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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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279
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41 <h2><a name="content000">ZF in Agda</a></h2>
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42
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43 <pre>
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44 zf.agda axiom of ZF
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45 zfc.agda axiom of choice
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46 Ordinals.agda axiom of Ordinals
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47 ordinal.agda countable model of Ordinals
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48 OD.agda model of ZF based on Ordinal Definable Set with assumptions
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49 ODC.agda Law of exclude middle from axiom of choice assumptions
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50 LEMC.agda model of choice with assumption of the Law of exclude middle
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51 OPair.agda ordered pair on OD
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52 BAlgbra.agda Boolean algebra on OD (not yet done)
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53 filter.agda Filter on OD (not yet done)
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54 cardinal.agda Caedinal number on OD (not yet done)
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55 logic.agda some basics on logic
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56 nat.agda some basics on Nat
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57
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58 </pre>
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59
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60 <hr/>
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61 <h2><a name="content001">Programming Mathematics</a></h2>
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62
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63 <p>
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273
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64 Programming is processing data structure with λ terms.
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65 <p>
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66 We are going to handle Mathematics in intuitionistic logic with λ terms.
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67 <p>
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68 Mathematics is a functional programming which values are proofs.
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69 <p>
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70 Programming ZF Set Theory in Agda
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71 <p>
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72
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73 <hr/>
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279
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74 <h2><a name="content002">Target</a></h2>
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273
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75
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76 <pre>
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77 Describe ZF axioms in Agda
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78 Construction a Model of ZF Set Theory in Agda
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79 Show necessary assumptions for the model
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80 Show validities of ZF axioms on the model
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81
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82 </pre>
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83 This shows consistency of Set Theory (with some assumptions),
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84 without circulating ZF Theory assumption.
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85 <p>
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86 <a href="https://github.com/shinji-kono/zf-in-agda">
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87 ZF in Agda https://github.com/shinji-kono/zf-in-agda
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88 </a>
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89 <p>
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90
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91 <hr/>
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279
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92 <h2><a name="content003">Why Set Theory</a></h2>
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273
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93 If we can formulate Set theory, it suppose to work on any mathematical theory.
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94 <p>
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95 Set Theory is a difficult point for beginners especially axiom of choice.
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96 <p>
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97 It has some amount of difficulty and self circulating discussion.
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98 <p>
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99 I'm planning to do it in my old age, but I'm enough age now.
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100 <p>
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101 This is done during from May to September.
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102 <p>
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103
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104 <hr/>
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279
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105 <h2><a name="content004">Agda and Intuitionistic Logic </a></h2>
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273
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106 Curry Howard Isomorphism
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107 <p>
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108
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109 <pre>
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110 Proposition : Proof ⇔ Type : Value
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111
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112 </pre>
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113 which means
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114 <p>
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115 constructing a typed lambda calculus which corresponds a logic
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116 <p>
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117 Typed lambda calculus which allows complex type as a value of a variable (System FC)
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118 <p>
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119 First class Type / Dependent Type
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120 <p>
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121 Agda is a such a programming language which has similar syntax of Haskell
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122 <p>
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123 Coq is specialized in proof assistance such as command and tactics .
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124 <p>
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125
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126 <hr/>
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279
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127 <h2><a name="content005">Introduction of Agda </a></h2>
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273
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128 A length of a list of type A.
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129 <p>
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130
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131 <pre>
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132 length : {A : Set } → List A → Nat
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133 length [] = zero
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134 length (_ ∷ t) = suc ( length t )
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135
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136 </pre>
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137 Simple functional programming language. Type declaration is mandatory.
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138 A colon means type, an equal means value. Indentation based.
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139 <p>
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140 Set is a base type (which may have a level ).
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141 <p>
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142 {} means implicit variable which can be omitted if Agda infers its value.
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143 <p>
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144
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145 <hr/>
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279
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146 <h2><a name="content006">data ( Sum type )</a></h2>
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273
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147 A data type which as exclusive multiple constructors. A similar one as
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148 union in C or case class in Scala.
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149 <p>
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150 It has a similar syntax as Haskell but it has a slight difference.
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151 <p>
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152
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153 <pre>
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154 data List (A : Set ) : Set where
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155 [] : List A
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156 _∷_ : A → List A → List A
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157
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158 </pre>
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159 _∷_ means infix operator. If use explicit _, it can be used in a normal function
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160 syntax.
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161 <p>
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162 Natural number can be defined as a usual way.
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163 <p>
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164
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165 <pre>
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166 data Nat : Set where
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167 zero : Nat
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168 suc : Nat → Nat
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169
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170 </pre>
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171
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172 <hr/>
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279
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173 <h2><a name="content007"> A → B means "A implies B"</a></h2>
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273
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174
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175 <p>
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176 In Agda, a type can be a value of a variable, which is usually called dependent type.
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177 Type has a name Set in Agda.
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178 <p>
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179
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180 <pre>
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181 ex3 : {A B : Set} → Set
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182 ex3 {A}{B} = A → B
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183
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184 </pre>
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185 ex3 is a type : A → B, which is a value of Set. It also means a formula : A implies B.
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186 <p>
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187
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188 <pre>
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189 A type is a formula, the value is the proof
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190
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191 </pre>
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192 A value of A → B can be interpreted as an inference from the formula A to the formula B, which
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193 can be a function from a proof of A to a proof of B.
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194 <p>
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195
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196 <hr/>
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279
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197 <h2><a name="content008">introduction と elimination</a></h2>
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273
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198 For a logical operator, there are two types of inference, an introduction and an elimination.
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199 <p>
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200
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201 <pre>
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202 intro creating symbol / constructor / introduction
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203 elim using symbolic / accessors / elimination
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204
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205 </pre>
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206 In Natural deduction, this can be written in proof schema.
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207 <p>
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208
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209 <pre>
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210 A
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211 :
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212 B A A → B
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213 ------------- →intro ------------------ →elim
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214 A → B B
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215
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216 </pre>
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217 In Agda, this is a pair of type and value as follows. Introduction of → uses λ.
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218 <p>
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219
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220 <pre>
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221 →intro : {A B : Set } → A → B → ( A → B )
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222 →intro _ b = λ x → b
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223 →elim : {A B : Set } → A → ( A → B ) → B
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224 →elim a f = f a
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225
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226 </pre>
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227 Important
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228 <p>
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229
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230 <pre>
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231 {A B : Set } → A → B → ( A → B )
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232
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233 </pre>
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234 is
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235 <p>
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236
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237 <pre>
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238 {A B : Set } → ( A → ( B → ( A → B ) ))
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239
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240 </pre>
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241 This makes currying of function easy.
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242 <p>
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243
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244 <hr/>
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279
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245 <h2><a name="content009"> To prove A → B </a></h2>
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273
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246 Make a left type as an argument. (intros in Coq)
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247 <p>
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248
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249 <pre>
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250 ex5 : {A B C : Set } → A → B → C → ?
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251 ex5 a b c = ?
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252
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253 </pre>
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254 ? is called a hole, which is unspecified part. Agda tell us which kind type is required for the Hole.
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255 <p>
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256 We are going to fill the holes, and if we have no warnings nor errors such as type conflict (Red),
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257 insufficient proof or instance (Yellow), Non-termination, the proof is completed.
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258 <p>
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259
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260 <hr/>
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279
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261 <h2><a name="content010"> A ∧ B</a></h2>
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273
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262 Well known conjunction's introduction and elimination is as follow.
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263 <p>
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264
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265 <pre>
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266 A B A ∧ B A ∧ B
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267 ------------- ----------- proj1 ---------- proj2
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268 A ∧ B A B
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269
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270 </pre>
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271 We can introduce a corresponding structure in our functional programming language.
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272 <p>
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273
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274 <hr/>
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279
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275 <h2><a name="content011"> record</a></h2>
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273
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276
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277 <pre>
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278 record _∧_ A B : Set
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279 field
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280 proj1 : A
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281 proj2 : B
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282
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283 </pre>
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284 _∧_ means infix operator. _∧_ A B can be written as A ∧ B (Haskell uses (∧) )
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285 <p>
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286 This a type which constructed from type A and type B. You may think this as an object
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287 or struct.
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288 <p>
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289
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290 <pre>
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291 record { proj1 = x ; proj2 = y }
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292
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293 </pre>
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294 is a constructor of _∧_.
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295 <p>
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296
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297 <pre>
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298 ex3 : {A B : Set} → A → B → ( A ∧ B )
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299 ex3 a b = record { proj1 = a ; proj2 = b }
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300 ex1 : {A B : Set} → ( A ∧ B ) → A
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301 ex1 a∧b = proj1 a∧b
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302
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303 </pre>
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304 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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305 except parenthesis or colons. A symbol requires space separation such as a type defining colon.
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306 <p>
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307 Defining record can be recursively, but we don't use the recursion here.
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308 <p>
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309
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310 <hr/>
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279
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311 <h2><a name="content012"> Mathematical structure</a></h2>
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273
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312 We have types of elements and the relationship in a mathematical structure.
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313 <p>
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314
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315 <pre>
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316 logical relation has no ordering
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317 there is a natural ordering in arguments and a value in a function
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318
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319 </pre>
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320 So we have typical definition style of mathematical structure with records.
