {x y z : A } → x ≡ y → y ≡ z → x ≡ z
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454 ex5 {A} {x} {y} {z} x≡y y≡z = subst ( λ k → x ≡ k ) y≡z x≡y
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455
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456 </pre>
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457
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458 <hr/>
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459 <h2><a name="content017">Equivalence relation</a></h2>
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|
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460
|
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461 <p>
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462
|
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463 <pre>
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464 refl' : {A : Set} {x : A } → x ≡ x
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465 refl' = ?
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466 sym : {A : Set} {x y : A } → x ≡ y → y ≡ x
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467 sym = ?
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468 trans : {A : Set} {x y z : A } → x ≡ y → y ≡ z → x ≡ z
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469 trans = ?
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470 cong : {A B : Set} {x y : A } { f : A → B } → x ≡ y → f x ≡ f y
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471 cong = ?
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472
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473 </pre>
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474
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475 <hr/>
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476 <h2><a name="content018">Ordering</a></h2>
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|
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477
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478 <p>
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479 Relation is a predicate on two value which has a same type.
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480 <p>
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481
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482 <pre>
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483 A → A → Set
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484
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485 </pre>
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486 Defining order is the definition of this type with predicate or a data.
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487 <p>
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488
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489 <pre>
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490 data _≤_ : Rel ℕ 0ℓ where
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491 z≤n : ∀ {n} → zero ≤ n
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492 s≤s : ∀ {m n} (m≤n : m ≤ n) → suc m ≤ suc n
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493
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494 </pre>
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|
495
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496 <hr/>
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497 <h2><a name="content019">Quantifier</a></h2>
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498
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499 <p>
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500 Handling quantifier in an intuitionistic logic requires special cares.
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501 <p>
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502 In the input of a function, there are no restriction on it, that is, it has
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503 a universal quantifier. (If we explicitly write ∀, Agda gives us a type inference on it)
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504 <p>
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505 There is no ∃ in agda, the one way is using negation like this.
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506 <p>
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507 ∃ (x : A ) → p x = ¬ ( ( x : A ) → ¬ ( p x ) )
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508 <p>
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|
|
509 On the another way, f : A can be used like this.
|
|
|
510 <p>
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511
|
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512 <pre>
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|
513 p f
|
|
|
514
|
|
|
515 </pre>
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|
516 If we use a function which can be defined globally which has stronger meaning the
|
|
|
517 usage of ∃ x in a logical expression.
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518 <p>
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|
519
|
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520 <hr/>
|
|
|
521 <h2><a name="content020">Can we do math in this way?</a></h2>
|
|
|
522 Yes, we can. Actually we have Principia Mathematica by Russell and Whitehead (with out computer support).
|
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|
523 <p>
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|
|
524 In some sense, this story is a reprinting of the work, (but Principia Mathematica has a different formulation than ZF).
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|
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525 <p>
|
|
|
526
|
|
|
527 <pre>
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|
528 define mathematical structure as a record
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|
|
529 program inferences as if we have proofs in variables
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|
|
530
|
|
|
531 </pre>
|
|
|
532
|
|
|
533 <hr/>
|
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|
534 <h2><a name="content021">Things which Agda cannot prove</a></h2>
|
|
|
535
|
|
|
536 <p>
|
|
|
537 The infamous Internal Parametricity is a limitation of Agda, it cannot prove so called Free Theorem, which
|
|
|
538 leads uniqueness of a functor in Category Theory.
|
|
|
539 <p>
|
|
|
540 Functional extensionality cannot be proved.
|
|
|
541 <pre>
|
|
|
542 (∀ x → f x ≡ g x) → f ≡ g
|
|
|
543
|
|
|
544 </pre>
|
|
|
545 Agda has no law of exclude middle.
|
|
|
546 <p>
|
|
|
547
|
|
|
548 <pre>
|
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|
549 a ∨ ( ¬ a )
|
|
|
550
|
|
|
551 </pre>
|
|
|
552 For example, (A → B) → ¬ B → ¬ A can be proved but, ( ¬ B → ¬ A ) → A → B cannot.
|
|
|
553 <p>
|
|
|
554 It also other problems such as termination, type inference or unification which we may overcome with
|
|
|
555 efforts or devices or may not.
|
|
|
556 <p>
|
|
|
557 If we cannot prove something, we can safely postulate it unless it leads a contradiction.
|
|
|
558 <pre>
|
|
|
559
|
|
|
560
|
|
|
561 </pre>
|
|
|
562
|
|
|
563 <hr/>
|
|
|
564 <h2><a name="content022">Classical story of ZF Set Theory</a></h2>
|
|
|
565
|
|
|
566 <p>
|
|
|
567 Assuming ZF, constructing a model of ZF is a flow of classical Set Theory, which leads
|
|
|
568 a relative consistency proof of the Set Theory.
|
|
|
569 Ordinal number is used in the flow.
|
|
|
570 <p>
|
|
|
571 In Agda, first we defines Ordinal numbers (Ordinals), then introduce Ordinal Definable Set (OD).
|
|
|
572 We need some non constructive assumptions in the construction. A model of Set theory is
|
|
|
573 constructed based on these assumptions.
|
|
|
574 <p>
|
|
|
575 <img src="fig/set-theory.svg">
|
|
|
576
|
|
|
577 <p>
|
|
|
578
|
|
|
579 <hr/>
|
|
|
580 <h2><a name="content023">Ordinals</a></h2>
|
|
|
581 Ordinals are our intuition of infinite things, which has ∅ and orders on the things.
|
|
|
582 It also has a successor osuc.
|
|
|
583 <p>
|
|
|
584
|
|
|
585 <pre>
|
|
|
586 record Ordinals {n : Level} : Set (suc (suc n)) where
|
|
|
587 field
|
|
|
588 ord : Set n
|
|
|
589 o∅ : ord
|
|
|
590 osuc : ord → ord
|
|
|
591 _o<_ : ord → ord → Set n
|
|
|
592 isOrdinal : IsOrdinals ord o∅ osuc _o<_
|
|
|
593
|
|
|
594 </pre>
|
|
|
595 It is different from natural numbers in way. The order of Ordinals is not defined in terms
|
|
|
596 of successor. It is given from outside, which make it possible to have higher order infinity.
|
|
|
597 <p>
|
|
|
598
|
|
|
599 <hr/>
|
|
|
600 <h2><a name="content024">Axiom of Ordinals</a></h2>
|
|
|
601 Properties of infinite things. We request a transfinite induction, which states that if
|
|
|
602 some properties are satisfied below all possible ordinals, the properties are true on all
|
|
|
603 ordinals.
|
|
|
604 <p>
|
|
|
605 Successor osuc has no ordinal between osuc and the base ordinal. There are some ordinals
|
|
|
606 which is not a successor of any ordinals. It is called limit ordinal.
