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273
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1 -title: Constructing ZF Set Theory in Agda
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2
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3 --author: Shinji KONO
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4
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279
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5 --ZF in Agda
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6
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7 zf.agda axiom of ZF
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8 zfc.agda axiom of choice
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9 Ordinals.agda axiom of Ordinals
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10 ordinal.agda countable model of Ordinals
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11 OD.agda model of ZF based on Ordinal Definable Set with assumptions
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12 ODC.agda Law of exclude middle from axiom of choice assumptions
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13 LEMC.agda model of choice with assumption of the Law of exclude middle
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14 OPair.agda ordered pair on OD
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15
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16 BAlgbra.agda Boolean algebra on OD (not yet done)
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17 filter.agda Filter on OD (not yet done)
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18 cardinal.agda Caedinal number on OD (not yet done)
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19
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20 logic.agda some basics on logic
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21 nat.agda some basics on Nat
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22
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273
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23 --Programming Mathematics
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24
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25 Programming is processing data structure with λ terms.
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26
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27 We are going to handle Mathematics in intuitionistic logic with λ terms.
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28
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29 Mathematics is a functional programming which values are proofs.
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30
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31 Programming ZF Set Theory in Agda
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32
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33 --Target
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34
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35 Describe ZF axioms in Agda
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36 Construction a Model of ZF Set Theory in Agda
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37 Show necessary assumptions for the model
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38 Show validities of ZF axioms on the model
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39
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40 This shows consistency of Set Theory (with some assumptions),
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41 without circulating ZF Theory assumption.
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42
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43 <a href="https://github.com/shinji-kono/zf-in-agda">
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44 ZF in Agda https://github.com/shinji-kono/zf-in-agda
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45 </a>
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46
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47 --Why Set Theory
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48
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49 If we can formulate Set theory, it suppose to work on any mathematical theory.
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50
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51 Set Theory is a difficult point for beginners especially axiom of choice.
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52
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53 It has some amount of difficulty and self circulating discussion.
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54
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55 I'm planning to do it in my old age, but I'm enough age now.
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56
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336
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57 if you familier with Agda, you can skip to
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58 <a href="#set-theory">
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59 there
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60 </a>
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273
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61
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62 --Agda and Intuitionistic Logic
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63
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64 Curry Howard Isomorphism
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65
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66 Proposition : Proof ⇔ Type : Value
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67
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68 which means
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69
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70 constructing a typed lambda calculus which corresponds a logic
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71
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72 Typed lambda calculus which allows complex type as a value of a variable (System FC)
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73
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74 First class Type / Dependent Type
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75
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76 Agda is a such a programming language which has similar syntax of Haskell
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77
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78 Coq is specialized in proof assistance such as command and tactics .
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79
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80 --Introduction of Agda
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81
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82 A length of a list of type A.
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83
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84 length : {A : Set } → List A → Nat
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85 length [] = zero
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86 length (_ ∷ t) = suc ( length t )
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87
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88 Simple functional programming language. Type declaration is mandatory.
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89 A colon means type, an equal means value. Indentation based.
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90
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91 Set is a base type (which may have a level ).
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92
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93 {} means implicit variable which can be omitted if Agda infers its value.
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94
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95 --data ( Sum type )
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96
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97 A data type which as exclusive multiple constructors. A similar one as
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98 union in C or case class in Scala.
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99
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100 It has a similar syntax as Haskell but it has a slight difference.
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101
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102 data List (A : Set ) : Set where
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103 [] : List A
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104 _∷_ : A → List A → List A
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105
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106 _∷_ means infix operator. If use explicit _, it can be used in a normal function
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107 syntax.
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108
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109 Natural number can be defined as a usual way.
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110
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111 data Nat : Set where
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112 zero : Nat
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113 suc : Nat → Nat
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114
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115 -- A → B means "A implies B"
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116
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117 In Agda, a type can be a value of a variable, which is usually called dependent type.
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118 Type has a name Set in Agda.
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119
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120 ex3 : {A B : Set} → Set
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121 ex3 {A}{B} = A → B
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122
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123 ex3 is a type : A → B, which is a value of Set. It also means a formula : A implies B.
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124
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125 A type is a formula, the value is the proof
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126
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127 A value of A → B can be interpreted as an inference from the formula A to the formula B, which
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128 can be a function from a proof of A to a proof of B.
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129
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130 --introduction と elimination
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131
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132 For a logical operator, there are two types of inference, an introduction and an elimination.
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133
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134 intro creating symbol / constructor / introduction
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135 elim using symbolic / accessors / elimination
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136
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137 In Natural deduction, this can be written in proof schema.
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138
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139 A
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140 :
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141 B A A → B
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142 ------------- →intro ------------------ →elim
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143 A → B B
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144
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145 In Agda, this is a pair of type and value as follows. Introduction of → uses λ.
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146
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147 →intro : {A B : Set } → A → B → ( A → B )
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148 →intro _ b = λ x → b
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149
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150 →elim : {A B : Set } → A → ( A → B ) → B
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151 →elim a f = f a
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152
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153 Important
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154
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155 {A B : Set } → A → B → ( A → B )
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156
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157 is
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158
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159 {A B : Set } → ( A → ( B → ( A → B ) ))
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160
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161 This makes currying of function easy.
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162
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163 -- To prove A → B
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164
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165 Make a left type as an argument. (intros in Coq)
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166
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167 ex5 : {A B C : Set } → A → B → C → ?
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168 ex5 a b c = ?
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169
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170 ? is called a hole, which is unspecified part. Agda tell us which kind type is required for the Hole.
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171
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172 We are going to fill the holes, and if we have no warnings nor errors such as type conflict (Red),
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173 insufficient proof or instance (Yellow), Non-termination, the proof is completed.
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174
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175 -- A ∧ B
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176
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177 Well known conjunction's introduction and elimination is as follow.
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178
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179 A B A ∧ B A ∧ B
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180 ------------- ----------- proj1 ---------- proj2
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181 A ∧ B A B
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182
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183 We can introduce a corresponding structure in our functional programming language.
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184
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185 -- record
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186
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187 record _∧_ A B : Set
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188 field
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189 proj1 : A
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190 proj2 : B
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191
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192 _∧_ means infix operator. _∧_ A B can be written as A ∧ B (Haskell uses (∧) )
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193
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194 This a type which constructed from type A and type B. You may think this as an object
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195 or struct.
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196
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197 record { proj1 = x ; proj2 = y }
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198
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199 is a constructor of _∧_.
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200
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201 ex3 : {A B : Set} → A → B → ( A ∧ B )
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202 ex3 a b = record { proj1 = a ; proj2 = b }
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203
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204 ex1 : {A B : Set} → ( A ∧ B ) → A
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205 ex1 a∧b = proj1 a∧b
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206
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207 a∧b is a variable name. If we have no spaces in a string, it is a word even if we have symbols
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208 except parenthesis or colons. A symbol requires space separation such as a type defining colon.
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209
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210 Defining record can be recursively, but we don't use the recursion here.
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211
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212 -- Mathematical structure
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213
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214 We have types of elements and the relationship in a mathematical structure.
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215
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216 logical relation has no ordering
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217 there is a natural ordering in arguments and a value in a function
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218
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219 So we have typical definition style of mathematical structure with records.
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220
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221 record IsOrdinals {n : Level} (ord : Set n)
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222 (_o<_ : ord → ord → Set n) : Set (suc (suc n)) where
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223 field
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224 Otrans : {x y z : ord } → x o< y → y o< z → x o< z
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225
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226 record Ordinals {n : Level} : Set (suc (suc n)) where
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227 field
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228 ord : Set n
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229 _o<_ : ord → ord → Set n
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230 isOrdinal : IsOrdinals ord _o<_
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231
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232 In IsOrdinals, axioms are written in flat way. In Ordinal, we may have
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233 inputs and outputs are put in the field including IsOrdinal.
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234
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235 Fields of Ordinal is existential objects in the mathematical structure.
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236
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359
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237 -- Limit Ordinal
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238
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239 If an ordinal is not a succesor of other, it is called limit ordinal. We need predicate to decide it.
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240
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241 not-limit-p : ( x : ord ) → Dec ( ¬ ((y : ord) → ¬ (x ≡ osuc y) ))
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242
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243 An ordinal may have an imeditate limit ordinal, we call it next x. Axiom of nrext is this.
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244
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245 record IsNext {n : Level } (ord : Set n) (o∅ : ord ) (osuc : ord → ord )
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246 (_o<_ : ord → ord → Set n) (next : ord → ord ) : Set (suc (suc n)) where
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247 field
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248 x<nx : { y : ord } → (y o< next y )
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249 osuc<nx : { x y : ord } → x o< next y → osuc x o< next y
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250 ¬nx<nx : {x y : ord} → y o< x → x o< next y → ¬ ((z : ord) → ¬ (x ≡ osuc z))
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251
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252 We show some intresting lemma.