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321 <p>
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322
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323 <pre>
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324 record IsOrdinals {n : Level} (ord : Set n)
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325 (_o<_ : ord → ord → Set n) : Set (suc (suc n)) where
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326 field
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327 Otrans : {x y z : ord } → x o< y → y o< z → x o< z
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328 record Ordinals {n : Level} : Set (suc (suc n)) where
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329 field
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330 ord : Set n
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331 _o<_ : ord → ord → Set n
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332 isOrdinal : IsOrdinals ord _o<_
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333
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334 </pre>
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335 In IsOrdinals, axioms are written in flat way. In Ordinal, we may have
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336 inputs and outputs are put in the field including IsOrdinal.
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337 <p>
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338 Fields of Ordinal is existential objects in the mathematical structure.
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339 <p>
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340
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341 <hr/>
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279
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342 <h2><a name="content013"> A Model and a theory</a></h2>
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273
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343 Agda record is a type, so we can write it in the argument, but is it really exists?
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344 <p>
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345 If we have a value of the record, it simply exists, that is, we need to create all the existence
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346 in the record satisfies all the axioms (= field of IsOrdinal) should be valid.
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347 <p>
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348
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349 <pre>
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350 type of record = theory
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351 value of record = model
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352
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353 </pre>
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354 We call the value of the record as a model. If mathematical structure has a
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355 model, it exists. Pretty Obvious.
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356 <p>
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357
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358 <hr/>
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279
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359 <h2><a name="content014"> postulate と module</a></h2>
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273
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360 Agda proofs are separated by modules, which are large records.
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361 <p>
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362 postulates are assumptions. We can assume a type without proofs.
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363 <p>
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364
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365 <pre>
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366 postulate
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367 sup-o : ( Ordinal → Ordinal ) → Ordinal
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368 sup-o< : { ψ : Ordinal → Ordinal } → ∀ {x : Ordinal } → ψ x o< sup-o ψ
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369
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370 </pre>
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371 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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372 <p>
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373 Writing some type in a module argument is the same as postulating a type, but
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374 postulate can be written the middle of a proof.
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375 <p>
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376 postulate can be constructive.
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377 <p>
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378 postulate can be inconsistent, which result everything has a proof.
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379 <p>
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380
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381 <hr/>
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279
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382 <h2><a name="content015"> A ∨ B</a></h2>
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273
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383
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384 <pre>
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385 data _∨_ (A B : Set) : Set where
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386 case1 : A → A ∨ B
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387 case2 : B → A ∨ B
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388
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389 </pre>
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390 As Haskell, case1/case2 are patterns.
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391 <p>
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392
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393 <pre>
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394 ex3 : {A B : Set} → ( A ∨ A ) → A
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395 ex3 = ?
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396
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397 </pre>
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398 In a case statement, Agda command C-C C-C generates possible cases in the head.
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399 <p>
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400
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401 <pre>
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402 ex3 : {A B : Set} → ( A ∨ A ) → A
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403 ex3 (case1 x) = ?
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404 ex3 (case2 x) = ?
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405
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406 </pre>
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407 Proof schema of ∨ is omit due to the complexity.
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408 <p>
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409
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410 <hr/>
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279
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411 <h2><a name="content016"> Negation</a></h2>
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273
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412
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413 <pre>
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414 ⊥
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415 ------------- ⊥-elim
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416 A
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417
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418 </pre>
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419 Anything can be derived from bottom, in this case a Set A. There is no introduction rule
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420 in ⊥, which can be implemented as data which has no constructor.
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421 <p>
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422
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423 <pre>
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424 data ⊥ : Set where
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425
|
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426 </pre>
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427 ⊥-elim can be proved like this.
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428 <p>
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429
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430 <pre>
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431 ⊥-elim : {A : Set } -> ⊥ -> A
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432 ⊥-elim ()
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433
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434 </pre>
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435 () means no match argument nor value.
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436 <p>
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437 A negation can be defined using ⊥ like this.
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438 <p>
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439
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440 <pre>
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441 ¬_ : Set → Set
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442 ¬ A = A → ⊥
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443
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444 </pre>
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445
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446 <hr/>
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279
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447 <h2><a name="content017">Equality </a></h2>
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273
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448
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449 <p>
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450 All the value in Agda are terms. If we have the same normalized form, two terms are equal.
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451 If we have variables in the terms, we will perform an unification. unifiable terms are equal.
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452 We don't go further on the unification.
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453 <p>
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454
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455 <pre>
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456 { x : A } x ≡ y f x y
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457 --------------- ≡-intro --------------------- ≡-elim
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458 x ≡ x f x x
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459
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460 </pre>
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461 equality _≡_ can be defined as a data.
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462 <p>
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463
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464 <pre>
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465 data _≡_ {A : Set } : A → A → Set where
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466 refl : {x : A} → x ≡ x
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467
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468 </pre>
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469 The elimination of equality is a substitution in a term.
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470 <p>
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471
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472 <pre>
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473 subst : {A : Set } → { x y : A } → ( f : A → Set ) → x ≡ y → f x → f y
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474 subst {A} {x} {y} f refl fx = fx
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475 ex5 : {A : Set} {x y z : A } → x ≡ y → y ≡ z → x ≡ z
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476 ex5 {A} {x} {y} {z} x≡y y≡z = subst ( λ k → x ≡ k ) y≡z x≡y
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477
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478 </pre>
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479
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480 <hr/>
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279
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481 <h2><a name="content018">Equivalence relation</a></h2>
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273
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482
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483 <p>
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484
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485 <pre>
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486 refl' : {A : Set} {x : A } → x ≡ x
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487 refl' = ?
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488 sym : {A : Set} {x y : A } → x ≡ y → y ≡ x
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489 sym = ?
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490 trans : {A : Set} {x y z : A } → x ≡ y → y ≡ z → x ≡ z
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491 trans = ?
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492 cong : {A B : Set} {x y : A } { f : A → B } → x ≡ y → f x ≡ f y
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493 cong = ?
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494
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495 </pre>
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496
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497 <hr/>
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279
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498 <h2><a name="content019">Ordering</a></h2>
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273
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499
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500 <p>
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501 Relation is a predicate on two value which has a same type.
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502 <p>
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503
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504 <pre>
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505 A → A → Set
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506
|
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507 </pre>
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508 Defining order is the definition of this type with predicate or a data.
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509 <p>
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510
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511 <pre>
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512 data _≤_ : Rel ℕ 0ℓ where
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513 z≤n : ∀ {n} → zero ≤ n
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514 s≤s : ∀ {m n} (m≤n : m ≤ n) → suc m ≤ suc n
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515
|
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|
516 </pre>
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517
|
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518 <hr/>
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279
|
519 <h2><a name="content020">Quantifier</a></h2>
|
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273
|
520
|
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521 <p>
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522 Handling quantifier in an intuitionistic logic requires special cares.
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523 <p>
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524 In the input of a function, there are no restriction on it, that is, it has
|
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525 a universal quantifier. (If we explicitly write ∀, Agda gives us a type inference on it)
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526 <p>
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527 There is no ∃ in agda, the one way is using negation like this.
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|
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528 <p>
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529 ∃ (x : A ) → p x = ¬ ( ( x : A ) → ¬ ( p x ) )
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530 <p>
|
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531 On the another way, f : A can be used like this.
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532 <p>
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533
|
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534 <pre>
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535 p f
|
|
|
536
|
|
|
537 </pre>
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538 If we use a function which can be defined globally which has stronger meaning the
|
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539 usage of ∃ x in a logical expression.
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|
|
540 <p>
|
|
|
541
|
|
|
542 <hr/>
|
|
279
|
543 <h2><a name="content021">Can we do math in this way?</a></h2>
|
|
273
|
544 Yes, we can. Actually we have Principia Mathematica by Russell and Whitehead (with out computer support).
|
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545 <p>
|
|
|
546 In some sense, this story is a reprinting of the work, (but Principia Mathematica has a different formulation than ZF).
|
|
|
547 <p>
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|
548
|
|
|
549 <pre>
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550 define mathematical structure as a record
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551 program inferences as if we have proofs in variables
|
|
|
552
|
|
|
553 </pre>
|
|
|
554
|
|
|
555 <hr/>
|
|
279
|
556 <h2><a name="content022">Things which Agda cannot prove</a></h2>
|
|
273
|
557
|
|
|
558 <p>
|
|
|
559 The infamous Internal Parametricity is a limitation of Agda, it cannot prove so called Free Theorem, which
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|
|
560 leads uniqueness of a functor in Category Theory.
|
|
|
561 <p>
|
|
|
562 Functional extensionality cannot be proved.
|
|
|
563 <pre>
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|
564 (∀ x → f x ≡ g x) → f ≡ g
|
|
|
565
|
|
|
566 </pre>
|
|
|
567 Agda has no law of exclude middle.
|
|
|
568 <p>
|
|
|
569
|
|
|
570 <pre>
|
|
|
571 a ∨ ( ¬ a )
|
|
|
572
|
|
|
573 </pre>
|
|
|
574 For example, (A → B) → ¬ B → ¬ A can be proved but, ( ¬ B → ¬ A ) → A → B cannot.
|
|
|
575 <p>
|
|
|
576 It also other problems such as termination, type inference or unification which we may overcome with
|
|
|
577 efforts or devices or may not.