|
|
|
607 <p>
|
|
|
608 Any two ordinal can be compared, that is less, equal or more, that is total order.
|
|
|
609 <p>
|
|
|
610
|
|
|
611 <pre>
|
|
|
612 record IsOrdinals {n : Level} (ord : Set n) (o∅ : ord )
|
|
|
613 (osuc : ord → ord )
|
|
|
614 (_o<_ : ord → ord → Set n) : Set (suc (suc n)) where
|
|
|
615 field
|
|
|
616 Otrans : {x y z : ord } → x o< y → y o< z → x o< z
|
|
|
617 OTri : Trichotomous {n} _≡_ _o<_
|
|
|
618 ¬x<0 : { x : ord } → ¬ ( x o< o∅ )
|
|
|
619 <-osuc : { x : ord } → x o< osuc x
|
|
|
620 osuc-≡< : { a x : ord } → x o< osuc a → (x ≡ a ) ∨ (x o< a)
|
|
|
621 TransFinite : { ψ : ord → Set (suc n) }
|
|
|
622 → ( (x : ord) → ( (y : ord ) → y o< x → ψ y ) → ψ x )
|
|
|
623 → ∀ (x : ord) → ψ x
|
|
|
624
|
|
|
625 </pre>
|
|
|
626
|
|
|
627 <hr/>
|
|
|
628 <h2><a name="content025">Concrete Ordinals</a></h2>
|
|
|
629
|
|
|
630 <p>
|
|
|
631 We can define a list like structure with level, which is a kind of two dimensional infinite array.
|
|
|
632 <p>
|
|
|
633
|
|
|
634 <pre>
|
|
|
635 data OrdinalD {n : Level} : (lv : Nat) → Set n where
|
|
|
636 Φ : (lv : Nat) → OrdinalD lv
|
|
|
637 OSuc : (lv : Nat) → OrdinalD {n} lv → OrdinalD lv
|
|
|
638
|
|
|
639 </pre>
|
|
|
640 The order of the OrdinalD can be defined in this way.
|
|
|
641 <p>
|
|
|
642
|
|
|
643 <pre>
|
|
|
644 data _d<_ {n : Level} : {lx ly : Nat} → OrdinalD {n} lx → OrdinalD {n} ly → Set n where
|
|
|
645 Φ< : {lx : Nat} → {x : OrdinalD {n} lx} → Φ lx d< OSuc lx x
|
|
|
646 s< : {lx : Nat} → {x y : OrdinalD {n} lx} → x d< y → OSuc lx x d< OSuc lx y
|
|
|
647
|
|
|
648 </pre>
|
|
|
649 This is a simple data structure, it has no abstract assumptions, and it is countable many data
|
|
|
650 structure.
|
|
|
651 <p>
|
|
|
652
|
|
|
653 <pre>
|
|
|
654 Φ 0
|
|
|
655 OSuc 2 ( Osuc 2 ( Osuc 2 (Φ 2)))
|
|
|
656 Osuc 0 (Φ 0) d< Φ 1
|
|
|
657
|
|
|
658 </pre>
|
|
|
659
|
|
|
660 <hr/>
|
|
|
661 <h2><a name="content026">Model of Ordinals</a></h2>
|
|
|
662
|
|
|
663 <p>
|
|
|
664 It is easy to show OrdinalD and its order satisfies the axioms of Ordinals.
|
|
|
665 <p>
|
|
|
666 So our Ordinals has a mode. This means axiom of Ordinals are consistent.
|
|
|
667 <p>
|
|
|
668
|
|
|
669 <hr/>
|
|
|
670 <h2><a name="content027">Debugging axioms using Model</a></h2>
|
|
|
671 Whether axiom is correct or not can be checked by a validity on a mode.
|
|
|
672 <p>
|
|
|
673 If not, we may fix the axioms or the model, such as the definitions of the order.
|
|
|
674 <p>
|
|
|
675 We can also ask whether the inputs exist.
|
|
|
676 <p>
|
|
|
677
|
|
|
678 <hr/>
|
|
|
679 <h2><a name="content028">Countable Ordinals can contains uncountable set?</a></h2>
|
|
|
680 Yes, the ordinals contains any level of infinite Set in the axioms.
|
|
|
681 <p>
|
|
|
682 If we handle real-number in the model, only countable number of real-number is used.
|
|
|
683 <p>
|
|
|
684
|
|
|
685 <pre>
|
|
|
686 from the outside view point, it is countable
|
|
|
687 from the internal view point, it is uncountable
|
|
|
688
|
|
|
689 </pre>
|
|
|
690 The definition of countable/uncountable is the same, but the properties are different
|
|
|
691 depending on the context.
|
|
|
692 <p>
|
|
|
693 We don't show the definition of cardinal number here.
|
|
|
694 <p>
|
|
|
695
|
|
|
696 <hr/>
|
|
|
697 <h2><a name="content029">What is Set</a></h2>
|
|
|
698 The word Set in Agda is not a Set of ZF Set, but it is a type (why it is named Set?).
|
|
|
699 <p>
|
|
|
700 From naive point view, a set i a list, but in Agda, elements have all the same type.
|
|
|
701 A set in ZF may contain other Sets in ZF, which not easy to implement it as a list.
|
|
|
702 <p>
|
|
|
703 Finite set may be written in finite series of ∨, but ...
|
|
|
704 <p>
|
|
|
705
|
|
|
706 <hr/>
|
|
|
707 <h2><a name="content030">We don't ask the contents of Set. It can be anything.</a></h2>
|
|
|
708 From empty set φ, we can think a set contains a φ, and a pair of φ and the set, and so on,
|
|
|
709 and all of them, and again we repeat this.
|
|
|
710 <p>
|
|
|
711
|
|
|
712 <pre>
|
|
|
713 φ {φ} {φ,{φ}}, {φ,{φ},...}
|
|
|
714
|
|
|
715 </pre>
|
|
|
716 It is called V.
|
|
|
717 <p>
|
|
|
718 This operation can be performed within a ZF Set theory. Classical Set Theory assumes
|
|
|
719 ZF, so this kind of thing is allowed.
|
|
|
720 <p>
|
|
|
721 But in our case, we have no ZF theory, so we are going to use Ordinals.
|
|
|
722 <p>
|
|
|
723
|
|
|
724 <hr/>
|
|
|
725 <h2><a name="content031">Ordinal Definable Set</a></h2>
|
|
|
726 We can define a sbuset of Ordinals using predicates. What is a subset?
|
|
|
727 <p>
|
|
|
728
|
|
|
729 <pre>
|
|
|
730 a predicate has an Ordinal argument
|
|
|
731
|
|
|
732 </pre>
|
|
|
733 is an Ordinal Definable Set (OD). In Agda, OD is defined as follows.
|
|
|
734 <p>
|
|
|
735
|
|
|
736 <pre>
|
|
|
737 record OD : Set (suc n ) where
|
|
|
738 field
|
|
|
739 def : (x : Ordinal ) → Set n
|
|
|
740
|
|
|
741 </pre>
|
|
|
742 Ordinals itself is not a set in a ZF Set theory but a class. In OD,
|
|
|
743 <p>
|
|
|
744
|
|
|
745 <pre>
|
|
|
746 record { def = λ x → true }
|
|
|
747
|
|
|
748 </pre>
|
|
|
749 means Ordinals itself, so ODs are larger than a notion of ZF Set Theory,
|
|
|
750 but we don't care about it.
|
|
|
751 <p>
|
|
|
752
|
|
|
753 <hr/>
|
|
|
754 <h2><a name="content032">∋ in OD</a></h2>
|
|
|
755 OD is a predicate on Ordinals and it does not contains OD directly. If we have a mapping
|
|
|
756 <p>
|
|
|
757
|
|
|
758 <pre>
|
|
|
759 od→ord : OD → Ordinal
|
|
|
760
|
|
|
761 </pre>
|
|
|
762 we may check an OD is an element of the OD using def.
|
|
|
763 <p>
|
|
|
764 A ∋ x can be define as follows.
|
|
|
765 <p>
|
|
|
766
|
|
|
767 <pre>
|
|
|
768 _∋_ : ( A x : OD ) → Set n
|
|
|
769 _∋_ A x = def A ( od→ord x )
|
|
|
770
|
|
|
771 </pre>
|
|
|
772 In ψ : Ordinal → Set, if A is a record { def = λ x → ψ x } , then
|
|
|
773 <p>
|
|
|
774
|
|
|
775 <pre>
|
|
|
776 A x = def A ( od→ord x ) = ψ (od→ord x)
|
|
|
777
|
|
|
778 </pre>
|
|
|
779
|
|
|
780 <hr/>
|
|
|
781 <h2><a name="content033">1 to 1 mapping between an OD and an Ordinal</a></h2>
|
|
|
782
|
|
|
783 <p>
|
|
|
784
|
|
|
785 <pre>
|
|
|
786 od→ord : OD → Ordinal
|
|
|
787 ord→od : Ordinal → OD
|
|
|
788 oiso : {x : OD } → ord→od ( od→ord x ) ≡ x
|
|
|
789 diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x
|
|
|
790
|
|
|
791 </pre>
|
|
|
792 They say the existing of the mappings can be proved in Classical Set Theory, but we
|
|
|
793 simply assumes these non constructively.