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253
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254 next< : {x y z : Ordinal} → x o< next z → y o< next x → y o< next z
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255 nexto=n : {x y : Ordinal} → x o< next (osuc y) → x o< next y
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256 nexto≡ : {x : Ordinal} → next x ≡ next (osuc x)
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257 next-is-limit : {x y : Ordinal} → ¬ (next x ≡ osuc y)
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258
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259 These are proved from the axiom. Our countable ordinal satisfies these.
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260
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273
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261 -- A Model and a theory
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262
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263 Agda record is a type, so we can write it in the argument, but is it really exists?
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264
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265 If we have a value of the record, it simply exists, that is, we need to create all the existence
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266 in the record satisfies all the axioms (= field of IsOrdinal) should be valid.
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267
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268 type of record = theory
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269 value of record = model
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270
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271 We call the value of the record as a model. If mathematical structure has a
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272 model, it exists. Pretty Obvious.
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273
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274 -- postulate と module
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275
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276 Agda proofs are separated by modules, which are large records.
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277
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278 postulates are assumptions. We can assume a type without proofs.
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279
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280 postulate
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281 sup-o : ( Ordinal → Ordinal ) → Ordinal
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282 sup-o< : { ψ : Ordinal → Ordinal } → ∀ {x : Ordinal } → ψ x o< sup-o ψ
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283
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284 sup-o is an example of upper bound of a function and sup-o< assumes it actually
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285 satisfies all the value is less than upper bound.
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286
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287 Writing some type in a module argument is the same as postulating a type, but
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288 postulate can be written the middle of a proof.
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289
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290 postulate can be constructive.
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291
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336
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292 postulate can be inconsistent, which result everything has a proof. Actualy this assumption
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293 doesnot work for Ordinals, we discuss this later.
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273
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294
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295 -- A ∨ B
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296
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297 data _∨_ (A B : Set) : Set where
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298 case1 : A → A ∨ B
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299 case2 : B → A ∨ B
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300
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301 As Haskell, case1/case2 are patterns.
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302
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303 ex3 : {A B : Set} → ( A ∨ A ) → A
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304 ex3 = ?
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305
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306 In a case statement, Agda command C-C C-C generates possible cases in the head.
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307
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308 ex3 : {A B : Set} → ( A ∨ A ) → A
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309 ex3 (case1 x) = ?
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310 ex3 (case2 x) = ?
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311
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312 Proof schema of ∨ is omit due to the complexity.
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313
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314 -- Negation
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315
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316 ⊥
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317 ------------- ⊥-elim
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318 A
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319
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320 Anything can be derived from bottom, in this case a Set A. There is no introduction rule
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321 in ⊥, which can be implemented as data which has no constructor.
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322
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323 data ⊥ : Set where
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324
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325 ⊥-elim can be proved like this.
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326
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327 ⊥-elim : {A : Set } -> ⊥ -> A
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328 ⊥-elim ()
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329
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330 () means no match argument nor value.
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331
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332 A negation can be defined using ⊥ like this.
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333
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334 ¬_ : Set → Set
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335 ¬ A = A → ⊥
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336
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337 --Equality
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338
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339 All the value in Agda are terms. If we have the same normalized form, two terms are equal.
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340 If we have variables in the terms, we will perform an unification. unifiable terms are equal.
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341 We don't go further on the unification.
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342
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343 { x : A } x ≡ y f x y
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344 --------------- ≡-intro --------------------- ≡-elim
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345 x ≡ x f x x
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346
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347 equality _≡_ can be defined as a data.
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348
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349 data _≡_ {A : Set } : A → A → Set where
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350 refl : {x : A} → x ≡ x
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351
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352 The elimination of equality is a substitution in a term.
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353
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354 subst : {A : Set } → { x y : A } → ( f : A → Set ) → x ≡ y → f x → f y
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355 subst {A} {x} {y} f refl fx = fx
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356
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357 ex5 : {A : Set} {x y z : A } → x ≡ y → y ≡ z → x ≡ z
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358 ex5 {A} {x} {y} {z} x≡y y≡z = subst ( λ k → x ≡ k ) y≡z x≡y
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359
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360
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361 --Equivalence relation
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362
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363 refl' : {A : Set} {x : A } → x ≡ x
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364 refl' = ?
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365 sym : {A : Set} {x y : A } → x ≡ y → y ≡ x
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366 sym = ?
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367 trans : {A : Set} {x y z : A } → x ≡ y → y ≡ z → x ≡ z
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368 trans = ?
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369 cong : {A B : Set} {x y : A } { f : A → B } → x ≡ y → f x ≡ f y
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370 cong = ?
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371
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372 --Ordering
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373
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374 Relation is a predicate on two value which has a same type.
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375
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376 A → A → Set
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377
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378 Defining order is the definition of this type with predicate or a data.
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379
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380 data _≤_ : Rel ℕ 0ℓ where
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381 z≤n : ∀ {n} → zero ≤ n
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382 s≤s : ∀ {m n} (m≤n : m ≤ n) → suc m ≤ suc n
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383
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384
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385 --Quantifier
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386
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387 Handling quantifier in an intuitionistic logic requires special cares.
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388
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389 In the input of a function, there are no restriction on it, that is, it has
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390 a universal quantifier. (If we explicitly write ∀, Agda gives us a type inference on it)
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391
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392 There is no ∃ in agda, the one way is using negation like this.
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393
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394 ∃ (x : A ) → p x = ¬ ( ( x : A ) → ¬ ( p x ) )
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395
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396 On the another way, f : A can be used like this.
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397
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398 p f
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399
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400 If we use a function which can be defined globally which has stronger meaning the
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401 usage of ∃ x in a logical expression.
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402
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403
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404 --Can we do math in this way?
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405
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406 Yes, we can. Actually we have Principia Mathematica by Russell and Whitehead (with out computer support).
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407
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408 In some sense, this story is a reprinting of the work, (but Principia Mathematica has a different formulation than ZF).
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409
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410 define mathematical structure as a record
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411 program inferences as if we have proofs in variables
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412
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413 --Things which Agda cannot prove
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414
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415 The infamous Internal Parametricity is a limitation of Agda, it cannot prove so called Free Theorem, which
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416 leads uniqueness of a functor in Category Theory.
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417
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418 Functional extensionality cannot be proved.
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419 (∀ x → f x ≡ g x) → f ≡ g
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420
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421 Agda has no law of exclude middle.
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422
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423 a ∨ ( ¬ a )
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424
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425 For example, (A → B) → ¬ B → ¬ A can be proved but, ( ¬ B → ¬ A ) → A → B cannot.
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426
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427 It also other problems such as termination, type inference or unification which we may overcome with
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428 efforts or devices or may not.
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429
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430 If we cannot prove something, we can safely postulate it unless it leads a contradiction.
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431
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432 --Classical story of ZF Set Theory
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433
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336
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434 <a name="set-theory">
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273
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435 Assuming ZF, constructing a model of ZF is a flow of classical Set Theory, which leads
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436 a relative consistency proof of the Set Theory.
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437 Ordinal number is used in the flow.
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438
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439 In Agda, first we defines Ordinal numbers (Ordinals), then introduce Ordinal Definable Set (OD).
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440 We need some non constructive assumptions in the construction. A model of Set theory is
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441 constructed based on these assumptions.
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442
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443 <center><img src="fig/set-theory.svg"></center>
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444
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445 --Ordinals
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446
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447 Ordinals are our intuition of infinite things, which has ∅ and orders on the things.
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448 It also has a successor osuc.
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449
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450 record Ordinals {n : Level} : Set (suc (suc n)) where
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451 field
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452 ord : Set n
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453 o∅ : ord
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454 osuc : ord → ord
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455 _o<_ : ord → ord → Set n
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456 isOrdinal : IsOrdinals ord o∅ osuc _o<_
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457
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458 It is different from natural numbers in way. The order of Ordinals is not defined in terms
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459 of successor. It is given from outside, which make it possible to have higher order infinity.
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460
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461 --Axiom of Ordinals
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462
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463 Properties of infinite things. We request a transfinite induction, which states that if
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464 some properties are satisfied below all possible ordinals, the properties are true on all
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465 ordinals.
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466
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467 Successor osuc has no ordinal between osuc and the base ordinal. There are some ordinals
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468 which is not a successor of any ordinals. It is called limit ordinal.
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469
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470 Any two ordinal can be compared, that is less, equal or more, that is total order.
