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1 Constructing ZF Set Theory in Agda
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2 ============
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3
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4 Shinji KONO (kono@ie.u-ryukyu.ac.jp), University of the Ryukyus
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5
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6 ## ZF in Agda
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7
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8 ```
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9 zf.agda axiom of ZF
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10 zfc.agda axiom of choice
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11 Ordinals.agda axiom of Ordinals
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12 ordinal.agda countable model of Ordinals
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13 OD.agda model of ZF based on Ordinal Definable Set with assumptions
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14 ODC.agda Law of exclude middle from axiom of choice assumptions
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15 LEMC.agda model of choice with assumption of the Law of exclude middle
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16 OPair.agda ordered pair on OD
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17
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18 BAlgbra.agda Boolean algebra on OD (not yet done)
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19 filter.agda Filter on OD (not yet done)
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20 cardinal.agda Caedinal number on OD (not yet done)
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21
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22 logic.agda some basics on logic
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23 nat.agda some basics on Nat
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24 ```
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25
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26 ## Ordinal Definable Set
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27
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28 It is a predicate has an Ordinal argument.
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29
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30 In Agda, OD is defined as follows.
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31
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32 ```
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33 record OD : Set (suc n ) where
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34 field
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35 def : (x : Ordinal ) → Set n
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36 ```
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37
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38 This is not a ZF Set, because it can contain entire Ordinals.
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39
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338
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40 ## HOD : Hereditarily Ordinal Definable
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337
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41
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42 What we need is a bounded OD, the containment is limited by an ordinal.
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43
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44 ```
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45 record HOD : Set (suc n) where
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46 field
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47 od : OD
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48 odmax : Ordinal
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49 <odmax : {y : Ordinal} → def od y → y o< odmax
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50 ```
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51
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52 In classical Set Theory, HOD stands for Hereditarily Ordinal Definable, which means
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53
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54 ```
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55 HOD = { x | TC x ⊆ OD }
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56 ```
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57
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58 TC x is all transitive closure of x, that is elements of x and following all elements of them are all OD. But
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59 what is x? In this case, x is an Set which we don't have yet. In our case, HOD is a bounded OD.
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60
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61 ## 1 to 1 mapping between an HOD and an Ordinal
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62
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63 HOD is a predicate on Ordinals and the solution is bounded by some ordinal. If we have a mapping
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64
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65 ```
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66 od→ord : HOD → Ordinal
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67 ord→od : Ordinal → HOD
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68 oiso : {x : HOD } → ord→od ( od→ord x ) ≡ x
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69 diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x
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70 ```
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71
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72 we can check an HOD is an element of the OD using def.
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73
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74 A ∋ x can be define as follows.
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75
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76 ```
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77 _∋_ : ( A x : HOD ) → Set n
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78 _∋_ A x = def (od A) ( od→ord x )
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79
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80 ```
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81 In ψ : Ordinal → Set, if A is a record { def = λ x → ψ x } , then
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82
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83 A x = def A ( od→ord x ) = ψ (od→ord x)
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84
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85 They say the existing of the mappings can be proved in Classical Set Theory, but we
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86 simply assumes these non constructively.
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87
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