|
|
|
794 <p>
|
|
|
795 We use postulate, it may contains additional restrictions which are not clear now. It look like the mapping should be a subset of Ordinals, but we don't explicitly
|
|
|
796 state it.
|
|
|
797 <p>
|
|
|
798 <img src="fig/ord-od-mapping.svg">
|
|
|
799
|
|
|
800 <p>
|
|
|
801
|
|
|
802 <hr/>
|
|
|
803 <h2><a name="content034">Order preserving in the mapping of OD and Ordinal</a></h2>
|
|
|
804 Ordinals have the order and OD has a natural order based on inclusion ( def / ∋ ).
|
|
|
805 <p>
|
|
|
806
|
|
|
807 <pre>
|
|
|
808 def y ( od→ord x )
|
|
|
809
|
|
|
810 </pre>
|
|
|
811 An elements of OD should be defined before the OD, that is, an ordinal corresponding an elements
|
|
|
812 have to be smaller than the corresponding ordinal of the containing OD.
|
|
|
813 <p>
|
|
|
814
|
|
|
815 <pre>
|
|
|
816 c<→o< : {x y : OD } → def y ( od→ord x ) → od→ord x o< od→ord y
|
|
|
817
|
|
|
818 </pre>
|
|
|
819 This is also said to be provable in classical Set Theory, but we simply assumes it.
|
|
|
820 <p>
|
|
|
821 OD has an order based on the corresponding ordinal, but it may not have corresponding def / ∋relation. This means the reverse order preservation,
|
|
|
822 <p>
|
|
|
823
|
|
|
824 <pre>
|
|
|
825 o<→c< : {n : Level} {x y : Ordinal } → x o< y → def (ord→od y) x
|
|
|
826
|
|
|
827 </pre>
|
|
|
828 is not valid. If we assumes it, ∀ x ∋ ∅ becomes true, which manes all OD becomes Ordinals in
|
|
|
829 the model.
|
|
|
830 <p>
|
|
|
831 <img src="fig/ODandOrdinals.svg">
|
|
|
832
|
|
|
833 <p>
|
|
|
834
|
|
|
835 <hr/>
|
|
|
836 <h2><a name="content035">ISO</a></h2>
|
|
|
837 It also requires isomorphisms,
|
|
|
838 <p>
|
|
|
839
|
|
|
840 <pre>
|
|
|
841 oiso : {x : OD } → ord→od ( od→ord x ) ≡ x
|
|
|
842 diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x
|
|
|
843
|
|
|
844 </pre>
|
|
|
845
|
|
|
846 <hr/>
|
|
|
847 <h2><a name="content036">Various Sets</a></h2>
|
|
|
848
|
|
|
849 <p>
|
|
|
850 In classical Set Theory, there is a hierarchy call L, which can be defined by a predicate.
|
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|
851 <p>
|
|
|
852
|
|
|
853 <pre>
|
|
|
854 Ordinal / things satisfies axiom of Ordinal / extension of natural number
|
|
|
855 V / hierarchical construction of Set from φ
|
|
|
856 L / hierarchical predicate definable construction of Set from φ
|
|
|
857 OD / equational formula on Ordinals
|
|
|
858 Agda Set / Type / it also has a level
|
|
|
859
|
|
|
860 </pre>
|
|
|
861
|
|
|
862 <hr/>
|
|
|
863 <h2><a name="content037">Fixes on ZF to intuitionistic logic</a></h2>
|
|
|
864
|
|
|
865 <p>
|
|
|
866 We use ODs as Sets in ZF, and defines record ZF, that is, we have to define
|
|
|
867 ZF axioms in Agda.
|
|
|
868 <p>
|
|
|
869 It may not valid in our model. We have to debug it.
|
|
|
870 <p>
|
|
|
871 Fixes are depends on axioms.
|
|
|
872 <p>
|
|
|
873 <img src="fig/axiom-type.svg">
|
|
|
874
|
|
|
875 <p>
|
|
|
876 <a href="fig/zf-record.html">
|
|
|
877 ZFのrecord </a>
|
|
|
878 <p>
|
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|
879
|
|
|
880 <hr/>
|
|
|
881 <h2><a name="content038">Pure logical axioms</a></h2>
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882
|
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|
883 <pre>
|
|
|
884 empty, pair, select, ε-inductioninfinity
|
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|
885
|
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|
886 </pre>
|
|
|
887 These are logical relations among OD.
|
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|
888 <p>
|
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|
889
|
|
|
890 <pre>
|
|
|
891 empty : ∀( x : ZFSet ) → ¬ ( ∅ ∋ x )
|
|
|
892 pair→ : ( x y t : ZFSet ) → (x , y) ∋ t → ( t ≈ x ) ∨ ( t ≈ y )
|
|
|
893 pair← : ( x y t : ZFSet ) → ( t ≈ x ) ∨ ( t ≈ y ) → (x , y) ∋ t
|
|
|
894 selection : { ψ : ZFSet → Set m } → ∀ { X y : ZFSet } → ( ( y ∈ X ) ∧ ψ y ) ⇔ (y ∈ Select X ψ )
|
|
|
895 infinity∅ : ∅ ∈ infinite
|
|
|
896 infinity : ∀( x : ZFSet ) → x ∈ infinite → ( x ∪ ( x , x ) ) ∈ infinite
|
|
|
897 ε-induction : { ψ : OD → Set (suc n)}
|
|
|
898 → ( {x : OD } → ({ y : OD } → x ∋ y → ψ y ) → ψ x )
|
|
|
899 → (x : OD ) → ψ x
|
|
|
900
|
|
|
901 </pre>
|
|
|
902 finitely can be define by Agda data.
|
|
|
903 <p>
|
|
|
904
|
|
|
905 <pre>
|
|
|
906 data infinite-d : ( x : Ordinal ) → Set n where
|
|
|
907 iφ : infinite-d o∅
|
|
|
908 isuc : {x : Ordinal } → infinite-d x →
|
|
|
909 infinite-d (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) ))
|
|
|
910
|
|
|
911 </pre>
|
|
|
912 Union (x , ( x , x )) should be an direct successor of x, but we cannot prove it in our model.
|
|
|
913 <p>
|
|
|
914
|
|
|
915 <hr/>
|
|
|
916 <h2><a name="content039">Axiom of Pair</a></h2>
|
|
|
917 In the Tanaka's book, axiom of pair is as follows.
|
|
|
918 <p>
|
|
|
919
|
|
|
920 <pre>
|
|
|
921 ∀ x ∀ y ∃ z ∀ t ( z ∋ t ↔ t ≈ x ∨ t ≈ y)
|
|
|
922
|
|
|
923 </pre>
|
|
|
924 We have fix ∃ z, a function (x , y) is defined, which is _,_ .
|
|
|
925 <p>
|
|
|
926
|
|
|
927 <pre>
|
|
|
928 _,_ : ( A B : ZFSet ) → ZFSet
|
|
|
929
|
|
|
930 </pre>
|
|
|
931 using this, we can define two directions in separates axioms, like this.