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471
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472 record IsOrdinals {n : Level} (ord : Set n) (o∅ : ord )
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473 (osuc : ord → ord )
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474 (_o<_ : ord → ord → Set n) : Set (suc (suc n)) where
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475 field
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476 Otrans : {x y z : ord } → x o< y → y o< z → x o< z
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477 OTri : Trichotomous {n} _≡_ _o<_
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478 ¬x<0 : { x : ord } → ¬ ( x o< o∅ )
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479 <-osuc : { x : ord } → x o< osuc x
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480 osuc-≡< : { a x : ord } → x o< osuc a → (x ≡ a ) ∨ (x o< a)
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481 TransFinite : { ψ : ord → Set (suc n) }
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482 → ( (x : ord) → ( (y : ord ) → y o< x → ψ y ) → ψ x )
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483 → ∀ (x : ord) → ψ x
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484
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336
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485 --Concrete Ordinals or Countable Ordinals
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273
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486
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487 We can define a list like structure with level, which is a kind of two dimensional infinite array.
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488
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489 data OrdinalD {n : Level} : (lv : Nat) → Set n where
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490 Φ : (lv : Nat) → OrdinalD lv
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491 OSuc : (lv : Nat) → OrdinalD {n} lv → OrdinalD lv
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492
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493 The order of the OrdinalD can be defined in this way.
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494
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495 data _d<_ {n : Level} : {lx ly : Nat} → OrdinalD {n} lx → OrdinalD {n} ly → Set n where
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496 Φ< : {lx : Nat} → {x : OrdinalD {n} lx} → Φ lx d< OSuc lx x
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497 s< : {lx : Nat} → {x y : OrdinalD {n} lx} → x d< y → OSuc lx x d< OSuc lx y
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498
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499 This is a simple data structure, it has no abstract assumptions, and it is countable many data
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500 structure.
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501
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502 Φ 0
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503 OSuc 2 ( Osuc 2 ( Osuc 2 (Φ 2)))
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504 Osuc 0 (Φ 0) d< Φ 1
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505
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506 --Model of Ordinals
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507
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508 It is easy to show OrdinalD and its order satisfies the axioms of Ordinals.
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509
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510 So our Ordinals has a mode. This means axiom of Ordinals are consistent.
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511
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512 --Debugging axioms using Model
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513
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514 Whether axiom is correct or not can be checked by a validity on a mode.
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515
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516 If not, we may fix the axioms or the model, such as the definitions of the order.
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517
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518 We can also ask whether the inputs exist.
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519
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520 --Countable Ordinals can contains uncountable set?
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521
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522 Yes, the ordinals contains any level of infinite Set in the axioms.
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523
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524 If we handle real-number in the model, only countable number of real-number is used.
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525
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526 from the outside view point, it is countable
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527 from the internal view point, it is uncountable
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528
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529 The definition of countable/uncountable is the same, but the properties are different
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530 depending on the context.
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531
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532 We don't show the definition of cardinal number here.
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533
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534 --What is Set
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535
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536 The word Set in Agda is not a Set of ZF Set, but it is a type (why it is named Set?).
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537
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538 From naive point view, a set i a list, but in Agda, elements have all the same type.
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539 A set in ZF may contain other Sets in ZF, which not easy to implement it as a list.
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540
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541 Finite set may be written in finite series of ∨, but ...
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542
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543 --We don't ask the contents of Set. It can be anything.
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544
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545 From empty set φ, we can think a set contains a φ, and a pair of φ and the set, and so on,
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546 and all of them, and again we repeat this.
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547
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548 φ {φ} {φ,{φ}}, {φ,{φ},...}
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549
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550 It is called V.
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551
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552 This operation can be performed within a ZF Set theory. Classical Set Theory assumes
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553 ZF, so this kind of thing is allowed.
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554
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555 But in our case, we have no ZF theory, so we are going to use Ordinals.
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556
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336
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557 The idea is to use an ordinal as a pointer to a record which defines a Set.
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558 If the recored defines a series of Ordinals which is a pointer to the Set.
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559 This record looks like a Set.
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273
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560
|
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561 --Ordinal Definable Set
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562
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|
563 We can define a sbuset of Ordinals using predicates. What is a subset?
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564
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565 a predicate has an Ordinal argument
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566
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567 is an Ordinal Definable Set (OD). In Agda, OD is defined as follows.
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568
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569 record OD : Set (suc n ) where
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570 field
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571 def : (x : Ordinal ) → Set n
|
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572
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|
573 Ordinals itself is not a set in a ZF Set theory but a class. In OD,
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574
|
|
336
|
575 data One : Set n where
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|
|
576 OneObj : One
|
|
273
|
577
|
|
336
|
578 record { def = λ x → One }
|
|
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579
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|
580 means it accepets all Ordinals, i.e. this is Ordinals itself, so ODs are larger than ZF Set.
|
|
|
581 You can say OD is a class in ZF Set Theory term.
|
|
273
|
582
|
|
|
583
|
|
336
|
584 --OD is not ZF Set
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585
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|
586 If we have 1 to 1 mapping between an OD and an Ordinal, OD contains several ODs and OD looks like
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587 a Set. The idea is to use an ordinal as a pointer to OD.
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|
|
588 Unfortunately this scheme does not work well. As we saw OD includes all Ordinals,
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589 which is a maximum of OD, but Ordinals has no maximum at all. So we have a contradction like
|
|
273
|
590
|
|
425
|
591 ¬OD-order : ( & : OD → Ordinal )
|
|
|
592 → ( * : Ordinal → OD ) → ( { x y : OD } → def y ( & x ) → & x o< & y) → ⊥
|
|
|
593 ¬OD-order & * c<→o< = ?
|
|
336
|
594
|
|
|
595 Actualy we can prove this contrdction, so we need some restrctions on OD.
|
|
|
596
|
|
|
597 This is a kind of Russel paradox, that is if OD contains everything, what happens if it contains itself.
|
|
|
598
|
|
|
599 -- HOD : Hereditarily Ordinal Definable
|
|
|
600
|
|
|
601 What we need is a bounded OD, the containment is limited by an ordinal.
|
|
273
|
602
|
|
336
|
603 record HOD : Set (suc n) where
|
|
|
604 field
|
|
|
605 od : OD
|
|
|
606 odmax : Ordinal
|
|
|
607 <odmax : {y : Ordinal} → def od y → y o< odmax
|
|
|
608
|
|
|
609 In classical Set Theory, HOD stands for Hereditarily Ordinal Definable, which means
|
|
|
610
|
|
|
611 HOD = { x | TC x ⊆ OD }
|
|
273
|
612
|
|
336
|
613 TC x is all transitive closure of x, that is elements of x and following all elements of them are all OD. But
|
|
|
614 what is x? In this case, x is an Set which we don't have yet. In our case, HOD is a bounded OD.
|
|
|
615
|
|
|
616 --1 to 1 mapping between an HOD and an Ordinal
|
|
|
617
|
|
|
618 HOD is a predicate on Ordinals and the solution is bounded by some ordinal. If we have a mapping
|
|
|
619
|
|
425
|
620 & : HOD → Ordinal
|
|
|
621 * : Ordinal → HOD
|
|
|
622 oiso : {x : HOD } → * ( & x ) ≡ x
|
|
|
623 diso : {x : Ordinal } → & ( * x ) ≡ x
|
|
336
|
624
|
|
|
625 we can check an HOD is an element of the OD using def.
|
|
273
|
626
|
|
|
627 A ∋ x can be define as follows.
|
|
|
628
|
|
336
|
629 _∋_ : ( A x : HOD ) → Set n
|
|
425
|
630 _∋_ A x = def (od A) ( & x )
|
|
273
|
631
|
|
|
632 In ψ : Ordinal → Set, if A is a record { def = λ x → ψ x } , then
|
|
|
633
|
|
425
|
634 A x = def A ( & x ) = ψ (& x)
|
|
273
|
635
|
|
|
636 They say the existing of the mappings can be proved in Classical Set Theory, but we
|
|
|
637 simply assumes these non constructively.
|
|
|
638
|
|
|
639 <center><img src="fig/ord-od-mapping.svg"></center>
|
|
|
640
|
|
|
641 --Order preserving in the mapping of OD and Ordinal
|
|
|
642
|
|
336
|
643 Ordinals have the order and HOD has a natural order based on inclusion ( def / ∋ ).
|
|
273
|
644
|
|
425
|
645 def (od y) ( & x )
|
|
273
|
646
|
|
336
|
647 An elements of HOD should be defined before the HOD, that is, an ordinal corresponding an elements
|
|
|
648 have to be smaller than the corresponding ordinal of the containing OD.
|
|
|
649 We also assumes subset is always smaller. This is necessary to make a limit of Power Set.
|
|
273
|
650
|
|
425
|
651 c<→o< : {x y : HOD } → def (od y) ( & x ) → & x o< & y
|
|
|
652 ⊆→o≤ : {y z : HOD } → ({x : Ordinal} → def (od y) x → def (od z) x ) → & y o< osuc (& z)
|
|
273
|
653
|
|
336
|
654 If wa assumes reverse order preservation,
|
|
273
|
655
|
|
425
|
656 o<→c< : {n : Level} {x y : Ordinal } → x o< y → def (* y) x
|
|
273
|
657
|
|
336
|
658 ∀ x ∋ ∅ becomes true, which manes all OD becomes Ordinals in the model.
