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author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Wed, 05 Jul 2023 12:32:31 +0900 |
parents | 003424a36fed |
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Constructing ZF Set Theory in Agda ============ Shinji KONO (kono@ie.u-ryukyu.ac.jp), University of the Ryukyus ## ZF in Agda [zfc](https://shinji-kono.github.io/zf-in-agda/html/zfc.html) axiom of choice [NSet](https://shinji-kono.github.io/zf-in-agda/html/NSet.html) Naive Set Theory [Ordinals](https://shinji-kono.github.io/zf-in-agda/html/Ordinals.html) axiom of Ordinals [OD](https://shinji-kono.github.io/zf-in-agda/html/OD.html) a model of ZF based on Ordinal Definable Set with assumptions [ODC](https://shinji-kono.github.io/zf-in-agda/html/ODC.html) Law of exclude middle from axiom of choice assumptions [LEMC](https://shinji-kono.github.io/zf-in-agda/html/LEMC.html) choice with assumption of the Law of exclude middle [BAlgebra](https://shinji-kono.github.io/zf-in-agda/html/BAlgebra.html) Boolean algebra on OD (not yet done) [zorn](https://shinji-kono.github.io/zf-in-agda/html/zorn.html) Zorn lemma [Topology](https://shinji-kono.github.io/zf-in-agda/html/Topology.html) Topology [Tychonoff](https://shinji-kono.github.io/zf-in-agda/html/Tychonoff.html) [VL](https://shinji-kono.github.io/zf-in-agda/html/VL.html) V and L [cardinal](https://shinji-kono.github.io/zf-in-agda/html/cardinal.html) Cardinals [filter](https://shinji-kono.github.io/zf-in-agda/html/filter.html) Filter [generic-filter](https://shinji-kono.github.io/zf-in-agda/html/generic-filter.html) Generic Filter [maximum-filter](https://shinji-kono.github.io/zf-in-agda/html/maximum-filter.html) Maximum filter by Zorn lemma [ordinal](https://shinji-kono.github.io/zf-in-agda/html/ordinal.html) countable model of Ordinals [OPair](https://shinji-kono.github.io/zf-in-agda/html/OPair.html) Orderd Pair and Direct Product [bijection](https://shinji-kono.github.io/zf-in-agda/html/bijection.html) Bijection without HOD ``` ## Ordinal Definable Set It is a predicate has an Ordinal argument. In Agda, OD is defined as follows. ``` record OD : Set (suc n ) where field def : (x : Ordinal ) → Set n ``` This is not a ZF Set, because it can contain entire Ordinals. ## HOD : Hereditarily Ordinal Definable What we need is a bounded OD, the containment is limited by an ordinal. ``` record HOD : Set (suc n) where field od : OD odmax : Ordinal <odmax : {y : Ordinal} → def od y → y o< odmax ``` In classical Set Theory, HOD stands for Hereditarily Ordinal Definable, which means ``` HOD = { x | TC x ⊆ OD } ``` TC x is all transitive closure of x, that is elements of x and following all elements of them are all OD. But what is x? In this case, x is an Set which we don't have yet. In our case, HOD is a bounded OD. ## 1 to 1 mapping between an HOD and an Ordinal HOD is a predicate on Ordinals and the solution is bounded by some ordinal. If we have a mapping ``` od→ord : HOD → Ordinal ord→od : Ordinal → HOD oiso : {x : HOD } → ord→od ( od→ord x ) ≡ x diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x ``` we can check an HOD is an element of the OD using def. A ∋ x can be define as follows. ``` _∋_ : ( A x : HOD ) → Set n _∋_ A x = def (od A) ( od→ord x ) ``` In ψ : Ordinal → Set, if A is a record { od = { def = λ x → ψ x } ...} , then A ∋ x = def (od A) ( od→ord x ) = ψ (od→ord x) They say the existing of the mappings can be proved in Classical Set Theory, but we simply assumes these non constructively.