|
|
|
932 <p>
|
|
|
933
|
|
|
934 <pre>
|
|
|
935 pair→ : ( x y t : ZFSet ) → (x , y) ∋ t → ( t ≈ x ) ∨ ( t ≈ y )
|
|
|
936 pair← : ( x y t : ZFSet ) → ( t ≈ x ) ∨ ( t ≈ y ) → (x , y) ∋ t
|
|
|
937
|
|
|
938 </pre>
|
|
|
939 This is already written in Agda, so we use these as axioms. All inputs have ∀.
|
|
|
940 <p>
|
|
|
941
|
|
|
942 <hr/>
|
|
|
943 <h2><a name="content040">pair in OD</a></h2>
|
|
|
944 OD is an equation on Ordinals, we can simply write axiom of pair in the OD.
|
|
|
945 <p>
|
|
|
946
|
|
|
947 <pre>
|
|
|
948 _,_ : OD → OD → OD
|
|
|
949 x , y = record { def = λ t → (t ≡ od→ord x ) ∨ ( t ≡ od→ord y ) }
|
|
|
950
|
|
|
951 </pre>
|
|
|
952 ≡ is an equality of λ terms, but please not that this is equality on Ordinals.
|
|
|
953 <p>
|
|
|
954
|
|
|
955 <hr/>
|
|
|
956 <h2><a name="content041">Validity of Axiom of Pair</a></h2>
|
|
|
957 Assuming ZFSet is OD, we are going to prove pair→ .
|
|
|
958 <p>
|
|
|
959
|
|
|
960 <pre>
|
|
|
961 pair→ : ( x y t : OD ) → (x , y) ∋ t → ( t == x ) ∨ ( t == y )
|
|
|
962 pair→ x y t p = ?
|
|
|
963
|
|
|
964 </pre>
|
|
|
965 In this program, type of p is ( x , y ) ∋ t , that is def ( x , y ) that is, (t ≡ od→ord x ) ∨ ( t ≡ od→ord y ) .
|
|
|
966 <p>
|
|
|
967 Since _∨_ is a data, it can be developed as (C-c C-c : agda2-make-case ).
|
|
|
968 <p>
|
|
|
969
|
|
|
970 <pre>
|
|
|
971 pair→ x y t (case1 t≡x ) = ?
|
|
|
972 pair→ x y t (case2 t≡y ) = ?
|
|
|
973
|
|
|
974 </pre>
|
|
|
975 The type of the ? is ( t == x ) ∨ ( t == y ), again it is data _∨_ .
|
|
|
976 <p>
|
|
|
977
|
|
|
978 <pre>
|
|
|
979 pair→ x y t (case1 t≡x ) = case1 ?
|
|
|
980 pair→ x y t (case2 t≡y ) = case2 ?
|
|
|
981
|
|
|
982 </pre>
|
|
|
983 The ? in case1 is t == x, so we have to create this from t≡x, which is a name of a variable
|
|
|
984 which type is
|
|
|
985 <p>
|
|
|
986
|
|
|
987 <pre>
|
|
|
988 t≡x : od→ord t ≡ od→ord x
|
|
|
989
|
|
|
990 </pre>
|
|
|
991 which is shown by an Agda command (C-C C-E : agda2-show-context ).
|
|
|
992 <p>
|
|
|
993 But we haven't defined == yet.
|
|
|
994 <p>
|
|
|
995
|
|
|
996 <hr/>
|
|
|
997 <h2><a name="content042">Equality of OD and Axiom of Extensionality </a></h2>
|
|
|
998 OD is defined by a predicates, if we compares normal form of the predicates, even if
|
|
|
999 it contains the same elements, it may be different, which is no good as an equality of
|
|
|
1000 Sets.
|
|
|
1001 <p>
|
|
|
1002 Axiom of Extensionality requires sets having the same elements are handled in the same way
|
|
|
1003 each other.
|
|
|
1004 <p>
|
|
|
1005
|
|
|
1006 <pre>
|
|
|
1007 ∀ z ( z ∈ x ⇔ z ∈ y ) ⇒ ∀ w ( x ∈ w ⇔ y ∈ w )
|
|
|
1008
|
|
|
1009 </pre>
|
|
|
1010 We can write this axiom in Agda as follows.
|
|
|
1011 <p>
|
|
|
1012
|
|
|
1013 <pre>
|
|
|
1014 extensionality : { A B w : ZFSet } → ( (z : ZFSet) → ( A ∋ z ) ⇔ (B ∋ z) ) → ( A ∈ w ⇔ B ∈ w )
|
|
|
1015
|
|
|
1016 </pre>
|
|
|
1017 So we use ( A ∋ z ) ⇔ (B ∋ z) as an equality (_==_) of our model. We have to show
|
|
|
1018 A ∈ w ⇔ B ∈ w from A == B.
|
|
|
1019 <p>
|
|
|
1020 x == y can be defined in this way.
|
|
|
1021 <p>
|
|
|
1022
|
|
|
1023 <pre>
|
|
|
1024 record _==_ ( a b : OD ) : Set n where
|
|
|
1025 field
|
|
|
1026 eq→ : ∀ { x : Ordinal } → def a x → def b x
|
|
|
1027 eq← : ∀ { x : Ordinal } → def b x → def a x
|
|
|
1028
|
|
|
1029 </pre>
|
|
|
1030 Actually, (z : OD) → (A ∋ z) ⇔ (B ∋ z) implies A == B.
|
|
|
1031 <p>
|
|
|
1032
|
|
|
1033 <pre>
|
|
|
1034 extensionality0 : {A B : OD } → ((z : OD) → (A ∋ z) ⇔ (B ∋ z)) → A == B
|
|
|
1035 eq→ (extensionality0 {A} {B} eq ) {x} d = ?
|
|
|
1036 eq← (extensionality0 {A} {B} eq ) {x} d = ?
|
|
|
1037
|
|
|
1038 </pre>
|
|
|
1039 ? are def B x and def A x and these are generated from eq : (z : OD) → (A ∋ z) ⇔ (B ∋ z) .
|
|
|
1040 <p>
|
|
|
1041 Actual proof is rather complicated.
|
|
|
1042 <p>
|
|
|
1043
|
|
|
1044 <pre>
|
|
|
1045 eq→ (extensionality0 {A} {B} eq ) {x} d = def-iso {A} {B} (sym diso) (proj1 (eq (ord→od x))) d
|
|
|
1046 eq← (extensionality0 {A} {B} eq ) {x} d = def-iso {B} {A} (sym diso) (proj2 (eq (ord→od x))) d
|
|
|
1047
|
|
|
1048 </pre>
|
|
|
1049 where
|
|
|
1050 <p>
|
|
|
1051
|
|
|
1052 <pre>
|
|
|
1053 def-iso : {A B : OD } {x y : Ordinal } → x ≡ y → (def A y → def B y) → def A x → def B x
|
|
|
1054 def-iso refl t = t
|
|
|
1055
|
|
|
1056 </pre>
|
|
|
1057
|
|
|
1058 <hr/>
|
|
|
1059 <h2><a name="content043">Validity of Axiom of Extensionality</a></h2>
|
|
|
1060
|
|
|
1061 <p>
|
|
|
1062 If we can derive (w ∋ A) ⇔ (w ∋ B) from A == B, the axiom becomes valid, but it seems impossible, so we assumes
|
|
|
1063 <p>
|
|
|
1064
|
|
|
1065 <pre>
|
|
|
1066 ==→o≡ : { x y : OD } → (x == y) → x ≡ y
|
|
|
1067
|
|
|
1068 </pre>
|
|
|
1069 Using this, we have
|
|
|
1070 <p>
|
|
|
1071
|
|
|
1072 <pre>
|
|
|
1073 extensionality : {A B w : OD } → ((z : OD ) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B)