|
|
273
|
659
|
|
|
660 <center><img src="fig/ODandOrdinals.svg"></center>
|
|
|
661
|
|
|
662 --Various Sets
|
|
|
663
|
|
|
664 In classical Set Theory, there is a hierarchy call L, which can be defined by a predicate.
|
|
|
665
|
|
|
666 Ordinal / things satisfies axiom of Ordinal / extension of natural number
|
|
|
667 V / hierarchical construction of Set from φ
|
|
|
668 L / hierarchical predicate definable construction of Set from φ
|
|
336
|
669 HOD / Hereditarily Ordinal Definable
|
|
273
|
670 OD / equational formula on Ordinals
|
|
|
671 Agda Set / Type / it also has a level
|
|
|
672
|
|
|
673
|
|
361
|
674 --Fixing ZF axom to fit intuitionistic logic
|
|
273
|
675
|
|
|
676 We use ODs as Sets in ZF, and defines record ZF, that is, we have to define
|
|
|
677 ZF axioms in Agda.
|
|
|
678
|
|
|
679 It may not valid in our model. We have to debug it.
|
|
|
680
|
|
|
681 Fixes are depends on axioms.
|
|
|
682
|
|
|
683 <center><img src="fig/axiom-type.svg"></center>
|
|
|
684
|
|
|
685 <a href="fig/zf-record.html">
|
|
|
686 ZFのrecord
|
|
|
687 </a>
|
|
|
688
|
|
|
689 --Pure logical axioms
|
|
|
690
|
|
279
|
691 empty, pair, select, ε-induction??infinity
|
|
273
|
692
|
|
|
693 These are logical relations among OD.
|
|
|
694
|
|
|
695 empty : ∀( x : ZFSet ) → ¬ ( ∅ ∋ x )
|
|
|
696 pair→ : ( x y t : ZFSet ) → (x , y) ∋ t → ( t ≈ x ) ∨ ( t ≈ y )
|
|
|
697 pair← : ( x y t : ZFSet ) → ( t ≈ x ) ∨ ( t ≈ y ) → (x , y) ∋ t
|
|
|
698 selection : { ψ : ZFSet → Set m } → ∀ { X y : ZFSet } → ( ( y ∈ X ) ∧ ψ y ) ⇔ (y ∈ Select X ψ )
|
|
|
699 infinity∅ : ∅ ∈ infinite
|
|
|
700 infinity : ∀( x : ZFSet ) → x ∈ infinite → ( x ∪ ( x , x ) ) ∈ infinite
|
|
|
701 ε-induction : { ψ : OD → Set (suc n)}
|
|
|
702 → ( {x : OD } → ({ y : OD } → x ∋ y → ψ y ) → ψ x )
|
|
|
703 → (x : OD ) → ψ x
|
|
|
704
|
|
|
705 finitely can be define by Agda data.
|
|
|
706
|
|
|
707 data infinite-d : ( x : Ordinal ) → Set n where
|
|
|
708 iφ : infinite-d o∅
|
|
|
709 isuc : {x : Ordinal } → infinite-d x →
|
|
425
|
710 infinite-d (& ( Union (* x , (* x , * x ) ) ))
|
|
273
|
711
|
|
|
712 Union (x , ( x , x )) should be an direct successor of x, but we cannot prove it in our model.
|
|
|
713
|
|
|
714 --Axiom of Pair
|
|
|
715
|
|
|
716 In the Tanaka's book, axiom of pair is as follows.
|
|
|
717
|
|
|
718 ∀ x ∀ y ∃ z ∀ t ( z ∋ t ↔ t ≈ x ∨ t ≈ y)
|
|
|
719
|
|
|
720 We have fix ∃ z, a function (x , y) is defined, which is _,_ .
|
|
|
721
|
|
|
722 _,_ : ( A B : ZFSet ) → ZFSet
|
|
|
723
|
|
|
724 using this, we can define two directions in separates axioms, like this.
|
|
|
725
|
|
|
726 pair→ : ( x y t : ZFSet ) → (x , y) ∋ t → ( t ≈ x ) ∨ ( t ≈ y )
|
|
|
727 pair← : ( x y t : ZFSet ) → ( t ≈ x ) ∨ ( t ≈ y ) → (x , y) ∋ t
|
|
|
728
|
|
|
729 This is already written in Agda, so we use these as axioms. All inputs have ∀.
|
|
|
730
|
|
|
731 --pair in OD
|
|
|
732
|
|
|
733 OD is an equation on Ordinals, we can simply write axiom of pair in the OD.
|
|
|
734
|
|
336
|
735 _,_ : HOD → HOD → HOD
|
|
425
|
736 x , y = record { od = record { def = λ t → (t ≡ & x ) ∨ ( t ≡ & y ) } ; odmax = ? ; <odmax = ? }
|
|
336
|
737
|
|
|
738 It is easy to find out odmax from odmax of x and y.
|
|
273
|
739
|
|
|
740 ≡ is an equality of λ terms, but please not that this is equality on Ordinals.
|
|
|
741
|
|
|
742 --Validity of Axiom of Pair
|
|
|
743
|
|
|
744 Assuming ZFSet is OD, we are going to prove pair→ .
|
|
|
745
|
|
|
746 pair→ : ( x y t : OD ) → (x , y) ∋ t → ( t == x ) ∨ ( t == y )
|
|
|
747 pair→ x y t p = ?
|
|
|
748
|
|
|
749 In this program, type of p is ( x , y ) ∋ t , that is
|
|
425
|
750 def ( x , y ) that is, (t ≡ & x ) ∨ ( t ≡ & y ) .
|
|
273
|
751
|
|
|
752 Since _∨_ is a data, it can be developed as (C-c C-c : agda2-make-case ).
|
|
|
753
|
|
|
754 pair→ x y t (case1 t≡x ) = ?
|
|
|
755 pair→ x y t (case2 t≡y ) = ?
|
|
|
756
|
|
|
757 The type of the ? is ( t == x ) ∨ ( t == y ), again it is data _∨_ .
|
|
|
758
|
|
|
759 pair→ x y t (case1 t≡x ) = case1 ?
|
|
|
760 pair→ x y t (case2 t≡y ) = case2 ?
|
|
|
761
|
|
|
762 The ? in case1 is t == x, so we have to create this from t≡x, which is a name of a variable
|
|
|
763 which type is
|
|
|
764
|
|
425
|
765 t≡x : & t ≡ & x
|
|
273
|
766
|
|
|
767 which is shown by an Agda command (C-C C-E : agda2-show-context ).
|
|
|
768
|
|
|
769 But we haven't defined == yet.
|
|
|
770
|
|
|
771 --Equality of OD and Axiom of Extensionality
|
|
|
772
|
|
|
773 OD is defined by a predicates, if we compares normal form of the predicates, even if
|
|
|
774 it contains the same elements, it may be different, which is no good as an equality of
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775 Sets.
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776
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777 Axiom of Extensionality requires sets having the same elements are handled in the same way
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778 each other.
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779
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780 ∀ z ( z ∈ x ⇔ z ∈ y ) ⇒ ∀ w ( x ∈ w ⇔ y ∈ w )
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781
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782 We can write this axiom in Agda as follows.
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783
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784 extensionality : { A B w : ZFSet } → ( (z : ZFSet) → ( A ∋ z ) ⇔ (B ∋ z) ) → ( A ∈ w ⇔ B ∈ w )
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785
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786 So we use ( A ∋ z ) ⇔ (B ∋ z) as an equality (_==_) of our model. We have to show
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787 A ∈ w ⇔ B ∈ w from A == B.
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788
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789 x == y can be defined in this way.
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790
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791 record _==_ ( a b : OD ) : Set n where
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792 field
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793 eq→ : ∀ { x : Ordinal } → def a x → def b x
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794 eq← : ∀ { x : Ordinal } → def b x → def a x
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795
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336
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796 Actually, (z : HOD) → (A ∋ z) ⇔ (B ∋ z) implies od A == od B.
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273
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797
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336
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798 extensionality0 : {A B : HOD } → ((z : HOD) → (A ∋ z) ⇔ (B ∋ z)) → od A == od B
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273
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799 eq→ (extensionality0 {A} {B} eq ) {x} d = ?
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800 eq← (extensionality0 {A} {B} eq ) {x} d = ?
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801
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802 ? are def B x and def A x and these are generated from eq : (z : OD) → (A ∋ z) ⇔ (B ∋ z) .
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803
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804 Actual proof is rather complicated.