|
|
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578 <p>
|
|
|
579 If we cannot prove something, we can safely postulate it unless it leads a contradiction.
|
|
|
580 <pre>
|
|
|
581
|
|
|
582
|
|
|
583 </pre>
|
|
|
584
|
|
|
585 <hr/>
|
|
279
|
586 <h2><a name="content023">Classical story of ZF Set Theory</a></h2>
|
|
273
|
587
|
|
|
588 <p>
|
|
|
589 Assuming ZF, constructing a model of ZF is a flow of classical Set Theory, which leads
|
|
|
590 a relative consistency proof of the Set Theory.
|
|
|
591 Ordinal number is used in the flow.
|
|
|
592 <p>
|
|
|
593 In Agda, first we defines Ordinal numbers (Ordinals), then introduce Ordinal Definable Set (OD).
|
|
|
594 We need some non constructive assumptions in the construction. A model of Set theory is
|
|
|
595 constructed based on these assumptions.
|
|
|
596 <p>
|
|
|
597 <img src="fig/set-theory.svg">
|
|
|
598
|
|
|
599 <p>
|
|
|
600
|
|
|
601 <hr/>
|
|
279
|
602 <h2><a name="content024">Ordinals</a></h2>
|
|
273
|
603 Ordinals are our intuition of infinite things, which has ∅ and orders on the things.
|
|
|
604 It also has a successor osuc.
|
|
|
605 <p>
|
|
|
606
|
|
|
607 <pre>
|
|
|
608 record Ordinals {n : Level} : Set (suc (suc n)) where
|
|
|
609 field
|
|
|
610 ord : Set n
|
|
|
611 o∅ : ord
|
|
|
612 osuc : ord → ord
|
|
|
613 _o<_ : ord → ord → Set n
|
|
|
614 isOrdinal : IsOrdinals ord o∅ osuc _o<_
|
|
|
615
|
|
|
616 </pre>
|
|
|
617 It is different from natural numbers in way. The order of Ordinals is not defined in terms
|
|
|
618 of successor. It is given from outside, which make it possible to have higher order infinity.
|
|
|
619 <p>
|
|
|
620
|
|
|
621 <hr/>
|
|
279
|
622 <h2><a name="content025">Axiom of Ordinals</a></h2>
|
|
273
|
623 Properties of infinite things. We request a transfinite induction, which states that if
|
|
|
624 some properties are satisfied below all possible ordinals, the properties are true on all
|
|
|
625 ordinals.
|
|
|
626 <p>
|
|
|
627 Successor osuc has no ordinal between osuc and the base ordinal. There are some ordinals
|
|
|
628 which is not a successor of any ordinals. It is called limit ordinal.
|
|
|
629 <p>
|
|
|
630 Any two ordinal can be compared, that is less, equal or more, that is total order.
|
|
|
631 <p>
|
|
|
632
|
|
|
633 <pre>
|
|
|
634 record IsOrdinals {n : Level} (ord : Set n) (o∅ : ord )
|
|
|
635 (osuc : ord → ord )
|
|
|
636 (_o<_ : ord → ord → Set n) : Set (suc (suc n)) where
|
|
|
637 field
|
|
|
638 Otrans : {x y z : ord } → x o< y → y o< z → x o< z
|
|
|
639 OTri : Trichotomous {n} _≡_ _o<_
|
|
|
640 ¬x<0 : { x : ord } → ¬ ( x o< o∅ )
|
|
|
641 <-osuc : { x : ord } → x o< osuc x
|
|
|
642 osuc-≡< : { a x : ord } → x o< osuc a → (x ≡ a ) ∨ (x o< a)
|
|
|
643 TransFinite : { ψ : ord → Set (suc n) }
|
|
|
644 → ( (x : ord) → ( (y : ord ) → y o< x → ψ y ) → ψ x )
|
|
|
645 → ∀ (x : ord) → ψ x
|
|
|
646
|
|
|
647 </pre>
|
|
|
648
|
|
|
649 <hr/>
|
|
279
|
650 <h2><a name="content026">Concrete Ordinals</a></h2>
|
|
273
|
651
|
|
|
652 <p>
|
|
|
653 We can define a list like structure with level, which is a kind of two dimensional infinite array.
|
|
|
654 <p>
|
|
|
655
|
|
|
656 <pre>
|
|
|
657 data OrdinalD {n : Level} : (lv : Nat) → Set n where
|
|
|
658 Φ : (lv : Nat) → OrdinalD lv
|
|
|
659 OSuc : (lv : Nat) → OrdinalD {n} lv → OrdinalD lv
|
|
|
660
|
|
|
661 </pre>
|
|
|
662 The order of the OrdinalD can be defined in this way.
|
|
|
663 <p>
|
|
|
664
|
|
|
665 <pre>
|
|
|
666 data _d<_ {n : Level} : {lx ly : Nat} → OrdinalD {n} lx → OrdinalD {n} ly → Set n where
|
|
|
667 Φ< : {lx : Nat} → {x : OrdinalD {n} lx} → Φ lx d< OSuc lx x
|
|
|
668 s< : {lx : Nat} → {x y : OrdinalD {n} lx} → x d< y → OSuc lx x d< OSuc lx y
|
|
|
669
|
|
|
670 </pre>
|
|
|
671 This is a simple data structure, it has no abstract assumptions, and it is countable many data
|
|
|
672 structure.
|
|
|
673 <p>
|
|
|
674
|
|
|
675 <pre>
|
|
|
676 Φ 0
|
|
|
677 OSuc 2 ( Osuc 2 ( Osuc 2 (Φ 2)))
|
|
|
678 Osuc 0 (Φ 0) d< Φ 1
|
|
|
679
|
|
|
680 </pre>
|
|
|
681
|
|
|
682 <hr/>
|
|
279
|
683 <h2><a name="content027">Model of Ordinals</a></h2>
|
|
273
|
684
|
|
|
685 <p>
|
|
|
686 It is easy to show OrdinalD and its order satisfies the axioms of Ordinals.
|
|
|
687 <p>
|
|
|
688 So our Ordinals has a mode. This means axiom of Ordinals are consistent.
|
|
|
689 <p>
|
|
|
690
|
|
|
691 <hr/>
|
|
279
|
692 <h2><a name="content028">Debugging axioms using Model</a></h2>
|
|
273
|
693 Whether axiom is correct or not can be checked by a validity on a mode.
|
|
|
694 <p>
|
|
|
695 If not, we may fix the axioms or the model, such as the definitions of the order.
|
|
|
696 <p>
|
|
|
697 We can also ask whether the inputs exist.
|
|
|
698 <p>
|
|
|
699
|
|
|
700 <hr/>
|
|
279
|
701 <h2><a name="content029">Countable Ordinals can contains uncountable set?</a></h2>
|
|
273
|
702 Yes, the ordinals contains any level of infinite Set in the axioms.
|
|
|
703 <p>
|
|
|
704 If we handle real-number in the model, only countable number of real-number is used.
|
|
|
705 <p>
|
|
|
706
|
|
|
707 <pre>
|
|
|
708 from the outside view point, it is countable
|
|
|
709 from the internal view point, it is uncountable
|
|
|
710
|
|
|
711 </pre>
|
|
|
712 The definition of countable/uncountable is the same, but the properties are different
|
|
|
713 depending on the context.
|
|
|
714 <p>
|
|
|
715 We don't show the definition of cardinal number here.
|
|
|
716 <p>
|
|
|
717
|
|
|
718 <hr/>
|
|
279
|
719 <h2><a name="content030">What is Set</a></h2>
|
|
273
|
720 The word Set in Agda is not a Set of ZF Set, but it is a type (why it is named Set?).
|
|
|
721 <p>
|
|
|
722 From naive point view, a set i a list, but in Agda, elements have all the same type.
|
|
|
723 A set in ZF may contain other Sets in ZF, which not easy to implement it as a list.
|
|
|
724 <p>
|
|
|
725 Finite set may be written in finite series of ∨, but ...
|
|
|
726 <p>
|
|
|
727
|
|
|
728 <hr/>
|
|
279
|
729 <h2><a name="content031">We don't ask the contents of Set. It can be anything.</a></h2>
|
|
273
|
730 From empty set φ, we can think a set contains a φ, and a pair of φ and the set, and so on,
|
|
|
731 and all of them, and again we repeat this.
|
|
|
732 <p>
|
|
|
733
|
|
|
734 <pre>
|
|
|
735 φ {φ} {φ,{φ}}, {φ,{φ},...}
|
|
|
736
|
|
|
737 </pre>
|
|
|
738 It is called V.
|
|
|
739 <p>
|
|
|
740 This operation can be performed within a ZF Set theory. Classical Set Theory assumes
|
|
|
741 ZF, so this kind of thing is allowed.
|
|
|
742 <p>
|
|
|
743 But in our case, we have no ZF theory, so we are going to use Ordinals.
|
|
|
744 <p>
|
|
|
745
|
|
|
746 <hr/>
|
|
279
|
747 <h2><a name="content032">Ordinal Definable Set</a></h2>
|
|
273
|
748 We can define a sbuset of Ordinals using predicates. What is a subset?
|
|
|
749 <p>
|
|
|
750
|
|
|
751 <pre>
|
|
|
752 a predicate has an Ordinal argument
|
|
|
753
|
|
|
754 </pre>
|
|
|
755 is an Ordinal Definable Set (OD). In Agda, OD is defined as follows.