|
|
|
1074 proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d
|
|
|
1075 proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d
|
|
|
1076
|
|
|
1077 </pre>
|
|
|
1078 This assumption means we may have an equivalence set on any predicates.
|
|
|
1079 <p>
|
|
|
1080
|
|
|
1081 <hr/>
|
|
|
1082 <h2><a name="content044">Non constructive assumptions so far</a></h2>
|
|
|
1083 We have correspondence between OD and Ordinals and one directional order preserving.
|
|
|
1084 <p>
|
|
|
1085 We also have equality assumption.
|
|
|
1086 <p>
|
|
|
1087
|
|
|
1088 <pre>
|
|
|
1089 od→ord : OD → Ordinal
|
|
|
1090 ord→od : Ordinal → OD
|
|
|
1091 oiso : {x : OD } → ord→od ( od→ord x ) ≡ x
|
|
|
1092 diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x
|
|
|
1093 c<→o< : {x y : OD } → def y ( od→ord x ) → od→ord x o< od→ord y
|
|
|
1094 ==→o≡ : { x y : OD } → (x == y) → x ≡ y
|
|
|
1095
|
|
|
1096 </pre>
|
|
|
1097
|
|
|
1098 <hr/>
|
|
|
1099 <h2><a name="content045">Axiom which have negation form</a></h2>
|
|
|
1100
|
|
|
1101 <p>
|
|
|
1102
|
|
|
1103 <pre>
|
|
|
1104 Union, Selection
|
|
|
1105
|
|
|
1106 </pre>
|
|
|
1107 These axioms contains ∃ x as a logical relation, which can be described in ¬ ( ∀ x ( ¬ p )).
|
|
|
1108 <p>
|
|
|
1109 Axiom of replacement uses upper bound of function on Ordinals, which makes it non-constructive.
|
|
|
1110 <p>
|
|
|
1111 Power Set axiom requires double negation,
|
|
|
1112 <p>
|
|
|
1113
|
|
|
1114 <pre>
|
|
|
1115 power→ : ∀( A t : ZFSet ) → Power A ∋ t → ∀ {x} → t ∋ x → ¬ ¬ ( A ∋ x )
|
|
|
1116 power← : ∀( A t : ZFSet ) → t ⊆_ A → Power A ∋ t
|
|
|
1117
|
|
|
1118 </pre>
|
|
|
1119 If we have an assumption of law of exclude middle, we can recover the original A ∋ x form.
|
|
|
1120 <p>
|
|
|
1121
|
|
|
1122 <hr/>
|
|
|
1123 <h2><a name="content046">Union </a></h2>
|
|
|
1124 The original form of the Axiom of Union is
|
|
|
1125 <p>
|
|
|
1126
|
|
|
1127 <pre>
|
|
|
1128 ∀ x ∃ y ∀ z (z ∈ y ⇔ ∃ u ∈ x ∧ (z ∈ u))
|
|
|
1129
|
|
|
1130 </pre>
|
|
|
1131 Union requires the existence of b in a ⊇ ∃ b ∋ x . We will use negation form of ∃.
|
|
|
1132 <p>
|
|
|
1133
|
|
|
1134 <pre>
|
|
|
1135 union→ : ( X z u : ZFSet ) → ( X ∋ u ) ∧ (u ∋ z ) → Union X ∋ z
|
|
|
1136 union← : ( X z : ZFSet ) → (X∋z : Union X ∋ z ) → ¬ ( (u : ZFSet ) → ¬ ((X ∋ u) ∧ (u ∋ z )))
|
|
|
1137
|
|
|
1138 </pre>
|
|
|
1139 The definition of Union in OD is like this.
|
|
|
1140 <p>
|
|
|
1141
|
|
|
1142 <pre>
|
|
|
1143 Union : OD → OD
|
|
|
1144 Union U = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((def U u) ∧ (def (ord→od u) x))) }
|
|
|
1145
|
|
|
1146 </pre>
|
|
|
1147 Proof of validity is straight forward.
|
|
|
1148 <p>
|
|
|
1149
|
|
|
1150 <pre>
|
|
|
1151 union→ : (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z
|
|
|
1152 union→ X z u xx not = ⊥-elim ( not (od→ord u) ( record { proj1 = proj1 xx
|
|
|
1153 ; proj2 = subst ( λ k → def k (od→ord z)) (sym oiso) (proj2 xx) } ))
|
|
|
1154 union← : (X z : OD) (X∋z : Union X ∋ z) → ¬ ( (u : OD ) → ¬ ((X ∋ u) ∧ (u ∋ z )))
|
|
|
1155 union← X z UX∋z = FExists _ lemma UX∋z where
|
|
|
1156 lemma : {y : Ordinal} → def X y ∧ def (ord→od y) (od→ord z) → ¬ ((u : OD) → ¬ (X ∋ u) ∧ (u ∋ z))
|
|
|
1157 lemma {y} xx not = not (ord→od y) record { proj1 = subst ( λ k → def X k ) (sym diso) (proj1 xx ) ; proj2 = proj2 xx }
|
|
|
1158
|
|
|
1159 </pre>
|
|
|
1160 where
|
|
|
1161 <p>
|
|
|
1162
|
|
|
1163 <pre>
|
|
|
1164 FExists : {m l : Level} → ( ψ : Ordinal → Set m )
|
|
|
1165 → {p : Set l} ( P : { y : Ordinal } → ψ y → ¬ p )
|
|
|
1166 → (exists : ¬ (∀ y → ¬ ( ψ y ) ))
|
|
|
1167 → ¬ p
|
|
|
1168 FExists {m} {l} ψ {p} P = contra-position ( λ p y ψy → P {y} ψy p )
|
|
|
1169
|
|
|
1170 </pre>
|
|
|
1171 which checks existence using contra-position.
|
|
|
1172 <p>
|
|
|
1173
|
|
|
1174 <hr/>
|
|
|
1175 <h2><a name="content047">Axiom of replacement</a></h2>
|
|
|
1176 We can replace the elements of a set by a function and it becomes a set. From the book,
|
|
|
1177 <p>
|
|
|
1178
|
|
|
1179 <pre>
|
|
|
1180 ∀ x ∀ y ∀ z ( ( ψ ( x , y ) ∧ ψ ( x , z ) ) → y = z ) → ∀ X ∃ A ∀ y ( y ∈ A ↔ ∃ x ∈ X ψ ( x , y ) )
|
|
|
1181
|
|
|
1182 </pre>
|
|
|
1183 The existential quantifier can be related by a function,
|
|
|
1184 <p>
|
|
|
1185
|
|
|
1186 <pre>
|
|
|
1187 Replace : OD → (OD → OD ) → OD
|
|
|
1188
|
|
|
1189 </pre>
|
|
|
1190 The axioms becomes as follows.
|
|
|
1191 <p>
|
|
|
1192
|
|
|
1193 <pre>
|
|
|
1194 replacement← : {ψ : ZFSet → ZFSet} → ∀ ( X x : ZFSet ) → x ∈ X → ψ x ∈ Replace X ψ
|
|
|
1195 replacement→ : {ψ : ZFSet → ZFSet} → ∀ ( X x : ZFSet ) → ( lt : x ∈ Replace X ψ ) → ¬ ( ∀ (y : ZFSet) → ¬ ( x ≈ ψ y ) )
|
|
|
1196
|
|
|
1197 </pre>
|
|
|
1198 In the axiom, the existence of the original elements is necessary. In order to do that we use OD which has
|
|
|
1199 negation form of existential quantifier in the definition.
