### changeset 337:de2c472bcbcd

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-zf-in-agda.ind
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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
+
+```
+    zf.agda            axiom of ZF
+    zfc.agda           axiom of choice
+    Ordinals.agda      axiom of Ordinals
+    ordinal.agda       countable model of Ordinals
+    OD.agda            model of ZF based on Ordinal Definable Set with assumptions
+    ODC.agda           Law of exclude middle from axiom of choice assumptions
+    LEMC.agda          model of choice with assumption of the Law of exclude middle
+    OPair.agda         ordered pair on OD
+
+    BAlgbra.agda       Boolean algebra on OD (not yet done)
+    filter.agda        Filter on OD (not yet done)
+    cardinal.agda      Caedinal number on OD (not yet done)
+
+    logic.agda         some basics on logic
+    nat.agda           some basics on Nat
+```
+
+## 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 { def = λ x → ψ x } , then
+
+    A x = def 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.
+
+## What have we done
+
+```
+    Axioms of Ordinals
+    An implementation of countable Ordinal
+    ZF Axioms
+    Model of ZF based on OD/HOD
+    LEM     axiom of choice from LEM
+    ODC     LEM from axiom of choice
+    OPair   classical ordered pair example
+    filter  definition of filter and ideal
+    cardinal  unfinished Cardinal number
+    BAlgebra  boolean algebra of OD
+
+```
+
+
+
+
+
+
+
+
+```