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805
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425
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806 eq→ (extensionality0 {A} {B} eq ) {x} d = odef-iso {A} {B} (sym diso) (proj1 (eq (* x))) d
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807 eq← (extensionality0 {A} {B} eq ) {x} d = odef-iso {B} {A} (sym diso) (proj2 (eq (* x))) d
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273
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808
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809 where
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810
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336
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811 odef-iso : {A B : HOD } {x y : Ordinal } → x ≡ y → (def A (od y) → def (od B) y) → def (od A) x → def (od B) x
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812 odef-iso refl t = t
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273
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813
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363
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814 --The uniqueness of HOD
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815
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816 To prove extensionality or other we need ==→o≡.
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817
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818 Since we have ==→o≡ , so odmax have to be unique. We should have odmaxmin in HOD.
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819 We can calculate the minimum using sup if we have bound but it is tedius.
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820 Only Select has non minimum odmax.
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821 We have the same problem on 'def' itself, but we leave it, that is we assume the
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822 assumption of predicates of Agda have a unique form, it is something like the
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823 functional extensionality assumption.
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824
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273
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825 --Validity of Axiom of Extensionality
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826
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336
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827 If we can derive (w ∋ A) ⇔ (w ∋ B) from od A == od B, the axiom becomes valid, but it seems impossible,
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273
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828 so we assumes
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829
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336
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830 ==→o≡ : { x y : HOD } → (od x == od y) → x ≡ y
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273
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831
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832 Using this, we have
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833
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336
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834 extensionality : {A B w : HOD } → ((z : HOD ) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B)
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273
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835 proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d
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336
|
836 proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d
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273
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837
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359
|
838 --Axiom of infinity
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839
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840 It means it has ω as a ZF Set. It is ususally written like this.
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841
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842 ∃ A ( ∅ ∈ A ∧ ∀ x ∈ A ( x ∪ { x } ∈ A ) )
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843
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844 x ∪ { x } is Union (x , (x , x)) in our notation. It contains existential quantifier, so we introduce a function
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845
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846 infinite : ZFSet
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847 infinity∅ : ∅ ∈ infinite
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848 infinity : ∀( x : ZFSet ) → x ∈ infinite → ( x ∪ ｛ x ｝) ∈ infinite
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849
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850 --ω in HOD
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851
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425
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852 To define an OD which arrows & (Union (x , (x , x))) as a predicate, we can use Agda data structure.
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359
|
853
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854 data infinite-d : ( x : Ordinal ) → Set n where
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855 iφ : infinite-d o∅
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856 isuc : {x : Ordinal } → infinite-d x →
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425
|
857 infinite-d (& ( Union (* x , (* x , * x ) ) ))
|
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359
|
858
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859 It has simular structure as Data.Nat in Agda, and it defines a correspendence of HOD and the data structure. Now
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860 we can define HOD like this.
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861
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862 infinite : HOD
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863 infinite = record { od = record { def = λ x → infinite-d x } ; odmax = ? ; <odmax = ? }
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864
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425
|
865 So what is the bound of ω? Analyzing the definition, it depends on the value of & (x , x), which
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866 cannot know the aboslute value. This is because we don't have constructive definition of & : HOD → Ordinal.
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867
|
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868 --HOD and Agda data structure
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869
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870 We may have
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871
|
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872 a HOD <=> Agda Data Strucure
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873
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874 as a data in Agda. Ex.
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875
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876 data ord-pair : (p : Ordinal) → Set n where
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877 pair : (x y : Ordinal ) → ord-pair ( & ( < * x , * y > ) )
|
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878
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|
879 ZFProduct : OD
|
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|
880 ZFProduct = record { def = λ x → ord-pair x }
|
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881
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882 pi1 : { p : Ordinal } → ord-pair p → Ordinal
|
|
|
883 pi1 ( pair x y) = x
|
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884
|
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885 π1 : { p : HOD } → def ZFProduct (& p) → HOD
|
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|
886 π1 lt = * (pi1 lt )
|
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887
|
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|
888 p-iso : { x : HOD } → (p : def ZFProduct (& x) ) → < π1 p , π2 p > ≡ x
|
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|
889 p-iso {x} p = ord≡→≡ (op-iso p)
|
|
|
890
|
|
359
|
891
|
|
361
|
892 --HOD Ordrinal mapping
|
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893
|
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894 We have large freedom on HOD. We reqest no minimality on odmax, so it may arbitrary larger.
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|
895 The address of HOD can be larger at least it have to be larger than the content's address.
|
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896 We only have a relative ordering such as
|
|
|
897
|
|
425
|
898 pair-xx<xy : {x y : HOD} → & (x , x) o< osuc (& (x , y) )
|
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899 pair<y : {x y : HOD } → y ∋ x → & (x , x) o< osuc (& y)
|
|
361
|
900
|
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|
901 Basicaly, we cannot know the concrete address of HOD other than empty set.
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|
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902
|
|
|
903 <center><img src="fig/address-of-HOD.svg"></center>
|
|
|
904
|
|
|
905 --Possible restriction on HOD
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906
|
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359
|
907 We need some restriction on the HOD-Ordinal mapping. Simple one is this.
|
|
|
908
|
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|
909 ωmax : Ordinal
|
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|
910 <ωmax : {y : Ordinal} → infinite-d y → y o< ωmax
|
|
|
911
|
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912 It depends on infinite-d and put no assuption on the other similar construction. A more general one may be
|
|
|
913
|
|
|
914 hod-ord< : {x : HOD } → Set n
|
|
425
|
915 hod-ord< {x} = & x o< next (odmax x)
|
|
359
|
916
|
|
|
917 next : Ordinal → Ordinal means imediate next limit ordinal of x. It supress unecessary space between address of HOD and
|
|
|
918 its bound.
|
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|
919
|
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|
920 In other words, the space between address of HOD and its bound is arbitrary, it is possible to assume ω has no bound.
|
|
|
921 This is the reason of necessity of Axiom of infinite.
|
|
|
922
|
|
|
923 --increment pase of address of HOD
|
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924
|
|
361
|
925 Assuming, hod-ord< , we have
|
|
359
|
926
|
|
425
|
927 pair-ord< : {x : HOD } → ( {y : HOD } → & y o< next (odmax y) ) → & ( x , x ) o< next (& x)
|
|
|
928 pair-ord< {x} ho< = subst (λ k → & (x , x) o< k ) lemmab0 lemmab1 where
|
|
|
929 lemmab0 : next (odmax (x , x)) ≡ next (& x)
|
|
359
|
930
|
|
361
|
931 So the address of ( x , x) and Union (x , (x , x)) is restricted in the limit ordinal. This makes ω bound. We can prove
|
|
|
932
|
|
425
|
933 infinite-bound : ({x : HOD} → & x o< next (odmax x)) → {y : Ordinal} → infinite-d y → y o< next o∅
|
|
359
|
934
|
|
425
|
935 We also notice that if we have & (x , x) ≡ osuc (& x), c<→o< can be drived from ⊆→o≤ .
|
|
359
|
936
|
|
425
|
937 ⊆→o≤→c<→o< : ({x : HOD} → & (x , x) ≡ osuc (& x) )
|
|
|
938 → ({y z : HOD } → ({x : Ordinal} → def (od y) x → def (od z) x ) → & y o< osuc (& z) )
|
|
|
939 → {x y : HOD } → def (od y) ( & x ) → & x o< & y
|
|
359
|
940
|
|
273
|
941 --Non constructive assumptions so far
|
|
|
942
|
|
425
|
943 & : HOD → Ordinal
|
|
|
944 * : Ordinal → HOD
|
|
|
945 c<→o< : {x y : HOD } → def (od y) ( & x ) → & x o< & y
|
|
|
946 ⊆→o≤ : {y z : HOD } → ({x : Ordinal} → def (od y) x → def (od z) x ) → & y o< osuc (& z)
|
|
|
947 oiso : {x : HOD } → * ( & x ) ≡ x
|
|
|
948 diso : {x : Ordinal } → & ( * x ) ≡ x
|
|
336
|
949 ==→o≡ : {x y : HOD } → (od x == od y) → x ≡ y
|
|
|
950 sup-o : (A : HOD) → ( ( x : Ordinal ) → def (od A) x → Ordinal ) → Ordinal
|
|
|
951 sup-o< : (A : HOD) → { ψ : ( x : Ordinal ) → def (od A) x → Ordinal } → ∀ {x : Ordinal } → (lt : def (od A) x ) → ψ x lt o< sup-o A ψ