|
|
|
1200 <p>
|
|
|
1201
|
|
|
1202 <pre>
|
|
|
1203 in-codomain : (X : OD ) → ( ψ : OD → OD ) → OD
|
|
|
1204 in-codomain X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( def X y ∧ ( x ≡ od→ord (ψ (ord→od y ))))) }
|
|
|
1205
|
|
|
1206 </pre>
|
|
|
1207 Besides this upper bounds is required.
|
|
|
1208 <p>
|
|
|
1209
|
|
|
1210 <pre>
|
|
|
1211 Replace : OD → (OD → OD ) → OD
|
|
|
1212 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 }
|
|
|
1213
|
|
|
1214 </pre>
|
|
|
1215 We omit the proof of the validity, but it is rather straight forward.
|
|
|
1216 <p>
|
|
|
1217
|
|
|
1218 <hr/>
|
|
|
1219 <h2><a name="content048">Validity of Power Set Axiom</a></h2>
|
|
|
1220 The original Power Set Axiom is this.
|
|
|
1221 <p>
|
|
|
1222
|
|
|
1223 <pre>
|
|
|
1224 ∀ X ∃ A ∀ t ( t ∈ A ↔ t ⊆ X ) )
|
|
|
1225
|
|
|
1226 </pre>
|
|
|
1227 The existential quantifier is replaced by a function
|
|
|
1228 <p>
|
|
|
1229
|
|
|
1230 <pre>
|
|
|
1231 Power : ( A : OD ) → OD
|
|
|
1232
|
|
|
1233 </pre>
|
|
|
1234 t ⊆ X is a record like this.
|
|
|
1235 <p>
|
|
|
1236
|
|
|
1237 <pre>
|
|
|
1238 record _⊆_ ( A B : OD ) : Set (suc n) where
|
|
|
1239 field
|
|
|
1240 incl : { x : OD } → A ∋ x → B ∋ x
|
|
|
1241
|
|
|
1242 </pre>
|
|
|
1243 Axiom becomes likes this.
|
|
|
1244 <p>
|
|
|
1245
|
|
|
1246 <pre>
|
|
|
1247 power→ : ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x)
|
|
|
1248 power← : (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t
|
|
|
1249
|
|
|
1250 </pre>
|
|
|
1251 The validity of the axioms are slight complicated, we have to define set of all subset. We define
|
|
|
1252 subset in a different form.
|
|
|
1253 <p>
|
|
|
1254
|
|
|
1255 <pre>
|
|
|
1256 ZFSubset : (A x : OD ) → OD
|
|
|
1257 ZFSubset A x = record { def = λ y → def A y ∧ def x y }
|
|
|
1258
|
|
|
1259 </pre>
|
|
|
1260 We can prove,
|
|
|
1261 <p>
|
|
|
1262
|
|
|
1263 <pre>
|
|
|
1264 ( {y : OD } → x ∋ y → ZFSubset A x ∋ y ) ⇔ ( x ⊆ A )
|
|
|
1265
|
|
|
1266 </pre>
|
|
|
1267 We only have upper bound as an ordinal, but we have an obvious OD based on the order of Ordinals,
|
|
|
1268 which is an Ordinals with our Model.
|
|
|
1269 <p>
|
|
|
1270
|
|
|
1271 <pre>
|
|
|
1272 Ord : ( a : Ordinal ) → OD
|
|
|
1273 Ord a = record { def = λ y → y o< a }
|
|
|
1274 Def : (A : OD ) → OD
|
|
|
1275 Def A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) )
|
|
|
1276
|
|
|
1277 </pre>
|
|
|
1278 This is slight larger than Power A, so we replace all elements x by A ∩ x (some of them may empty).
|
|
|
1279 <p>
|
|
|
1280
|
|
|
1281 <pre>
|
|
|
1282 Power : OD → OD
|
|
|
1283 Power A = Replace (Def (Ord (od→ord A))) ( λ x → A ∩ x )
|
|
|
1284
|
|
|
1285 </pre>
|
|
|
1286 Creating Power Set of Ordinals is rather easy, then we use replacement axiom on A ∩ x since we have this.
|
|
|
1287 <p>
|
|
|
1288
|
|
|
1289 <pre>
|
|
|
1290 ∩-≡ : { a b : OD } → ({x : OD } → (a ∋ x → b ∋ x)) → a == ( b ∩ a )
|
|
|
1291
|
|
|
1292 </pre>
|
|
|
1293 In case of Ord a intro of Power Set axiom becomes valid.
|
|
|
1294 <p>
|
|
|
1295
|
|
|
1296 <pre>
|
|
|
1297 ord-power← : (a : Ordinal ) (t : OD) → ({x : OD} → (t ∋ x → (Ord a) ∋ x)) → Def (Ord a) ∋ t
|
|
|
1298
|
|
|
1299 </pre>
|
|
|
1300 Using this, we can prove,
|
|
|
1301 <p>
|
|
|
1302
|
|
|
1303 <pre>
|
|
|
1304 power→ : ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x)
|
|
|
1305 power← : (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t
|
|
|
1306
|
|
|
1307 </pre>
|
|
|
1308
|
|
|
1309 <hr/>
|
|
|
1310 <h2><a name="content049">Axiom of regularity, Axiom of choice, ε-induction</a></h2>
|
|
|
1311
|
|
|
1312 <p>
|
|
|
1313 Axiom of regularity requires non self intersectable elements (which is called minimum), if we
|
|
|
1314 replace it by a function, it becomes a choice function. It makes axiom of choice valid.
|
|
|
1315 <p>
|
|
|
1316 This means we cannot prove axiom regularity form our model, and if we postulate this, axiom of
|
|
|
1317 choice also becomes valid.
|
|
|
1318 <p>
|
|
|
1319
|
|
|
1320 <pre>
|
|
|
1321 minimal : (x : OD ) → ¬ (x == od∅ )→ OD
|
|
|
1322 x∋minimal : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → def x ( od→ord ( minimal x ne ) )
|
|
|
1323 minimal-1 : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → (y : OD ) → ¬ ( def (minimal x ne) (od→ord y)) ∧ (def x (od→ord y) )
|
|
|
1324
|
|
|
1325 </pre>
|
|
|
1326 We can avoid this using ε-induction (a predicate is valid on all set if the predicate is true on some element of set).
|
|
|
1327 Assuming law of exclude middle, they say axiom of regularity will be proved, but we haven't check it yet.
|
|
|
1328 <p>
|
|
|
1329
|
|
|
1330 <pre>
|
|
|
1331 ε-induction : { ψ : OD → Set (suc n)}
|
|
|
1332 → ( {x : OD } → ({ y : OD } → x ∋ y → ψ y ) → ψ x )
|
|
|
1333 → (x : OD ) → ψ x
|
|
|
1334
|
|
|
1335 </pre>
|
|
|
1336 In our model, we assumes the mapping between Ordinals and OD, this is actually the TransFinite induction in Ordinals.
|
|
|
1337 <p>
|
|
|
1338 The axiom of choice in the book is complicated using any pair in a set, so we use use a form in the Wikipedia.
|
|
|
1339 <p>
|
|
|
1340
|
|
|
1341 <pre>
|
|
|
1342 ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ]
|
|
|
1343
|
|
|
1344 </pre>
|
|
|
1345 We can formulate like this.
|
|
|
1346 <p>
|
|
|
1347
|
|
|
1348 <pre>
|
|
|
1349 choice-func : (X : ZFSet ) → {x : ZFSet } → ¬ ( x ≈ ∅ ) → ( X ∋ x ) → ZFSet
|
|
|
1350 choice : (X : ZFSet ) → {A : ZFSet } → ( X∋A : X ∋ A ) → (not : ¬ ( A ≈ ∅ )) → A ∋ choice-func X not X∋A
|
|
|
1351
|
|
|
1352 </pre>
|
|
|
1353 It does not requires ∅ ∉ X .
|
|
|
1354 <p>
|
|
|
1355
|
|
|
1356 <hr/>
|
|
|
1357 <h2><a name="content050">Axiom of choice and Law of Excluded Middle</a></h2>
|
|
|
1358 In our model, since OD has a mapping to Ordinals, it has evident order, which means well ordering theorem is valid,
|
|
|
1359 but it don't have correct form of the axiom yet. They say well ordering axiom is equivalent to the axiom of choice,
|
|
|
1360 but it requires law of the exclude middle.