|
|
|
756 <p>
|
|
|
757
|
|
|
758 <pre>
|
|
|
759 record OD : Set (suc n ) where
|
|
|
760 field
|
|
|
761 def : (x : Ordinal ) → Set n
|
|
|
762
|
|
|
763 </pre>
|
|
|
764 Ordinals itself is not a set in a ZF Set theory but a class. In OD,
|
|
|
765 <p>
|
|
|
766
|
|
|
767 <pre>
|
|
|
768 record { def = λ x → true }
|
|
|
769
|
|
|
770 </pre>
|
|
|
771 means Ordinals itself, so ODs are larger than a notion of ZF Set Theory,
|
|
|
772 but we don't care about it.
|
|
|
773 <p>
|
|
|
774
|
|
|
775 <hr/>
|
|
279
|
776 <h2><a name="content033">∋ in OD</a></h2>
|
|
273
|
777 OD is a predicate on Ordinals and it does not contains OD directly. If we have a mapping
|
|
|
778 <p>
|
|
|
779
|
|
|
780 <pre>
|
|
|
781 od→ord : OD → Ordinal
|
|
|
782
|
|
|
783 </pre>
|
|
|
784 we may check an OD is an element of the OD using def.
|
|
|
785 <p>
|
|
|
786 A ∋ x can be define as follows.
|
|
|
787 <p>
|
|
|
788
|
|
|
789 <pre>
|
|
|
790 _∋_ : ( A x : OD ) → Set n
|
|
|
791 _∋_ A x = def A ( od→ord x )
|
|
|
792
|
|
|
793 </pre>
|
|
|
794 In ψ : Ordinal → Set, if A is a record { def = λ x → ψ x } , then
|
|
|
795 <p>
|
|
|
796
|
|
|
797 <pre>
|
|
|
798 A x = def A ( od→ord x ) = ψ (od→ord x)
|
|
|
799
|
|
|
800 </pre>
|
|
|
801
|
|
|
802 <hr/>
|
|
279
|
803 <h2><a name="content034">1 to 1 mapping between an OD and an Ordinal</a></h2>
|
|
273
|
804
|
|
|
805 <p>
|
|
|
806
|
|
|
807 <pre>
|
|
|
808 od→ord : OD → Ordinal
|
|
|
809 ord→od : Ordinal → OD
|
|
|
810 oiso : {x : OD } → ord→od ( od→ord x ) ≡ x
|
|
|
811 diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x
|
|
|
812
|
|
|
813 </pre>
|
|
|
814 They say the existing of the mappings can be proved in Classical Set Theory, but we
|
|
|
815 simply assumes these non constructively.
|
|
|
816 <p>
|
|
|
817 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
|
|
|
818 state it.
|
|
|
819 <p>
|
|
|
820 <img src="fig/ord-od-mapping.svg">
|
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821
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822 <p>
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823
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824 <hr/>
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279
|
825 <h2><a name="content035">Order preserving in the mapping of OD and Ordinal</a></h2>
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273
|
826 Ordinals have the order and OD has a natural order based on inclusion ( def / ∋ ).
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827 <p>
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828
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829 <pre>
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830 def y ( od→ord x )
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831
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832 </pre>
|
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833 An elements of OD should be defined before the OD, that is, an ordinal corresponding an elements
|
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834 have to be smaller than the corresponding ordinal of the containing OD.
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835 <p>
|
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836
|
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|
837 <pre>
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838 c<→o< : {x y : OD } → def y ( od→ord x ) → od→ord x o< od→ord y
|
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839
|
|
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840 </pre>
|
|
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841 This is also said to be provable in classical Set Theory, but we simply assumes it.
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842 <p>
|
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|
843 OD has an order based on the corresponding ordinal, but it may not have corresponding def / ∋relation. This means the reverse order preservation,
|
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|
844 <p>
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845
|
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846 <pre>
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847 o<→c< : {n : Level} {x y : Ordinal } → x o< y → def (ord→od y) x
|
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848
|
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849 </pre>
|
|
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850 is not valid. If we assumes it, ∀ x ∋ ∅ becomes true, which manes all OD becomes Ordinals in
|
|
|
851 the model.
|
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|
852 <p>
|
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|
853 <img src="fig/ODandOrdinals.svg">
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854
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855 <p>
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856
|
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857 <hr/>
|
|
279
|
858 <h2><a name="content036">ISO</a></h2>
|
|
273
|
859 It also requires isomorphisms,
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860 <p>
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861
|
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|
862 <pre>
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863 oiso : {x : OD } → ord→od ( od→ord x ) ≡ x
|
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864 diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x
|
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865
|
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866 </pre>
|
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867
|
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868 <hr/>
|
|
279
|
869 <h2><a name="content037">Various Sets</a></h2>
|
|
273
|
870
|
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871 <p>
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872 In classical Set Theory, there is a hierarchy call L, which can be defined by a predicate.
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873 <p>
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874
|
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875 <pre>
|
|
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876 Ordinal / things satisfies axiom of Ordinal / extension of natural number
|
|
|
877 V / hierarchical construction of Set from φ
|
|
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878 L / hierarchical predicate definable construction of Set from φ
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|
879 OD / equational formula on Ordinals
|
|
|
880 Agda Set / Type / it also has a level
|
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881
|
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|
882 </pre>
|
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883
|
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|
884 <hr/>
|
|
279
|
885 <h2><a name="content038">Fixes on ZF to intuitionistic logic</a></h2>
|
|
273
|
886
|
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|
887 <p>
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|
888 We use ODs as Sets in ZF, and defines record ZF, that is, we have to define
|
|
|
889 ZF axioms in Agda.
|
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890 <p>
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|
891 It may not valid in our model. We have to debug it.
|
|
|
892 <p>
|
|
|
893 Fixes are depends on axioms.
|
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|
894 <p>
|
|
|
895 <img src="fig/axiom-type.svg">
|
|
|
896
|
|
|
897 <p>
|
|
|
898 <a href="fig/zf-record.html">
|
|
|
899 ZFのrecord </a>
|
|
|
900 <p>
|
|
|
901
|
|
|
902 <hr/>
|
|
279
|
903 <h2><a name="content039">Pure logical axioms</a></h2>
|
|
273
|
904
|
|
|
905 <pre>
|
|
279
|
906 empty, pair, select, ε-induction??infinity
|
|
273
|
907
|
|
|
908 </pre>
|
|
|
909 These are logical relations among OD.
|
|
|
910 <p>
|
|
|
911
|
|
|
912 <pre>
|
|
|
913 empty : ∀( x : ZFSet ) → ¬ ( ∅ ∋ x )
|
|
|
914 pair→ : ( x y t : ZFSet ) → (x , y) ∋ t → ( t ≈ x ) ∨ ( t ≈ y )
|
|
|
915 pair← : ( x y t : ZFSet ) → ( t ≈ x ) ∨ ( t ≈ y ) → (x , y) ∋ t
|
|
|
916 selection : { ψ : ZFSet → Set m } → ∀ { X y : ZFSet } → ( ( y ∈ X ) ∧ ψ y ) ⇔ (y ∈ Select X ψ )
|
|
|
917 infinity∅ : ∅ ∈ infinite
|
|
|
918 infinity : ∀( x : ZFSet ) → x ∈ infinite → ( x ∪ ( x , x ) ) ∈ infinite
|
|
|
919 ε-induction : { ψ : OD → Set (suc n)}
|
|
|
920 → ( {x : OD } → ({ y : OD } → x ∋ y → ψ y ) → ψ x )
|
|
|
921 → (x : OD ) → ψ x
|
|
|
922
|
|
|
923 </pre>
|
|
|
924 finitely can be define by Agda data.
|
|
|
925 <p>
|
|
|
926
|
|
|
927 <pre>
|
|
|
928 data infinite-d : ( x : Ordinal ) → Set n where
|
|
|
929 iφ : infinite-d o∅
|
|
|
930 isuc : {x : Ordinal } → infinite-d x →
|
|
|
931 infinite-d (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) ))
|
|
|
932
|
|
|
933 </pre>
|
|
|
934 Union (x , ( x , x )) should be an direct successor of x, but we cannot prove it in our model.
|
|
|
935 <p>
|
|
|
936
|
|
|
937 <hr/>
|
|
279
|
938 <h2><a name="content040">Axiom of Pair</a></h2>
|
|
273
|
939 In the Tanaka's book, axiom of pair is as follows.
|
|
|
940 <p>
|
|
|
941
|
|
|
942 <pre>
|
|
|
943 ∀ x ∀ y ∃ z ∀ t ( z ∋ t ↔ t ≈ x ∨ t ≈ y)
|
|
|
944
|
|
|
945 </pre>
|
|
|
946 We have fix ∃ z, a function (x , y) is defined, which is _,_ .
|
|
|
947 <p>
|
|
|
948
|
|
|
949 <pre>
|
|
|
950 _,_ : ( A B : ZFSet ) → ZFSet
|
|
|
951
|
|
|
952 </pre>
|
|
|
953 using this, we can define two directions in separates axioms, like this.