|
|
273
|
952
|
|
|
953 --Axiom which have negation form
|
|
|
954
|
|
|
955 Union, Selection
|
|
|
956
|
|
|
957 These axioms contains ∃ x as a logical relation, which can be described in ¬ ( ∀ x ( ¬ p )).
|
|
|
958
|
|
|
959 Axiom of replacement uses upper bound of function on Ordinals, which makes it non-constructive.
|
|
|
960
|
|
|
961 Power Set axiom requires double negation,
|
|
|
962
|
|
|
963 power→ : ∀( A t : ZFSet ) → Power A ∋ t → ∀ {x} → t ∋ x → ¬ ¬ ( A ∋ x )
|
|
|
964 power← : ∀( A t : ZFSet ) → t ⊆_ A → Power A ∋ t
|
|
|
965
|
|
|
966 If we have an assumption of law of exclude middle, we can recover the original A ∋ x form.
|
|
|
967
|
|
|
968 --Union
|
|
|
969
|
|
|
970 The original form of the Axiom of Union is
|
|
|
971
|
|
|
972 ∀ x ∃ y ∀ z (z ∈ y ⇔ ∃ u ∈ x ∧ (z ∈ u))
|
|
|
973
|
|
|
974 Union requires the existence of b in a ⊇ ∃ b ∋ x . We will use negation form of ∃.
|
|
|
975
|
|
|
976 union→ : ( X z u : ZFSet ) → ( X ∋ u ) ∧ (u ∋ z ) → Union X ∋ z
|
|
|
977 union← : ( X z : ZFSet ) → (X∋z : Union X ∋ z ) → ¬ ( (u : ZFSet ) → ¬ ((X ∋ u) ∧ (u ∋ z )))
|
|
|
978
|
|
359
|
979 The definition of Union in HOD is like this.
|
|
273
|
980
|
|
359
|
981 Union : HOD → HOD
|
|
425
|
982 Union U = record { od = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((odef U u) ∧ (odef (* u) x))) }
|
|
|
983 ; odmax = osuc (& U) ; <odmax = ? }
|
|
359
|
984
|
|
|
985 The bound of Union is evident, but its proof is rather complicated.
|
|
273
|
986
|
|
|
987 Proof of validity is straight forward.
|
|
|
988
|
|
359
|
989 union→ : (X z u : HOD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z
|
|
425
|
990 union→ X z u xx not = ⊥-elim ( not (& u) ( record { proj1 = proj1 xx
|
|
|
991 ; proj2 = subst ( λ k → odef k (& z)) (sym oiso) (proj2 xx) } ))
|
|
359
|
992 union← : (X z : HOD) (X∋z : Union X ∋ z) → ¬ ( (u : HOD ) → ¬ ((X ∋ u) ∧ (u ∋ z )))
|
|
273
|
993 union← X z UX∋z = FExists _ lemma UX∋z where
|
|
425
|
994 lemma : {y : Ordinal} → odef X y ∧ odef (* y) (& z) → ¬ ((u : HOD) → ¬ (X ∋ u) ∧ (u ∋ z))
|
|
|
995 lemma {y} xx not = not (* y) record { proj1 = subst ( λ k → odef X k ) (sym diso) (proj1 xx ) ; proj2 = proj2 xx }
|
|
273
|
996
|
|
|
997 where
|
|
|
998
|
|
|
999 FExists : {m l : Level} → ( ψ : Ordinal → Set m )
|
|
|
1000 → {p : Set l} ( P : { y : Ordinal } → ψ y → ¬ p )
|
|
|
1001 → (exists : ¬ (∀ y → ¬ ( ψ y ) ))
|
|
|
1002 → ¬ p
|
|
|
1003 FExists {m} {l} ψ {p} P = contra-position ( λ p y ψy → P {y} ψy p )
|
|
|
1004
|
|
|
1005 which checks existence using contra-position.
|
|
|
1006
|
|
|
1007 --Axiom of replacement
|
|
|
1008
|
|
|
1009 We can replace the elements of a set by a function and it becomes a set. From the book,
|
|
|
1010
|
|
|
1011 ∀ x ∀ y ∀ z ( ( ψ ( x , y ) ∧ ψ ( x , z ) ) → y = z ) → ∀ X ∃ A ∀ y ( y ∈ A ↔ ∃ x ∈ X ψ ( x , y ) )
|
|
|
1012
|
|
|
1013 The existential quantifier can be related by a function,
|
|
|
1014
|
|
359
|
1015 Replace : HOD → (HOD → HOD ) → HOD
|
|
273
|
1016
|
|
|
1017 The axioms becomes as follows.
|
|
|
1018
|
|
|
1019 replacement← : {ψ : ZFSet → ZFSet} → ∀ ( X x : ZFSet ) → x ∈ X → ψ x ∈ Replace X ψ
|
|
|
1020 replacement→ : {ψ : ZFSet → ZFSet} → ∀ ( X x : ZFSet ) → ( lt : x ∈ Replace X ψ ) → ¬ ( ∀ (y : ZFSet) → ¬ ( x ≈ ψ y ) )
|
|
|
1021
|
|
|
1022 In the axiom, the existence of the original elements is necessary. In order to do that we use OD which has
|
|
|
1023 negation form of existential quantifier in the definition.
|
|
|
1024
|
|
|
1025 in-codomain : (X : OD ) → ( ψ : OD → OD ) → OD
|
|
425
|
1026 in-codomain X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( def X y ∧ ( x ≡ & (ψ (* y ))))) }
|
|
273
|
1027
|
|
359
|
1028 in-codomain is a logical relation-ship, and it is written in OD.
|
|
273
|
1029
|
|
359
|
1030 Replace : HOD → (HOD → HOD) → HOD
|
|
425
|
1031 Replace X ψ = record { od = record { def = λ x → (x o< sup-o X (λ y X∋y → & (ψ (* y)))) ∧ def (in-codomain X ψ) x }
|
|
359
|
1032 ; odmax = rmax ; <odmax = rmax<} where
|
|
|
1033 rmax : Ordinal
|
|
425
|
1034 rmax = sup-o X (λ y X∋y → & (ψ (* y)))
|
|
359
|
1035 rmax< : {y : Ordinal} → (y o< rmax) ∧ def (in-codomain X ψ) y → y o< rmax
|
|
|
1036 rmax< lt = proj1 lt
|
|
273
|
1037
|
|
359
|
1038 The bbound of Replace is defined by supremum, that is, it is not limited in a limit ordinal of original ZF Set.
|
|
|
1039
|
|
|
1040 Once we have a bound, validity of the axiom is an easy task to check the logical relation-ship.
|
|
273
|
1041
|
|
|
1042 --Validity of Power Set Axiom
|
|
|
1043
|
|
|
1044 The original Power Set Axiom is this.
|
|
|
1045
|
|
|
1046 ∀ X ∃ A ∀ t ( t ∈ A ↔ t ⊆ X ) )
|
|
|
1047
|
|
|
1048 The existential quantifier is replaced by a function
|
|
|
1049
|
|
|
1050 Power : ( A : OD ) → OD
|
|
|
1051
|
|
|
1052 t ⊆ X is a record like this.
|
|
|
1053
|
|
|
1054 record _⊆_ ( A B : OD ) : Set (suc n) where
|
|
|
1055 field
|
|
|
1056 incl : { x : OD } → A ∋ x → B ∋ x
|
|
|
1057
|
|
|
1058 Axiom becomes likes this.
|
|
|
1059
|
|
|
1060 power→ : ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x)
|
|
|
1061 power← : (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t
|
|
|
1062
|
|
|
1063 The validity of the axioms are slight complicated, we have to define set of all subset. We define
|
|
|
1064 subset in a different form.
|
|
|
1065
|
|
359
|
1066 ZFSubset : (A x : HOD ) → HOD
|
|
|
1067 ZFSubset A x = record { od = record { def = λ y → odef A y ∧ odef x y } ; odmax = omin (odmax A) (odmax x) ; <odmax = lemma } where
|
|
|
1068 lemma : {y : Ordinal} → def (od A) y ∧ def (od x) y → y o< omin (odmax A) (odmax x)
|
|
|
1069 lemma {y} and = min1 (<odmax A (proj1 and)) (<odmax x (proj2 and))
|
|
273
|
1070
|
|
|
1071 We can prove,
|
|
|
1072
|
|
359
|
1073 ( {y : HOD } → x ∋ y → ZFSubset A x ∋ y ) ⇔ ( x ⊆ A )
|
|
273
|
1074
|
|
|
1075 We only have upper bound as an ordinal, but we have an obvious OD based on the order of Ordinals,
|
|
|
1076 which is an Ordinals with our Model.