|
|
|
1361 <p>
|
|
|
1362 Actually, it is well known to prove law of the exclude middle from axiom of choice in intuitionistic logic, and we can
|
|
|
1363 perform the proof in our mode. Using the definition like this, predicates and ODs are related and we can ask the
|
|
|
1364 set is empty or not if we have an axiom of choice, so we have the law of the exclude middle p ∨ ( ¬ p ) .
|
|
|
1365 <p>
|
|
|
1366
|
|
|
1367 <pre>
|
|
|
1368 ppp : { p : Set n } { a : OD } → record { def = λ x → p } ∋ a → p
|
|
|
1369 ppp {p} {a} d = d
|
|
|
1370
|
|
|
1371 </pre>
|
|
|
1372 We can prove axiom of choice from law excluded middle since we have TransFinite induction. So Axiom of choice
|
|
|
1373 and Law of Excluded Middle is equivalent in our mode.
|
|
|
1374 <p>
|
|
|
1375
|
|
|
1376 <hr/>
|
|
|
1377 <h2><a name="content051">Relation-ship among ZF axiom</a></h2>
|
|
|
1378 <img src="fig/axiom-dependency.svg">
|
|
|
1379
|
|
|
1380 <p>
|
|
|
1381
|
|
|
1382 <hr/>
|
|
|
1383 <h2><a name="content052">Non constructive assumption in our model</a></h2>
|
|
|
1384 mapping between OD and Ordinals
|
|
|
1385 <p>
|
|
|
1386
|
|
|
1387 <pre>
|
|
|
1388 od→ord : OD → Ordinal
|
|
|
1389 ord→od : Ordinal → OD
|
|
|
1390 oiso : {x : OD } → ord→od ( od→ord x ) ≡ x
|
|
|
1391 diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x
|
|
|
1392 c<→o< : {x y : OD } → def y ( od→ord x ) → od→ord x o< od→ord y
|
|
|
1393
|
|
|
1394 </pre>
|
|
|
1395 Equivalence on OD
|
|
|
1396 <p>
|
|
|
1397
|
|
|
1398 <pre>
|
|
|
1399 ==→o≡ : { x y : OD } → (x == y) → x ≡ y
|
|
|
1400
|
|
|
1401 </pre>
|
|
|
1402 Upper bound
|
|
|
1403 <p>
|
|
|
1404
|
|
|
1405 <pre>
|
|
|
1406 sup-o : ( Ordinal → Ordinal ) → Ordinal
|
|
|
1407 sup-o< : { ψ : Ordinal → Ordinal } → ∀ {x : Ordinal } → ψ x o< sup-o ψ
|
|
|
1408
|
|
|
1409 </pre>
|
|
|
1410 Axiom of choice and strong axiom of regularity.
|
|
|
1411 <p>
|
|
|
1412
|
|
|
1413 <pre>
|
|
|
1414 minimal : (x : OD ) → ¬ (x == od∅ )→ OD
|
|
|
1415 x∋minimal : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → def x ( od→ord ( minimal x ne ) )
|
|
|
1416 minimal-1 : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → (y : OD ) → ¬ ( def (minimal x ne) (od→ord y)) ∧ (def x (od→ord y) )
|
|
|
1417
|
|
|
1418 </pre>
|
|
|
1419
|
|
|
1420 <hr/>
|
|
|
1421 <h2><a name="content053">So it this correct?</a></h2>
|
|
|
1422
|
|
|
1423 <p>
|
|
|
1424 Our axiom are syntactically the same in the text book, but negations are slightly different.
|
|
|
1425 <p>
|
|
|
1426 If we assumes excluded middle, these are exactly same.
|
|
|
1427 <p>
|
|
|
1428 Even if we assumes excluded middle, intuitionistic logic itself remains consistent, but we cannot prove it.
|
|
|
1429 <p>
|
|
|
1430 Except the upper bound, axioms are simple logical relation.
|
|
|
1431 <p>
|
|
|
1432 Proof of existence of mapping between OD and Ordinals are not obvious. We don't know we prove it or not.
|
|
|
1433 <p>
|
|
|
1434 Existence of the Upper bounds is a pure assumption, if we have not limit on Ordinals, it may contradicts,
|
|
|
1435 but we don't have explicit upper limit on Ordinals.
|
|
|
1436 <p>
|
|
|
1437 Several inference on our model or our axioms are basically parallel to the set theory text book, so it looks like correct.
|
|
|
1438 <p>
|
|
|
1439
|
|
|
1440 <hr/>
|
|
|
1441 <h2><a name="content054">How to use Agda Set Theory</a></h2>
|
|
|
1442 Assuming record ZF, classical set theory can be developed. If necessary, axiom of choice can be
|
|
|
1443 postulated or assuming law of excluded middle.
|
|
|
1444 <p>
|
|
|
1445 Instead, simply assumes non constructive assumption, various theory can be developed. We haven't check
|
|
|
1446 these assumptions are proved in record ZF, so we are not sure, these development is a result of ZF Set theory.
|
|
|
1447 <p>
|
|
|
1448 ZF record itself is not necessary, for example, topology theory without ZF can be possible.
|
|
|
1449 <p>
|
|
|
1450
|
|
|
1451 <hr/>
|
|
|
1452 <h2><a name="content055">Topos and Set Theory</a></h2>
|
|
|
1453 Topos is a mathematical structure in Category Theory, which is a Cartesian closed category which has a
|
|
|
1454 sub-object classifier.
|
|
|
1455 <p>
|
|
|
1456 Topos itself is model of intuitionistic logic.
|
|
|
1457 <p>
|
|
|
1458 Transitive Sets are objects of Cartesian closed category.
|
|
|
1459 It is possible to introduce Power Set Functor on it
|
|
|
1460 <p>
|
|
|
1461 We can use replacement A ∩ x for each element in Transitive Set,
|
|
|
1462 in the similar way of our power set axiom. I
|
|
|
1463 <p>
|
|
|
1464 A model of ZF Set theory can be constructed on top of the Topos which is shown in Oisus.
|
|
|
1465 <p>
|
|
|
1466 Our Agda model is a proof theoretic version of it.
|
|
|
1467 <p>
|
|
|
1468
|
|
|
1469 <hr/>
|
|
|
1470 <h2><a name="content056">Cardinal number and Continuum hypothesis</a></h2>
|
|
|
1471 Axiom of choice is required to define cardinal number
|
|
|
1472 <p>
|
|
|
1473 definition of cardinal number is not yet done
|
|
|
1474 <p>
|
|
|
1475 definition of filter is not yet done
|
|
|
1476 <p>
|
|
|
1477 we may have a model without axiom of choice or without continuum hypothesis
|
|
|
1478 <p>
|
|
|
1479 Possible representation of continuum hypothesis is this.
|
|
|
1480 <p>
|
|
|
1481
|
|
|
1482 <pre>
|
|
|
1483 continuum-hyphotheis : (a : Ordinal) → Power (Ord a) ⊆ Ord (osuc a)
|
|
|
1484
|
|
|
1485 </pre>
|
|
|
1486
|
|
|
1487 <hr/>
|
|
|
1488 <h2><a name="content057">Filter</a></h2>
|
|
|
1489
|
|
|
1490 <p>
|
|
|
1491 filter is a dual of ideal on boolean algebra or lattice. Existence on natural number
|
|
|
1492 is depends on axiom of choice.