|
|
|
954 <p>
|
|
|
955
|
|
|
956 <pre>
|
|
|
957 pair→ : ( x y t : ZFSet ) → (x , y) ∋ t → ( t ≈ x ) ∨ ( t ≈ y )
|
|
|
958 pair← : ( x y t : ZFSet ) → ( t ≈ x ) ∨ ( t ≈ y ) → (x , y) ∋ t
|
|
|
959
|
|
|
960 </pre>
|
|
|
961 This is already written in Agda, so we use these as axioms. All inputs have ∀.
|
|
|
962 <p>
|
|
|
963
|
|
|
964 <hr/>
|
|
279
|
965 <h2><a name="content041">pair in OD</a></h2>
|
|
273
|
966 OD is an equation on Ordinals, we can simply write axiom of pair in the OD.
|
|
|
967 <p>
|
|
|
968
|
|
|
969 <pre>
|
|
|
970 _,_ : OD → OD → OD
|
|
|
971 x , y = record { def = λ t → (t ≡ od→ord x ) ∨ ( t ≡ od→ord y ) }
|
|
|
972
|
|
|
973 </pre>
|
|
|
974 ≡ is an equality of λ terms, but please not that this is equality on Ordinals.
|
|
|
975 <p>
|
|
|
976
|
|
|
977 <hr/>
|
|
279
|
978 <h2><a name="content042">Validity of Axiom of Pair</a></h2>
|
|
273
|
979 Assuming ZFSet is OD, we are going to prove pair→ .
|
|
|
980 <p>
|
|
|
981
|
|
|
982 <pre>
|
|
|
983 pair→ : ( x y t : OD ) → (x , y) ∋ t → ( t == x ) ∨ ( t == y )
|
|
|
984 pair→ x y t p = ?
|
|
|
985
|
|
|
986 </pre>
|
|
|
987 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 ) .
|
|
|
988 <p>
|
|
|
989 Since _∨_ is a data, it can be developed as (C-c C-c : agda2-make-case ).
|
|
|
990 <p>
|
|
|
991
|
|
|
992 <pre>
|
|
|
993 pair→ x y t (case1 t≡x ) = ?
|
|
|
994 pair→ x y t (case2 t≡y ) = ?
|
|
|
995
|
|
|
996 </pre>
|
|
|
997 The type of the ? is ( t == x ) ∨ ( t == y ), again it is data _∨_ .
|
|
|
998 <p>
|
|
|
999
|
|
|
1000 <pre>
|
|
|
1001 pair→ x y t (case1 t≡x ) = case1 ?
|
|
|
1002 pair→ x y t (case2 t≡y ) = case2 ?
|
|
|
1003
|
|
|
1004 </pre>
|
|
|
1005 The ? in case1 is t == x, so we have to create this from t≡x, which is a name of a variable
|
|
|
1006 which type is
|
|
|
1007 <p>
|
|
|
1008
|
|
|
1009 <pre>
|
|
|
1010 t≡x : od→ord t ≡ od→ord x
|
|
|
1011
|
|
|
1012 </pre>
|
|
|
1013 which is shown by an Agda command (C-C C-E : agda2-show-context ).
|
|
|
1014 <p>
|
|
|
1015 But we haven't defined == yet.
|
|
|
1016 <p>
|
|
|
1017
|
|
|
1018 <hr/>
|
|
279
|
1019 <h2><a name="content043">Equality of OD and Axiom of Extensionality </a></h2>
|
|
273
|
1020 OD is defined by a predicates, if we compares normal form of the predicates, even if
|
|
|
1021 it contains the same elements, it may be different, which is no good as an equality of
|
|
|
1022 Sets.
|
|
|
1023 <p>
|
|
|
1024 Axiom of Extensionality requires sets having the same elements are handled in the same way
|
|
|
1025 each other.
|
|
|
1026 <p>
|
|
|
1027
|
|
|
1028 <pre>
|
|
|
1029 ∀ z ( z ∈ x ⇔ z ∈ y ) ⇒ ∀ w ( x ∈ w ⇔ y ∈ w )
|
|
|
1030
|
|
|
1031 </pre>
|
|
|
1032 We can write this axiom in Agda as follows.
|
|
|
1033 <p>
|
|
|
1034
|
|
|
1035 <pre>
|
|
|
1036 extensionality : { A B w : ZFSet } → ( (z : ZFSet) → ( A ∋ z ) ⇔ (B ∋ z) ) → ( A ∈ w ⇔ B ∈ w )
|
|
|
1037
|
|
|
1038 </pre>
|
|
|
1039 So we use ( A ∋ z ) ⇔ (B ∋ z) as an equality (_==_) of our model. We have to show
|
|
|
1040 A ∈ w ⇔ B ∈ w from A == B.
|
|
|
1041 <p>
|
|
|
1042 x == y can be defined in this way.
|
|
|
1043 <p>
|
|
|
1044
|
|
|
1045 <pre>
|
|
|
1046 record _==_ ( a b : OD ) : Set n where
|
|
|
1047 field
|
|
|
1048 eq→ : ∀ { x : Ordinal } → def a x → def b x
|
|
|
1049 eq← : ∀ { x : Ordinal } → def b x → def a x
|
|
|
1050
|
|
|
1051 </pre>
|
|
|
1052 Actually, (z : OD) → (A ∋ z) ⇔ (B ∋ z) implies A == B.
|
|
|
1053 <p>
|
|
|
1054
|
|
|
1055 <pre>
|
|
|
1056 extensionality0 : {A B : OD } → ((z : OD) → (A ∋ z) ⇔ (B ∋ z)) → A == B
|
|
|
1057 eq→ (extensionality0 {A} {B} eq ) {x} d = ?
|
|
|
1058 eq← (extensionality0 {A} {B} eq ) {x} d = ?
|
|
|
1059
|
|
|
1060 </pre>
|
|
|
1061 ? are def B x and def A x and these are generated from eq : (z : OD) → (A ∋ z) ⇔ (B ∋ z) .
|
|
|
1062 <p>
|
|
|
1063 Actual proof is rather complicated.
|
|
|
1064 <p>
|
|
|
1065
|
|
|
1066 <pre>
|
|
|
1067 eq→ (extensionality0 {A} {B} eq ) {x} d = def-iso {A} {B} (sym diso) (proj1 (eq (ord→od x))) d
|
|
|
1068 eq← (extensionality0 {A} {B} eq ) {x} d = def-iso {B} {A} (sym diso) (proj2 (eq (ord→od x))) d
|
|
|
1069
|
|
|
1070 </pre>
|
|
|
1071 where
|
|
|
1072 <p>
|
|
|
1073
|
|
|
1074 <pre>
|
|
|
1075 def-iso : {A B : OD } {x y : Ordinal } → x ≡ y → (def A y → def B y) → def A x → def B x
|
|
|
1076 def-iso refl t = t
|
|
|
1077
|
|
|
1078 </pre>
|
|
|
1079
|
|
|
1080 <hr/>
|
|
279
|
1081 <h2><a name="content044">Validity of Axiom of Extensionality</a></h2>
|
|
273
|
1082
|
|
|
1083 <p>
|
|
|
1084 If we can derive (w ∋ A) ⇔ (w ∋ B) from A == B, the axiom becomes valid, but it seems impossible, so we assumes
|
|
|
1085 <p>
|
|
|
1086
|
|
|
1087 <pre>
|
|
|
1088 ==→o≡ : { x y : OD } → (x == y) → x ≡ y
|
|
|
1089
|
|
|
1090 </pre>
|
|
|
1091 Using this, we have
|
|
|
1092 <p>
|
|
|
1093
|
|
|
1094 <pre>
|
|
|
1095 extensionality : {A B w : OD } → ((z : OD ) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B)
|
|
|
1096 proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d
|
|
|
1097 proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d
|
|
|
1098
|
|
|
1099 </pre>
|
|
|
1100 This assumption means we may have an equivalence set on any predicates.
|
|
|
1101 <p>
|
|
|
1102
|
|
|
1103 <hr/>
|
|
279
|
1104 <h2><a name="content045">Non constructive assumptions so far</a></h2>
|
|
273
|
1105 We have correspondence between OD and Ordinals and one directional order preserving.
|
|
|
1106 <p>
|
|
|
1107 We also have equality assumption.
|
|
|
1108 <p>
|
|
|
1109
|
|
|
1110 <pre>
|
|
|
1111 od→ord : OD → Ordinal
|
|
|
1112 ord→od : Ordinal → OD
|
|
|
1113 oiso : {x : OD } → ord→od ( od→ord x ) ≡ x
|
|
|
1114 diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x
|
|
|
1115 c<→o< : {x y : OD } → def y ( od→ord x ) → od→ord x o< od→ord y
|
|
|
1116 ==→o≡ : { x y : OD } → (x == y) → x ≡ y
|
|
|
1117
|
|
|
1118 </pre>
|
|
|
1119
|
|
|
1120 <hr/>
|
|
279
|
1121 <h2><a name="content046">Axiom which have negation form</a></h2>
|
|
273
|
1122
|
|
|
1123 <p>
|
|
|
1124
|
|
|
1125 <pre>
|
|
|
1126 Union, Selection
|
|
|
1127
|
|
|
1128 </pre>
|
|
|
1129 These axioms contains ∃ x as a logical relation, which can be described in ¬ ( ∀ x ( ¬ p )).
|
|
|
1130 <p>
|
|
|
1131 Axiom of replacement uses upper bound of function on Ordinals, which makes it non-constructive.
|
|
|
1132 <p>
|
|
|
1133 Power Set axiom requires double negation,
|
|
|
1134 <p>
|
|
|
1135
|
|
|
1136 <pre>
|
|
|
1137 power→ : ∀( A t : ZFSet ) → Power A ∋ t → ∀ {x} → t ∋ x → ¬ ¬ ( A ∋ x )