|
|
|
1077
|
|
|
1078 Ord : ( a : Ordinal ) → OD
|
|
|
1079 Ord a = record { def = λ y → y o< a }
|
|
|
1080
|
|
|
1081 Def : (A : OD ) → OD
|
|
425
|
1082 Def A = Ord ( sup-o ( λ x → & ( ZFSubset A (* x )) ) )
|
|
273
|
1083
|
|
|
1084 This is slight larger than Power A, so we replace all elements x by A ∩ x (some of them may empty).
|
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1085
|
|
|
1086 Power : OD → OD
|
|
425
|
1087 Power A = Replace (Def (Ord (& A))) ( λ x → A ∩ x )
|
|
273
|
1088
|
|
|
1089 Creating Power Set of Ordinals is rather easy, then we use replacement axiom on A ∩ x since we have this.
|
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1090
|
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359
|
1091 ∩-≡ : { a b : HOD } → ({x : HOD } → (a ∋ x → b ∋ x)) → od a == od ( b ∩ a )
|
|
273
|
1092
|
|
|
1093 In case of Ord a intro of Power Set axiom becomes valid.
|
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1094
|
|
359
|
1095 ord-power← : (a : Ordinal ) (t : HOD) → ({x : HOD} → (t ∋ x → (Ord a) ∋ x)) → OPwr (Ord a) ∋ t
|
|
273
|
1096
|
|
|
1097 Using this, we can prove,
|
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1098
|
|
359
|
1099 power→ : ( A t : HOD) → Power A ∋ t → {x : HOD} → t ∋ x → ¬ ¬ (A ∋ x)
|
|
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1100 power← : (A t : HOD) → ({x : HOD} → (t ∋ x → A ∋ x)) → Power A ∋ t
|
|
273
|
1101
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1102
|
|
|
1103 --Axiom of regularity, Axiom of choice, ε-induction
|
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1104
|
|
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1105 Axiom of regularity requires non self intersectable elements (which is called minimum), if we
|
|
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1106 replace it by a function, it becomes a choice function. It makes axiom of choice valid.
|
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1107
|
|
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1108 This means we cannot prove axiom regularity form our model, and if we postulate this, axiom of
|
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1109 choice also becomes valid.
|
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|
1110
|
|
359
|
1111 minimal : (x : HOD ) → ¬ (x == od∅ )→ OD
|
|
425
|
1112 x∋minimal : (x : HOD ) → ( ne : ¬ (x == od∅ ) ) → def x ( & ( minimal x ne ) )
|
|
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1113 minimal-1 : (x : HOD ) → ( ne : ¬ (x == od∅ ) ) → (y : HOD ) → ¬ ( def (minimal x ne) (& y)) ∧ (def x (& y) )
|
|
273
|
1114
|
|
|
1115 We can avoid this using ε-induction (a predicate is valid on all set if the predicate is true on some element of set).
|
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|
1116 Assuming law of exclude middle, they say axiom of regularity will be proved, but we haven't check it yet.
|
|
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1117
|
|
359
|
1118 ε-induction : { ψ : HOD → Set (suc n)}
|
|
|
1119 → ( {x : HOD } → ({ y : HOD } → x ∋ y → ψ y ) → ψ x )
|
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1120 → (x : HOD ) → ψ x
|
|
273
|
1121
|
|
|
1122 In our model, we assumes the mapping between Ordinals and OD, this is actually the TransFinite induction in Ordinals.
|
|
|
1123
|
|
|
1124 The axiom of choice in the book is complicated using any pair in a set, so we use use a form in the Wikipedia.
|
|
|
1125
|
|
|
1126 ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ]
|
|
|
1127
|
|
|
1128 We can formulate like this.
|
|
|
1129
|
|
|
1130 choice-func : (X : ZFSet ) → {x : ZFSet } → ¬ ( x ≈ ∅ ) → ( X ∋ x ) → ZFSet
|
|
|
1131 choice : (X : ZFSet ) → {A : ZFSet } → ( X∋A : X ∋ A ) → (not : ¬ ( A ≈ ∅ )) → A ∋ choice-func X not X∋A
|
|
|
1132
|
|
|
1133 It does not requires ∅ ∉ X .
|
|
|
1134
|
|
|
1135 --Axiom of choice and Law of Excluded Middle
|
|
|
1136
|
|
359
|
1137 In our model, since HOD has a mapping to Ordinals, it has evident order, which means well ordering theorem is valid,
|
|
273
|
1138 but it don't have correct form of the axiom yet. They say well ordering axiom is equivalent to the axiom of choice,
|
|
|
1139 but it requires law of the exclude middle.
|
|
|
1140
|
|
|
1141 Actually, it is well known to prove law of the exclude middle from axiom of choice in intuitionistic logic, and we can
|
|
|
1142 perform the proof in our mode. Using the definition like this, predicates and ODs are related and we can ask the
|
|
|
1143 set is empty or not if we have an axiom of choice, so we have the law of the exclude middle p ∨ ( ¬ p ) .
|
|
|
1144
|
|
359
|
1145 ppp : { p : Set n } { a : HOD } → record { def = λ x → p } ∋ a → p
|
|
273
|
1146 ppp {p} {a} d = d
|
|
|
1147
|
|
|
1148 We can prove axiom of choice from law excluded middle since we have TransFinite induction. So Axiom of choice
|
|
|
1149 and Law of Excluded Middle is equivalent in our mode.
|
|
|
1150
|
|
|
1151 --Relation-ship among ZF axiom
|
|
|
1152
|
|
|
1153 <center><img src="fig/axiom-dependency.svg"></center>
|
|
|
1154
|
|
|
1155 --Non constructive assumption in our model
|
|
|
1156
|
|
359
|
1157 mapping between HOD and Ordinals
|
|
273
|
1158
|
|
425
|
1159 & : HOD → Ordinal
|
|
|
1160 * : Ordinal → OD
|
|
|
1161 oiso : {x : HOD } → * ( & x ) ≡ x
|
|
|
1162 diso : {x : Ordinal } → & ( * x ) ≡ x
|
|
|
1163 c<→o< : {x y : HOD } → def y ( & x ) → & x o< & y
|
|
|
1164 ⊆→o≤ : {y z : HOD } → ({x : Ordinal} → def (od y) x → def (od z) x ) → & y o< osuc (& z)
|
|
273
|
1165
|
|
|
1166 Equivalence on OD
|
|
|
1167
|
|
359
|
1168 ==→o≡ : { x y : HOD } → (od x == od y) → x ≡ y
|
|
273
|
1169
|
|
|
1170 Upper bound
|
|
|
1171
|
|
359
|
1172 sup-o : (A : HOD) → ( ( x : Ordinal ) → def (od A) x → Ordinal ) → Ordinal
|
|
|
1173 sup-o< : (A : HOD) → { ψ : ( x : Ordinal ) → def (od A) x → Ordinal } → ∀ {x : Ordinal } → (lt : def (od A) x ) → ψ x lt o< sup-o A ψ
|
|
|
1174
|
|
|
1175 In order to have bounded ω,
|
|
|
1176
|
|
425
|
1177 hod-ord< : {x : HOD} → & x o< next (odmax x)
|
|
273
|
1178
|
|
|
1179 Axiom of choice and strong axiom of regularity.
|
|
|
1180
|
|
359
|
1181 minimal : (x : HOD ) → ¬ (x =h= od∅ )→ HOD
|
|
425
|
1182 x∋minimal : (x : HOD ) → ( ne : ¬ (x =h= od∅ ) ) → odef x ( & ( minimal x ne ) )
|
|
|
1183 minimal-1 : (x : HOD ) → ( ne : ¬ (x =h= od∅ ) ) → (y : HOD ) → ¬ ( odef (minimal x ne) (& y)) ∧ (odef x (& y) )
|
|
|
1184
|
|
|
1185 --V
|
|
|
1186
|
|
|
1187 The cumulative hierarchy
|
|
|
1188 V 0 := ∅
|
|
|
1189 V α + 1 := P ( V α )
|
|
|
1190 V α := ⋃ { V β | β < α }
|
|
|
1191
|
|
|
1192 Using TransFinite induction
|
|
|
1193
|
|
|
1194 V : ( β : Ordinal ) → HOD
|
|
|
1195 V β = TransFinite V1 β where
|
|
|
1196 V1 : (x : Ordinal ) → ( ( y : Ordinal) → y o< x → HOD ) → HOD
|
|
|
1197 V1 x V0 with trio< x o∅
|
|
|
1198 V1 x V0 | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a)
|
|
|
1199 V1 x V0 | tri≈ ¬a refl ¬c = Ord o∅
|
|
|
1200 V1 x V0 | tri> ¬a ¬b c with Oprev-p x
|
|
|
1201 V1 x V0 | tri> ¬a ¬b c | yes p = Power ( V0 (Oprev.oprev p ) (subst (λ k → _ o< k) (Oprev.oprev=x p) <-osuc ))
|
|
|
1202 V1 x V0 | tri> ¬a ¬b c | no ¬p =
|
|
|
1203 record { od = record { def = λ y → (y o< x ) ∧ ((lt : y o< x ) → odef (V0 y lt) x ) } ;