|
|
|
1493 <p>
|
|
|
1494
|
|
|
1495 <pre>
|
|
|
1496 record Filter ( L : OD ) : Set (suc n) where
|
|
|
1497 field
|
|
|
1498 filter : OD
|
|
|
1499 proper : ¬ ( filter ∋ od∅ )
|
|
|
1500 inL : filter ⊆ L
|
|
|
1501 filter1 : { p q : OD } → q ⊆ L → filter ∋ p → p ⊆ q → filter ∋ q
|
|
|
1502 filter2 : { p q : OD } → filter ∋ p → filter ∋ q → filter ∋ (p ∩ q)
|
|
|
1503
|
|
|
1504 </pre>
|
|
|
1505 We may construct a model of non standard analysis or set theory.
|
|
|
1506 <p>
|
|
|
1507 This may be simpler than classical forcing theory ( not yet done).
|
|
|
1508 <p>
|
|
|
1509
|
|
|
1510 <hr/>
|
|
|
1511 <h2><a name="content058">Programming Mathematics</a></h2>
|
|
|
1512 Mathematics is a functional programming in Agda where proof is a value of a variable. The mathematical
|
|
|
1513 structure are
|
|
|
1514 <p>
|
|
|
1515
|
|
|
1516 <pre>
|
|
|
1517 record and data
|
|
|
1518
|
|
|
1519 </pre>
|
|
|
1520 Proof is check by type consistency not by the computation, but it may include some normalization.
|
|
|
1521 <p>
|
|
|
1522 Type inference and termination is not so clear in multi recursions.
|
|
|
1523 <p>
|
|
|
1524 Defining Agda record is a good way to understand mathematical theory, for examples,
|
|
|
1525 <p>
|
|
|
1526
|
|
|
1527 <pre>
|
|
|
1528 Category theory ( Yoneda lemma, Floyd Adjunction functor theorem, Applicative functor )
|
|
|
1529 Automaton ( Subset construction、Language containment)
|
|
|
1530
|
|
|
1531 </pre>
|
|
|
1532 are proved in Agda.
|
|
|
1533 <p>
|
|
|
1534
|
|
|
1535 <hr/>
|
|
|
1536 <h2><a name="content059">link</a></h2>
|
|
|
1537 Summer school of foundation of mathematics (in Japanese)<br> <a href="https://www.sci.shizuoka.ac.jp/~math/yorioka/ss2019/">https://www.sci.shizuoka.ac.jp/~math/yorioka/ss2019/</a>
|
|
|
1538 <p>
|
|
|
1539 Foundation of axiomatic set theory (in Japanese)<br> <a href="https://www.sci.shizuoka.ac.jp/~math/yorioka/ss2019/sakai0.pdf">https://www.sci.shizuoka.ac.jp/~math/yorioka/ss2019/sakai0.pdf
|
|
|
1540 </a>
|
|
|
1541 <p>
|
|
|
1542 Agda
|
|
|
1543 <br> <a href="https://agda.readthedocs.io/en/v2.6.0.1/">https://agda.readthedocs.io/en/v2.6.0.1/</a>
|
|
|
1544 <p>
|
|
|
1545 ZF-in-Agda source
|
|
|
1546 <br> <a href="https://github.com/shinji-kono/zf-in-agda.git">https://github.com/shinji-kono/zf-in-agda.git
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1547 </a>
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1548 <p>
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1549 Category theory in Agda source
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1550 <br> <a href="https://github.com/shinji-kono/category-exercise-in-agda">https://github.com/shinji-kono/category-exercise-in-agda
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1551 </a>
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1552 <p>
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1553 </div>
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1554 <ol class="side" id="menu">
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1555 Constructing ZF Set Theory in Agda
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1556 <li><a href="#content000"> Programming Mathematics</a>
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1557 <li><a href="#content001"> Target</a>
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1558 <li><a href="#content002"> Why Set Theory</a>
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1559 <li><a href="#content003"> Agda and Intuitionistic Logic </a>
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1560 <li><a href="#content004"> Introduction of Agda </a>
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1561 <li><a href="#content005"> data ( Sum type )</a>
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1562 <li><a href="#content006"> A → B means "A implies B"</a>
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1563 <li><a href="#content007"> introduction と elimination</a>
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1564 <li><a href="#content008"> To prove A → B </a>
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1565 <li><a href="#content009"> A ∧ B</a>
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1566 <li><a href="#content010"> record</a>
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1567 <li><a href="#content011"> Mathematical structure</a>
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1568 <li><a href="#content012"> A Model and a theory</a>
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1569 <li><a href="#content013"> postulate と module</a>
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1570 <li><a href="#content014"> A ∨ B</a>
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1571 <li><a href="#content015"> Negation</a>
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1572 <li><a href="#content016"> Equality </a>
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1573 <li><a href="#content017"> Equivalence relation</a>
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1574 <li><a href="#content018"> Ordering</a>
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1575 <li><a href="#content019"> Quantifier</a>
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1576 <li><a href="#content020"> Can we do math in this way?</a>
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1577 <li><a href="#content021"> Things which Agda cannot prove</a>
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1578 <li><a href="#content022"> Classical story of ZF Set Theory</a>
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1579 <li><a href="#content023"> Ordinals</a>
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1580 <li><a href="#content024"> Axiom of Ordinals</a>
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1581 <li><a href="#content025"> Concrete Ordinals</a>
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1582 <li><a href="#content026"> Model of Ordinals</a>
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1583 <li><a href="#content027"> Debugging axioms using Model</a>
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1584 <li><a href="#content028"> Countable Ordinals can contains uncountable set?</a>
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1585 <li><a href="#content029"> What is Set</a>
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1586 <li><a href="#content030"> We don't ask the contents of Set. It can be anything.</a>
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1587 <li><a href="#content031"> Ordinal Definable Set</a>
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1588 <li><a href="#content032"> ∋ in OD</a>
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1589 <li><a href="#content033"> 1 to 1 mapping between an OD and an Ordinal</a>
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1590 <li><a href="#content034"> Order preserving in the mapping of OD and Ordinal</a>
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1591 <li><a href="#content035"> ISO</a>
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1592 <li><a href="#content036"> Various Sets</a>
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1593 <li><a href="#content037"> Fixes on ZF to intuitionistic logic</a>
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1594 <li><a href="#content038"> Pure logical axioms</a>
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1595 <li><a href="#content039"> Axiom of Pair</a>
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1596 <li><a href="#content040"> pair in OD</a>
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1597 <li><a href="#content041"> Validity of Axiom of Pair</a>
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1598 <li><a href="#content042"> Equality of OD and Axiom of Extensionality </a>
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1599 <li><a href="#content043"> Validity of Axiom of Extensionality</a>
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1600 <li><a href="#content044"> Non constructive assumptions so far</a>
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1601 <li><a href="#content045"> Axiom which have negation form</a>
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1602 <li><a href="#content046"> Union </a>
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1603 <li><a href="#content047"> Axiom of replacement</a>
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1604 <li><a href="#content048"> Validity of Power Set Axiom</a>
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1605 <li><a href="#content049"> Axiom of regularity, Axiom of choice, ε-induction</a>
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1606 <li><a href="#content050"> Axiom of choice and Law of Excluded Middle</a>
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1607 <li><a href="#content051"> Relation-ship among ZF axiom</a>
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1608 <li><a href="#content052"> Non constructive assumption in our model</a>
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1609 <li><a href="#content053"> So it this correct?</a>
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1610 <li><a href="#content054"> How to use Agda Set Theory</a>
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1611 <li><a href="#content055"> Topos and Set Theory</a>
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1612 <li><a href="#content056"> Cardinal number and Continuum hypothesis</a>
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1613 <li><a href="#content057"> Filter</a>
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1614 <li><a href="#content058"> Programming Mathematics</a>
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1615 <li><a href="#content059"> link</a>
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1616 </ol>
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1617
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1618 <hr/> Shinji KONO / Sat Jan 11 20:04:01 2020
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1619 </body></html>
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