|
|
|
1138 power← : ∀( A t : ZFSet ) → t ⊆_ A → Power A ∋ t
|
|
|
1139
|
|
|
1140 </pre>
|
|
|
1141 If we have an assumption of law of exclude middle, we can recover the original A ∋ x form.
|
|
|
1142 <p>
|
|
|
1143
|
|
|
1144 <hr/>
|
|
279
|
1145 <h2><a name="content047">Union </a></h2>
|
|
273
|
1146 The original form of the Axiom of Union is
|
|
|
1147 <p>
|
|
|
1148
|
|
|
1149 <pre>
|
|
|
1150 ∀ x ∃ y ∀ z (z ∈ y ⇔ ∃ u ∈ x ∧ (z ∈ u))
|
|
|
1151
|
|
|
1152 </pre>
|
|
|
1153 Union requires the existence of b in a ⊇ ∃ b ∋ x . We will use negation form of ∃.
|
|
|
1154 <p>
|
|
|
1155
|
|
|
1156 <pre>
|
|
|
1157 union→ : ( X z u : ZFSet ) → ( X ∋ u ) ∧ (u ∋ z ) → Union X ∋ z
|
|
|
1158 union← : ( X z : ZFSet ) → (X∋z : Union X ∋ z ) → ¬ ( (u : ZFSet ) → ¬ ((X ∋ u) ∧ (u ∋ z )))
|
|
|
1159
|
|
|
1160 </pre>
|
|
|
1161 The definition of Union in OD is like this.
|
|
|
1162 <p>
|
|
|
1163
|
|
|
1164 <pre>
|
|
|
1165 Union : OD → OD
|
|
|
1166 Union U = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((def U u) ∧ (def (ord→od u) x))) }
|
|
|
1167
|
|
|
1168 </pre>
|
|
|
1169 Proof of validity is straight forward.
|
|
|
1170 <p>
|
|
|
1171
|
|
|
1172 <pre>
|
|
|
1173 union→ : (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z
|
|
|
1174 union→ X z u xx not = ⊥-elim ( not (od→ord u) ( record { proj1 = proj1 xx
|
|
|
1175 ; proj2 = subst ( λ k → def k (od→ord z)) (sym oiso) (proj2 xx) } ))
|
|
|
1176 union← : (X z : OD) (X∋z : Union X ∋ z) → ¬ ( (u : OD ) → ¬ ((X ∋ u) ∧ (u ∋ z )))
|
|
|
1177 union← X z UX∋z = FExists _ lemma UX∋z where
|
|
|
1178 lemma : {y : Ordinal} → def X y ∧ def (ord→od y) (od→ord z) → ¬ ((u : OD) → ¬ (X ∋ u) ∧ (u ∋ z))
|
|
|
1179 lemma {y} xx not = not (ord→od y) record { proj1 = subst ( λ k → def X k ) (sym diso) (proj1 xx ) ; proj2 = proj2 xx }
|
|
|
1180
|
|
|
1181 </pre>
|
|
|
1182 where
|
|
|
1183 <p>
|
|
|
1184
|
|
|
1185 <pre>
|
|
|
1186 FExists : {m l : Level} → ( ψ : Ordinal → Set m )
|
|
|
1187 → {p : Set l} ( P : { y : Ordinal } → ψ y → ¬ p )
|
|
|
1188 → (exists : ¬ (∀ y → ¬ ( ψ y ) ))
|
|
|
1189 → ¬ p
|
|
|
1190 FExists {m} {l} ψ {p} P = contra-position ( λ p y ψy → P {y} ψy p )
|
|
|
1191
|
|
|
1192 </pre>
|
|
|
1193 which checks existence using contra-position.
|
|
|
1194 <p>
|
|
|
1195
|
|
|
1196 <hr/>
|
|
279
|
1197 <h2><a name="content048">Axiom of replacement</a></h2>
|
|
273
|
1198 We can replace the elements of a set by a function and it becomes a set. From the book,
|
|
|
1199 <p>
|
|
|
1200
|
|
|
1201 <pre>
|
|
|
1202 ∀ x ∀ y ∀ z ( ( ψ ( x , y ) ∧ ψ ( x , z ) ) → y = z ) → ∀ X ∃ A ∀ y ( y ∈ A ↔ ∃ x ∈ X ψ ( x , y ) )
|
|
|
1203
|
|
|
1204 </pre>
|
|
|
1205 The existential quantifier can be related by a function,
|
|
|
1206 <p>
|
|
|
1207
|
|
|
1208 <pre>
|
|
|
1209 Replace : OD → (OD → OD ) → OD
|
|
|
1210
|
|
|
1211 </pre>
|
|
|
1212 The axioms becomes as follows.
|
|
|
1213 <p>
|
|
|
1214
|
|
|
1215 <pre>
|
|
|
1216 replacement← : {ψ : ZFSet → ZFSet} → ∀ ( X x : ZFSet ) → x ∈ X → ψ x ∈ Replace X ψ
|
|
|
1217 replacement→ : {ψ : ZFSet → ZFSet} → ∀ ( X x : ZFSet ) → ( lt : x ∈ Replace X ψ ) → ¬ ( ∀ (y : ZFSet) → ¬ ( x ≈ ψ y ) )
|
|
|
1218
|
|
|
1219 </pre>
|
|
|
1220 In the axiom, the existence of the original elements is necessary. In order to do that we use OD which has
|
|
|
1221 negation form of existential quantifier in the definition.
|
|
|
1222 <p>
|
|
|
1223
|
|
|
1224 <pre>
|
|
|
1225 in-codomain : (X : OD ) → ( ψ : OD → OD ) → OD
|
|
|
1226 in-codomain X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( def X y ∧ ( x ≡ od→ord (ψ (ord→od y ))))) }
|
|
|
1227
|
|
|
1228 </pre>
|
|
|
1229 Besides this upper bounds is required.
|
|
|
1230 <p>
|
|
|
1231
|
|
|
1232 <pre>
|
|
|
1233 Replace : OD → (OD → OD ) → OD
|
|
|
1234 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 }
|
|
|
1235
|
|
|
1236 </pre>
|
|
|
1237 We omit the proof of the validity, but it is rather straight forward.
|
|
|
1238 <p>
|
|
|
1239
|
|
|
1240 <hr/>
|
|
279
|
1241 <h2><a name="content049">Validity of Power Set Axiom</a></h2>
|
|
273
|
1242 The original Power Set Axiom is this.
|
|
|
1243 <p>
|
|
|
1244
|
|
|
1245 <pre>
|
|
|
1246 ∀ X ∃ A ∀ t ( t ∈ A ↔ t ⊆ X ) )
|
|
|
1247
|
|
|
1248 </pre>
|
|
|
1249 The existential quantifier is replaced by a function
|
|
|
1250 <p>
|
|
|
1251
|
|
|
1252 <pre>
|
|
|
1253 Power : ( A : OD ) → OD
|
|
|
1254
|
|
|
1255 </pre>
|
|
|
1256 t ⊆ X is a record like this.
|
|
|
1257 <p>
|
|
|
1258
|
|
|
1259 <pre>
|
|
|
1260 record _⊆_ ( A B : OD ) : Set (suc n) where
|
|
|
1261 field
|
|
|
1262 incl : { x : OD } → A ∋ x → B ∋ x
|
|
|
1263
|
|
|
1264 </pre>
|
|
|
1265 Axiom becomes likes this.
|
|
|
1266 <p>
|
|
|
1267
|
|
|
1268 <pre>
|
|
|
1269 power→ : ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x)
|
|
|
1270 power← : (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t
|
|
|
1271
|
|
|
1272 </pre>
|
|
|
1273 The validity of the axioms are slight complicated, we have to define set of all subset. We define
|
|
|
1274 subset in a different form.
|
|
|
1275 <p>
|
|
|
1276
|
|
|
1277 <pre>
|
|
|
1278 ZFSubset : (A x : OD ) → OD
|
|
|
1279 ZFSubset A x = record { def = λ y → def A y ∧ def x y }
|
|
|
1280
|
|
|
1281 </pre>
|
|
|
1282 We can prove,
|
|
|
1283 <p>
|
|
|
1284
|
|
|
1285 <pre>
|
|
|
1286 ( {y : OD } → x ∋ y → ZFSubset A x ∋ y ) ⇔ ( x ⊆ A )
|
|
|
1287
|
|
|
1288 </pre>
|
|
|
1289 We only have upper bound as an ordinal, but we have an obvious OD based on the order of Ordinals,
|
|
|
1290 which is an Ordinals with our Model.