|
|
|
1204 odmax = x; <odmax = λ {x} lt → proj1 lt }
|
|
|
1205
|
|
|
1206 In our system, clearly V ⊆ HOD。
|
|
|
1207
|
|
|
1208 --L
|
|
|
1209
|
|
|
1210 The constructible Sets
|
|
|
1211 L 0 := ∅
|
|
|
1212 L α + 1 := Df (L α) -- Definable Power Set
|
|
|
1213 V α := ⋃ { L β | β < α }
|
|
|
1214
|
|
|
1215 What is Df? In our system, Power x is definable by Sup.
|
|
|
1216
|
|
|
1217 record Definitions : Set (suc n) where
|
|
|
1218 field
|
|
|
1219 Definition : HOD → HOD
|
|
|
1220
|
|
|
1221 open Definitions
|
|
|
1222
|
|
|
1223 Df : Definitions → (x : HOD) → HOD
|
|
|
1224 Df D x = Power x ∩ Definition D x
|
|
|
1225
|
|
|
1226 --L
|
|
|
1227
|
|
|
1228 L : ( β : Ordinal ) → Definitions → HOD
|
|
|
1229 L β D = TransFinite L1 β where
|
|
|
1230 L1 : (x : Ordinal ) → ( ( y : Ordinal) → y o< x → HOD ) → HOD
|
|
|
1231 L1 x L0 with trio< x o∅
|
|
|
1232 L1 x L0 | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a)
|
|
|
1233 L1 x L0 | tri≈ ¬a refl ¬c = Ord o∅
|
|
|
1234 L1 x L0 | tri> ¬a ¬b c with Oprev-p x
|
|
|
1235 L1 x L0 | tri> ¬a ¬b c | yes p = Df D ( L0 (Oprev.oprev p ) (subst (λ k → _ o< k) (Oprev.oprev=x p) <-osuc ))
|
|
|
1236 L1 x L0 | tri> ¬a ¬b c | no ¬p =
|
|
|
1237 record { od = record { def = λ y → (y o< x ) ∧ ((lt : y o< x ) → odef (L0 y lt) x ) } ;
|
|
|
1238 odmax = x; <odmax = λ {x} lt → proj1 lt }
|
|
|
1239
|
|
|
1240 --V=L
|
|
|
1241
|
|
|
1242 V=L0 : Set (suc n)
|
|
|
1243 V=L0 = (x : Ordinal) → V x ≡ L x record { Definition = λ y → y }
|
|
|
1244
|
|
|
1245 is obvious. Definitions should have some restrictions, such as it includes Nat.
|
|
|
1246
|
|
|
1247 --Some other ...
|
|
273
|
1248
|
|
|
1249 --So it this correct?
|
|
|
1250
|
|
|
1251 Our axiom are syntactically the same in the text book, but negations are slightly different.
|
|
|
1252
|
|
|
1253 If we assumes excluded middle, these are exactly same.
|
|
|
1254
|
|
|
1255 Even if we assumes excluded middle, intuitionistic logic itself remains consistent, but we cannot prove it.
|
|
|
1256
|
|
|
1257 Except the upper bound, axioms are simple logical relation.
|
|
|
1258
|
|
359
|
1259 Proof of existence of mapping between HOD and Ordinals are not obvious. We don't know we prove it or not.
|
|
273
|
1260
|
|
|
1261 Existence of the Upper bounds is a pure assumption, if we have not limit on Ordinals, it may contradicts,
|
|
|
1262 but we don't have explicit upper limit on Ordinals.
|
|
|
1263
|
|
|
1264 Several inference on our model or our axioms are basically parallel to the set theory text book, so it looks like correct.
|
|
|
1265
|
|
|
1266 --How to use Agda Set Theory
|
|
|
1267
|
|
|
1268 Assuming record ZF, classical set theory can be developed. If necessary, axiom of choice can be
|
|
|
1269 postulated or assuming law of excluded middle.
|
|
|
1270
|
|
|
1271 Instead, simply assumes non constructive assumption, various theory can be developed. We haven't check
|
|
|
1272 these assumptions are proved in record ZF, so we are not sure, these development is a result of ZF Set theory.
|
|
|
1273
|
|
|
1274 ZF record itself is not necessary, for example, topology theory without ZF can be possible.
|
|
|
1275
|
|
|
1276 --Topos and Set Theory
|
|
|
1277
|
|
|
1278 Topos is a mathematical structure in Category Theory, which is a Cartesian closed category which has a
|
|
|
1279 sub-object classifier.
|
|
|
1280
|
|
|
1281 Topos itself is model of intuitionistic logic.
|
|
|
1282
|
|
|
1283 Transitive Sets are objects of Cartesian closed category.
|
|
|
1284 It is possible to introduce Power Set Functor on it
|
|
|
1285
|
|
|
1286 We can use replacement A ∩ x for each element in Transitive Set,
|
|
|
1287 in the similar way of our power set axiom. I
|
|
|
1288
|
|
|
1289 A model of ZF Set theory can be constructed on top of the Topos which is shown in Oisus.
|
|
|
1290
|
|
|
1291 Our Agda model is a proof theoretic version of it.
|
|
|
1292
|
|
|
1293 --Cardinal number and Continuum hypothesis
|
|
|
1294
|
|
|
1295 Axiom of choice is required to define cardinal number
|
|
|
1296
|
|
|
1297 definition of cardinal number is not yet done
|
|
|
1298
|
|
|
1299 definition of filter is not yet done
|
|
|
1300
|
|
|
1301 we may have a model without axiom of choice or without continuum hypothesis
|
|
|
1302
|
|
|
1303 Possible representation of continuum hypothesis is this.
|
|
|
1304
|
|
|
1305 continuum-hyphotheis : (a : Ordinal) → Power (Ord a) ⊆ Ord (osuc a)
|
|
|
1306
|
|
|
1307 --Filter
|
|
|
1308
|
|
|
1309 filter is a dual of ideal on boolean algebra or lattice. Existence on natural number
|
|
|
1310 is depends on axiom of choice.
|
|
|
1311
|
|
359
|
1312 record Filter ( L : HOD ) : Set (suc n) where
|
|
273
|
1313 field
|
|
|
1314 filter : OD
|
|
|
1315 proper : ¬ ( filter ∋ od∅ )
|
|
|
1316 inL : filter ⊆ L
|
|
359
|
1317 filter1 : { p q : HOD } → q ⊆ L → filter ∋ p → p ⊆ q → filter ∋ q
|
|
|
1318 filter2 : { p q : HOD } → filter ∋ p → filter ∋ q → filter ∋ (p ∩ q)
|
|
273
|
1319
|
|
|
1320 We may construct a model of non standard analysis or set theory.
|
|
|
1321
|
|
|
1322 This may be simpler than classical forcing theory ( not yet done).
|
|
|
1323
|
|
|
1324 --Programming Mathematics
|
|
|
1325
|
|
|
1326 Mathematics is a functional programming in Agda where proof is a value of a variable. The mathematical
|
|
|
1327 structure are
|
|
|
1328
|
|
|
1329 record and data
|
|
|
1330
|
|
|
1331 Proof is check by type consistency not by the computation, but it may include some normalization.
|
|
|
1332
|
|
|
1333 Type inference and termination is not so clear in multi recursions.
|
|
|
1334
|
|
|
1335 Defining Agda record is a good way to understand mathematical theory, for examples,
|
|
|
1336
|
|
|
1337 Category theory ( Yoneda lemma, Floyd Adjunction functor theorem, Applicative functor )
|
|
|
1338 Automaton ( Subset construction、Language containment)
|
|
|
1339
|
|
|
1340 are proved in Agda.
|
|
|
1341
|
|
|
1342 --link
|
|
|
1343
|
|
|
1344 Summer school of foundation of mathematics (in Japanese)
|
|
|
1345 <br> <a href="https://www.sci.shizuoka.ac.jp/~math/yorioka/ss2019/">
|
|
|
1346 https://www.sci.shizuoka.ac.jp/~math/yorioka/ss2019/
|
|
|
1347 </a>
|
|
|
1348
|
|
|
1349 Foundation of axiomatic set theory (in Japanese)
|
|
|
1350 <br> <a href="https://www.sci.shizuoka.ac.jp/~math/yorioka/ss2019/sakai0.pdf">
|
|
|
1351 https://www.sci.shizuoka.ac.jp/~math/yorioka/ss2019/sakai0.pdf
|
|
|
1352 </a>
|
|
|
1353
|
|
|
1354 Agda
|
|
|
1355 <br> <a href="https://agda.readthedocs.io/en/v2.6.0.1/">
|
|
|
1356 https://agda.readthedocs.io/en/v2.6.0.1/
|
|
|
1357 </a>
|
|
|
1358
|
|
|
1359 ZF-in-Agda source
|
|
|
1360 <br> <a href="https://github.com/shinji-kono/zf-in-agda.git">
|
|
|
1361 https://github.com/shinji-kono/zf-in-agda.git
|
|
|
1362 </a>
|
|
|
1363
|
|
|
1364 Category theory in Agda source
|
|
|
1365 <br> <a href="https://github.com/shinji-kono/category-exercise-in-agda">
|
|
|
1366 https://github.com/shinji-kono/category-exercise-in-agda
|
|
|
1367 </a>
|
|
|
1368
|
|
|
1369
|
|
|
1370
|
