Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate README.md @ 648:3821048a8552
...
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Mon, 27 Jun 2022 11:11:56 +0900 |
| parents | a5f8084b8368 |
| children | 42000f20fdbe |
| rev | line source |
|---|---|
| 337 | 1 Constructing ZF Set Theory in Agda |
| 2 ============ | |
| 3 | |
| 4 Shinji KONO (kono@ie.u-ryukyu.ac.jp), University of the Ryukyus | |
| 5 | |
| 6 ## ZF in Agda | |
| 7 | |
| 8 ``` | |
| 9 zf.agda axiom of ZF | |
| 10 zfc.agda axiom of choice | |
| 11 Ordinals.agda axiom of Ordinals | |
| 12 ordinal.agda countable model of Ordinals | |
| 13 OD.agda model of ZF based on Ordinal Definable Set with assumptions | |
| 14 ODC.agda Law of exclude middle from axiom of choice assumptions | |
| 15 LEMC.agda model of choice with assumption of the Law of exclude middle | |
| 16 OPair.agda ordered pair on OD | |
| 17 | |
| 18 BAlgbra.agda Boolean algebra on OD (not yet done) | |
| 19 filter.agda Filter on OD (not yet done) | |
| 20 cardinal.agda Caedinal number on OD (not yet done) | |
| 21 | |
| 22 logic.agda some basics on logic | |
| 23 nat.agda some basics on Nat | |
| 24 ``` | |
| 25 | |
| 26 ## Ordinal Definable Set | |
| 27 | |
| 28 It is a predicate has an Ordinal argument. | |
| 29 | |
| 30 In Agda, OD is defined as follows. | |
| 31 | |
| 32 ``` | |
| 33 record OD : Set (suc n ) where | |
| 34 field | |
| 35 def : (x : Ordinal ) → Set n | |
| 36 ``` | |
| 37 | |
| 38 This is not a ZF Set, because it can contain entire Ordinals. | |
| 39 | |
| 338 | 40 ## HOD : Hereditarily Ordinal Definable |
| 337 | 41 |
| 42 What we need is a bounded OD, the containment is limited by an ordinal. | |
| 43 | |
| 44 ``` | |
| 45 record HOD : Set (suc n) where | |
| 46 field | |
| 47 od : OD | |
| 48 odmax : Ordinal | |
| 49 <odmax : {y : Ordinal} → def od y → y o< odmax | |
| 50 ``` | |
| 51 | |
| 52 In classical Set Theory, HOD stands for Hereditarily Ordinal Definable, which means | |
| 53 | |
| 54 ``` | |
| 55 HOD = { x | TC x ⊆ OD } | |
| 56 ``` | |
| 57 | |
| 58 TC x is all transitive closure of x, that is elements of x and following all elements of them are all OD. But | |
| 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. | |
| 60 | |
| 61 ## 1 to 1 mapping between an HOD and an Ordinal | |
| 62 | |
| 63 HOD is a predicate on Ordinals and the solution is bounded by some ordinal. If we have a mapping | |
| 64 | |
| 65 ``` | |
| 66 od→ord : HOD → Ordinal | |
| 67 ord→od : Ordinal → HOD | |
| 68 oiso : {x : HOD } → ord→od ( od→ord x ) ≡ x | |
| 69 diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x | |
| 70 ``` | |
| 71 | |
| 72 we can check an HOD is an element of the OD using def. | |
| 73 | |
| 74 A ∋ x can be define as follows. | |
| 75 | |
| 76 ``` | |
| 77 _∋_ : ( A x : HOD ) → Set n | |
| 78 _∋_ A x = def (od A) ( od→ord x ) | |
| 79 | |
| 80 ``` | |
|
431
a5f8084b8368
reorganiztion for apkg
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
359
diff
changeset
|
81 In ψ : Ordinal → Set, if A is a record { od = { def = λ x → ψ x } ...} , then |
| 337 | 82 |
|
431
a5f8084b8368
reorganiztion for apkg
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
359
diff
changeset
|
83 A ∋ x = def (od A) ( od→ord x ) = ψ (od→ord x) |
| 337 | 84 |
| 85 They say the existing of the mappings can be proved in Classical Set Theory, but we | |
| 86 simply assumes these non constructively. | |
| 87 |