|
|
|
1291 <p>
|
|
|
1292
|
|
|
1293 <pre>
|
|
|
1294 Ord : ( a : Ordinal ) → OD
|
|
|
1295 Ord a = record { def = λ y → y o< a }
|
|
|
1296 Def : (A : OD ) → OD
|
|
|
1297 Def A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) )
|
|
|
1298
|
|
|
1299 </pre>
|
|
|
1300 This is slight larger than Power A, so we replace all elements x by A ∩ x (some of them may empty).
|
|
|
1301 <p>
|
|
|
1302
|
|
|
1303 <pre>
|
|
|
1304 Power : OD → OD
|
|
|
1305 Power A = Replace (Def (Ord (od→ord A))) ( λ x → A ∩ x )
|
|
|
1306
|
|
|
1307 </pre>
|
|
|
1308 Creating Power Set of Ordinals is rather easy, then we use replacement axiom on A ∩ x since we have this.
|
|
|
1309 <p>
|
|
|
1310
|
|
|
1311 <pre>
|
|
|
1312 ∩-≡ : { a b : OD } → ({x : OD } → (a ∋ x → b ∋ x)) → a == ( b ∩ a )
|
|
|
1313
|
|
|
1314 </pre>
|
|
|
1315 In case of Ord a intro of Power Set axiom becomes valid.
|
|
|
1316 <p>
|
|
|
1317
|
|
|
1318 <pre>
|
|
|
1319 ord-power← : (a : Ordinal ) (t : OD) → ({x : OD} → (t ∋ x → (Ord a) ∋ x)) → Def (Ord a) ∋ t
|
|
|
1320
|
|
|
1321 </pre>
|
|
|
1322 Using this, we can prove,
|
|
|
1323 <p>
|
|
|
1324
|
|
|
1325 <pre>
|
|
|
1326 power→ : ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x)
|
|
|
1327 power← : (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t
|
|
|
1328
|
|
|
1329 </pre>
|
|
|
1330
|
|
|
1331 <hr/>
|
|
279
|
1332 <h2><a name="content050">Axiom of regularity, Axiom of choice, ε-induction</a></h2>
|
|
273
|
1333
|
|
|
1334 <p>
|
|
|
1335 Axiom of regularity requires non self intersectable elements (which is called minimum), if we
|
|
|
1336 replace it by a function, it becomes a choice function. It makes axiom of choice valid.
|
|
|
1337 <p>
|
|
|
1338 This means we cannot prove axiom regularity form our model, and if we postulate this, axiom of
|
|
|
1339 choice also becomes valid.
|
|
|
1340 <p>
|
|
|
1341
|
|
|
1342 <pre>
|
|
|
1343 minimal : (x : OD ) → ¬ (x == od∅ )→ OD
|
|
|
1344 x∋minimal : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → def x ( od→ord ( minimal x ne ) )
|
|
|
1345 minimal-1 : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → (y : OD ) → ¬ ( def (minimal x ne) (od→ord y)) ∧ (def x (od→ord y) )
|
|
|
1346
|
|
|
1347 </pre>
|
|
|
1348 We can avoid this using ε-induction (a predicate is valid on all set if the predicate is true on some element of set).
|
|
|
1349 Assuming law of exclude middle, they say axiom of regularity will be proved, but we haven't check it yet.
|
|
|
1350 <p>
|
|
|
1351
|
|
|
1352 <pre>
|
|
|
1353 ε-induction : { ψ : OD → Set (suc n)}
|
|
|
1354 → ( {x : OD } → ({ y : OD } → x ∋ y → ψ y ) → ψ x )
|
|
|
1355 → (x : OD ) → ψ x
|
|
|
1356
|
|
|
1357 </pre>
|
|
|
1358 In our model, we assumes the mapping between Ordinals and OD, this is actually the TransFinite induction in Ordinals.
|
|
|
1359 <p>
|
|
|
1360 The axiom of choice in the book is complicated using any pair in a set, so we use use a form in the Wikipedia.
|
|
|
1361 <p>
|
|
|
1362
|
|
|
1363 <pre>
|
|
|
1364 ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ]
|
|
|
1365
|
|
|
1366 </pre>
|
|
|
1367 We can formulate like this.
|
|
|
1368 <p>
|
|
|
1369
|
|
|
1370 <pre>
|
|
|
1371 choice-func : (X : ZFSet ) → {x : ZFSet } → ¬ ( x ≈ ∅ ) → ( X ∋ x ) → ZFSet
|
|
|
1372 choice : (X : ZFSet ) → {A : ZFSet } → ( X∋A : X ∋ A ) → (not : ¬ ( A ≈ ∅ )) → A ∋ choice-func X not X∋A
|
|
|
1373
|
|
|
1374 </pre>
|
|
|
1375 It does not requires ∅ ∉ X .
|
|
|
1376 <p>
|
|
|
1377
|
|
|
1378 <hr/>
|
|
279
|
1379 <h2><a name="content051">Axiom of choice and Law of Excluded Middle</a></h2>
|
|
273
|
1380 In our model, since OD has a mapping to Ordinals, it has evident order, which means well ordering theorem is valid,
|
|
|
1381 but it don't have correct form of the axiom yet. They say well ordering axiom is equivalent to the axiom of choice,
|
|
|
1382 but it requires law of the exclude middle.
|
|
|
1383 <p>
|
|
|
1384 Actually, it is well known to prove law of the exclude middle from axiom of choice in intuitionistic logic, and we can
|
|
|
1385 perform the proof in our mode. Using the definition like this, predicates and ODs are related and we can ask the
|
|
|
1386 set is empty or not if we have an axiom of choice, so we have the law of the exclude middle p ∨ ( ¬ p ) .
|
|
|
1387 <p>
|
|
|
1388
|
|
|
1389 <pre>
|
|
|
1390 ppp : { p : Set n } { a : OD } → record { def = λ x → p } ∋ a → p
|
|
|
1391 ppp {p} {a} d = d
|
|
|
1392
|
|
|
1393 </pre>
|
|
|
1394 We can prove axiom of choice from law excluded middle since we have TransFinite induction. So Axiom of choice
|
|
|
1395 and Law of Excluded Middle is equivalent in our mode.
|
|
|
1396 <p>
|
|
|
1397
|
|
|
1398 <hr/>
|
|
279
|
1399 <h2><a name="content052">Relation-ship among ZF axiom</a></h2>
|
|
273
|
1400 <img src="fig/axiom-dependency.svg">
|
|
|
1401
|
|
|
1402 <p>
|
|
|
1403
|
|
|
1404 <hr/>
|
|
279
|
1405 <h2><a name="content053">Non constructive assumption in our model</a></h2>
|
|
273
|
1406 mapping between OD and Ordinals
|
|
|
1407 <p>
|
|
|
1408
|
|
|
1409 <pre>
|
|
|
1410 od→ord : OD → Ordinal
|
|
|
1411 ord→od : Ordinal → OD
|
|
|
1412 oiso : {x : OD } → ord→od ( od→ord x ) ≡ x
|
|
|
1413 diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x
|
|
|
1414 c<→o< : {x y : OD } → def y ( od→ord x ) → od→ord x o< od→ord y
|
|
|
1415
|
|
|
1416 </pre>
|
|
|
1417 Equivalence on OD
|
|
|
1418 <p>
|
|
|
1419
|
|
|
1420 <pre>
|
|
|
1421 ==→o≡ : { x y : OD } → (x == y) → x ≡ y
|
|
|
1422
|
|
|
1423 </pre>
|
|
|
1424 Upper bound
|
|
|
1425 <p>
|
|
|
1426
|
|
|
1427 <pre>
|
|
|
1428 sup-o : ( Ordinal → Ordinal ) → Ordinal
|
|
|
1429 sup-o< : { ψ : Ordinal → Ordinal } → ∀ {x : Ordinal } → ψ x o< sup-o ψ
|
|
|
1430
|
|
|
1431 </pre>
|
|
|
1432 Axiom of choice and strong axiom of regularity.
|
|
|
1433 <p>
|
|
|
1434
|
|
|
1435 <pre>
|
|
|
1436 minimal : (x : OD ) → ¬ (x == od∅ )→ OD
|
|
|
1437 x∋minimal : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → def x ( od→ord ( minimal x ne ) )
|
|
|
1438 minimal-1 : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → (y : OD ) → ¬ ( def (minimal x ne) (od→ord y)) ∧ (def x (od→ord y) )
|
|
|
1439
|
|
|
1440 </pre>
|
|
|
1441
|
|
|
1442 <hr/>
|
|
279
|
1443 <h2><a name="content054">So it this correct?</a></h2>
|
|
273
|
1444
|
|
|
1445 <p>
|
|
|
1446 Our axiom are syntactically the same in the text book, but negations are slightly different.
|
|
|
1447 <p>
|
|
|
1448 If we assumes excluded middle, these are exactly same.
|
|
|
1449 <p>
|
|
|
1450 Even if we assumes excluded middle, intuitionistic logic itself remains consistent, but we cannot prove it.
|
|
|
1451 <p>
|
|
|
1452 Except the upper bound, axioms are simple logical relation.
|
|
|
1453 <p>
|
|
|
1454 Proof of existence of mapping between OD and Ordinals are not obvious. We don't know we prove it or not.
|
|
|
1455 <p>
|
|
|
1456 Existence of the Upper bounds is a pure assumption, if we have not limit on Ordinals, it may contradicts,
|
|
|
1457 but we don't have explicit upper limit on Ordinals.
|
|
|
1458 <p>
|
|
|
1459 Several inference on our model or our axioms are basically parallel to the set theory text book, so it looks like correct.
|
|
|
1460 <p>
|
|
|
1461
|
|
|
1462 <hr/>
|
|
279
|
1463 <h2><a name="content055">How to use Agda Set Theory</a></h2>
|
|
273
|
1464 Assuming record ZF, classical set theory can be developed. If necessary, axiom of choice can be
|
|
|
1465 postulated or assuming law of excluded middle.
|
|
|
1466 <p>
|
|
|
1467 Instead, simply assumes non constructive assumption, various theory can be developed. We haven't check
|
|
|
1468 these assumptions are proved in record ZF, so we are not sure, these development is a result of ZF Set theory.
|
|
|
1469 <p>
|
|
|
1470 ZF record itself is not necessary, for example, topology theory without ZF can be possible.
|
|
|
1471 <p>
|
|
|
1472
|
|
|
1473 <hr/>
|
|
279
|
1474 <h2><a name="content056">Topos and Set Theory</a></h2>
|
|
273
|
1475 Topos is a mathematical structure in Category Theory, which is a Cartesian closed category which has a
|
|
|
1476 sub-object classifier.
|
|
|
1477 <p>
|
|
|
1478 Topos itself is model of intuitionistic logic.
|
|
|
1479 <p>
|
|
|
1480 Transitive Sets are objects of Cartesian closed category.
|
|
|
1481 It is possible to introduce Power Set Functor on it
|
|
|
1482 <p>
|
|
|
1483 We can use replacement A ∩ x for each element in Transitive Set,
|
|
|
1484 in the similar way of our power set axiom. I
|
|
|
1485 <p>
|
|
|
1486 A model of ZF Set theory can be constructed on top of the Topos which is shown in Oisus.
|
|
|
1487 <p>
|
|
|
1488 Our Agda model is a proof theoretic version of it.
|
|
|
1489 <p>
|
|
|
1490
|
|
|
1491 <hr/>
|
|
279
|
1492 <h2><a name="content057">Cardinal number and Continuum hypothesis</a></h2>
|
|
273
|
1493 Axiom of choice is required to define cardinal number
|
|
|
1494 <p>
|
|
|
1495 definition of cardinal number is not yet done
|
|
|
1496 <p>
|
|
|
1497 definition of filter is not yet done
|
|
|
1498 <p>
|
|
|
1499 we may have a model without axiom of choice or without continuum hypothesis
|
|
|
1500 <p>
|
|
|
1501 Possible representation of continuum hypothesis is this.
|
|
|
1502 <p>
|
|
|
1503
|
|
|
1504 <pre>
|
|
|
1505 continuum-hyphotheis : (a : Ordinal) → Power (Ord a) ⊆ Ord (osuc a)
|
|
|
1506
|
|
|
1507 </pre>
|
|
|
1508
|
|
|
1509 <hr/>
|
|
279
|
1510 <h2><a name="content058">Filter</a></h2>
|
|
273
|
1511
|
|
|
1512 <p>
|
|
|
1513 filter is a dual of ideal on boolean algebra or lattice. Existence on natural number
|
|
|
1514 is depends on axiom of choice.
|
|
|
1515 <p>
|
|
|
1516
|
|
|
1517 <pre>
|
|
|
1518 record Filter ( L : OD ) : Set (suc n) where
|
|
|
1519 field
|
|
|
1520 filter : OD
|
|
|
1521 proper : ¬ ( filter ∋ od∅ )
|
|
|
1522 inL : filter ⊆ L
|
|
|
1523 filter1 : { p q : OD } → q ⊆ L → filter ∋ p → p ⊆ q → filter ∋ q
|
|
|
1524 filter2 : { p q : OD } → filter ∋ p → filter ∋ q → filter ∋ (p ∩ q)
|
|
|
1525
|
|
|
1526 </pre>
|
|
|
1527 We may construct a model of non standard analysis or set theory.
|
|
|
1528 <p>
|
|
|
1529 This may be simpler than classical forcing theory ( not yet done).
|
|
|
1530 <p>
|
|
|
1531
|
|
|
1532 <hr/>
|
|
279
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1533 <h2><a name="content059">Programming Mathematics</a></h2>
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273
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1534 Mathematics is a functional programming in Agda where proof is a value of a variable. The mathematical
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1535 structure are
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1536 <p>
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1537
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1538 <pre>
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1539 record and data
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1540
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1541 </pre>
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1542 Proof is check by type consistency not by the computation, but it may include some normalization.
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1543 <p>
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1544 Type inference and termination is not so clear in multi recursions.
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1545 <p>
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1546 Defining Agda record is a good way to understand mathematical theory, for examples,
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1547 <p>
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1548
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1549 <pre>
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1550 Category theory ( Yoneda lemma, Floyd Adjunction functor theorem, Applicative functor )
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1551 Automaton ( Subset construction、Language containment)
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1552
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1553 </pre>
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1554 are proved in Agda.
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1555 <p>
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1556
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1557 <hr/>
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279
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1558 <h2><a name="content060">link</a></h2>
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273
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1559 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>
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1560 <p>
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1561 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
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1562 </a>
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1563 <p>
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1564 Agda
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1565 <br> <a href="https://agda.readthedocs.io/en/v2.6.0.1/">https://agda.readthedocs.io/en/v2.6.0.1/</a>
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1566 <p>
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1567 ZF-in-Agda source
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1568 <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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1569 </a>
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1570 <p>
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1571 Category theory in Agda source
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1572 <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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1573 </a>
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1574 <p>
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1575 </div>
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1576 <ol class="side" id="menu">
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1577 Constructing ZF Set Theory in Agda
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279
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1578 <li><a href="#content000"> ZF in Agda</a>
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1579 <li><a href="#content001"> Programming Mathematics</a>
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1580 <li><a href="#content002"> Target</a>
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1581 <li><a href="#content003"> Why Set Theory</a>
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1582 <li><a href="#content004"> Agda and Intuitionistic Logic </a>
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1583 <li><a href="#content005"> Introduction of Agda </a>
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1584 <li><a href="#content006"> data ( Sum type )</a>
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1585 <li><a href="#content007"> A → B means "A implies B"</a>
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1586 <li><a href="#content008"> introduction と elimination</a>
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1587 <li><a href="#content009"> To prove A → B </a>
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1588 <li><a href="#content010"> A ∧ B</a>
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1589 <li><a href="#content011"> record</a>
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1590 <li><a href="#content012"> Mathematical structure</a>
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1591 <li><a href="#content013"> A Model and a theory</a>
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1592 <li><a href="#content014"> postulate と module</a>
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1593 <li><a href="#content015"> A ∨ B</a>
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1594 <li><a href="#content016"> Negation</a>
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1595 <li><a href="#content017"> Equality </a>
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1596 <li><a href="#content018"> Equivalence relation</a>
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1597 <li><a href="#content019"> Ordering</a>
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1598 <li><a href="#content020"> Quantifier</a>
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1599 <li><a href="#content021"> Can we do math in this way?</a>
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1600 <li><a href="#content022"> Things which Agda cannot prove</a>
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1601 <li><a href="#content023"> Classical story of ZF Set Theory</a>
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1602 <li><a href="#content024"> Ordinals</a>
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1603 <li><a href="#content025"> Axiom of Ordinals</a>
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1604 <li><a href="#content026"> Concrete Ordinals</a>
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1605 <li><a href="#content027"> Model of Ordinals</a>
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1606 <li><a href="#content028"> Debugging axioms using Model</a>
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1607 <li><a href="#content029"> Countable Ordinals can contains uncountable set?</a>
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1608 <li><a href="#content030"> What is Set</a>
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1609 <li><a href="#content031"> We don't ask the contents of Set. It can be anything.</a>
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1610 <li><a href="#content032"> Ordinal Definable Set</a>
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1611 <li><a href="#content033"> ∋ in OD</a>
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1612 <li><a href="#content034"> 1 to 1 mapping between an OD and an Ordinal</a>
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1613 <li><a href="#content035"> Order preserving in the mapping of OD and Ordinal</a>
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1614 <li><a href="#content036"> ISO</a>
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1615 <li><a href="#content037"> Various Sets</a>
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1616 <li><a href="#content038"> Fixes on ZF to intuitionistic logic</a>
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1617 <li><a href="#content039"> Pure logical axioms</a>
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1618 <li><a href="#content040"> Axiom of Pair</a>
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1619 <li><a href="#content041"> pair in OD</a>
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1620 <li><a href="#content042"> Validity of Axiom of Pair</a>
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1621 <li><a href="#content043"> Equality of OD and Axiom of Extensionality </a>
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1622 <li><a href="#content044"> Validity of Axiom of Extensionality</a>
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1623 <li><a href="#content045"> Non constructive assumptions so far</a>
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1624 <li><a href="#content046"> Axiom which have negation form</a>
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1625 <li><a href="#content047"> Union </a>
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1626 <li><a href="#content048"> Axiom of replacement</a>
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1627 <li><a href="#content049"> Validity of Power Set Axiom</a>
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1628 <li><a href="#content050"> Axiom of regularity, Axiom of choice, ε-induction</a>
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1629 <li><a href="#content051"> Axiom of choice and Law of Excluded Middle</a>
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1630 <li><a href="#content052"> Relation-ship among ZF axiom</a>
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1631 <li><a href="#content053"> Non constructive assumption in our model</a>
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1632 <li><a href="#content054"> So it this correct?</a>
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1633 <li><a href="#content055"> How to use Agda Set Theory</a>
|
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1634 <li><a href="#content056"> Topos and Set Theory</a>
|
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|
1635 <li><a href="#content057"> Cardinal number and Continuum hypothesis</a>
|
|
|
1636 <li><a href="#content058"> Filter</a>
|
|
|
1637 <li><a href="#content059"> Programming Mathematics</a>
|
|
|
1638 <li><a href="#content060"> link</a>
|
|
273
|
1639 </ol>
|
|
|
1640
|
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279
|
1641 <hr/> Shinji KONO / Sat May 9 16:41:10 2020
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273
|
1642 </body></html>
|
