### changeset 302:304c271b3d47

HOD and reduction mapping of Ordinals
author Shinji KONO Sun, 28 Jun 2020 18:09:04 +0900 b012a915bbb5 7963b76df6e1 OD.agda Ordinals.agda 2 files changed, 38 insertions(+), 0 deletions(-) [+]
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```--- a/OD.agda	Wed Jun 24 14:05:38 2020 +0900
+++ b/OD.agda	Sun Jun 28 18:09:04 2020 +0900
@@ -67,6 +67,24 @@
--  In classical Set Theory, sup is defined by Uion. Since we are working on constructive logic,
--  we need explict assumption on sup.

+record HOD (odmax : Ordinal) : Set (suc n) where
+  field
+    hmax : {x : Ordinal } → x o< odmax
+    hdef : Ordinal  → Set n
+
+record OrdinalSubset (maxordinal : Ordinal) : Set (suc n) where
+  field
+    os→ : (x : Ordinal) → x o< maxordinal → Ordinal
+    os← : Ordinal → Ordinal
+    os←limit : (x : Ordinal) → os← x o< maxordinal
+    os-iso← : {x : Ordinal} →  os→  (os← x) (os←limit x) ≡ x
+    os-iso→ : {x : Ordinal} → (lt : x o< maxordinal ) →  os← (os→ x lt) ≡ x
+
+open HOD
+
+-- HOD→OD : {x : Ordinal} → HOD x → OD
+-- HOD→OD hod = record { def = hdef {!!} }
+
record ODAxiom : Set (suc n) where
-- OD can be iso to a subset of Ordinal ( by means of Godel Set )
field
@@ -83,6 +101,25 @@
-- sup-x  : {n : Level } → ( OD → Ordinal ) →  Ordinal
-- sup-lb : {n : Level } → { ψ : OD →  Ordinal } → {z : Ordinal }  →  z o< sup-o ψ → z o< osuc (ψ (sup-x ψ))

+record HODAxiom : Set (suc n) where
+  -- OD can be iso to a subset of Ordinal ( by means of Godel Set )
+ field
+  mod : Ordinal
+  mod-limit :  ¬ ((y : Ordinal) → mod ≡ osuc y)
+  os : OrdinalSubset mod
+  od→ord : HOD mod → Ordinal
+  ord→od : Ordinal  → HOD mod
+  c<→o<  :  {x y : HOD mod }   → hdef y (od→ord x) → od→ord x o< od→ord y
+  oiso   :  {x : HOD mod }      → ord→od ( od→ord x ) ≡ x
+  diso   :  {x : Ordinal } → od→ord ( ord→od x ) ≡ x
+  ==→o≡ : { x y : OD  } → (x == y) → x ≡ y
+  -- supermum as Replacement Axiom ( corresponding Ordinal of OD has maximum )
+  sup-o  :  ( HOD mod → Ordinal ) →  Ordinal
+  sup-o< :  { ψ : HOD mod →  Ordinal } → ∀ {x : HOD mod } → ψ x  o<  sup-o ψ
+  -- contra-position of mimimulity of supermum required in Power Set Axiom which we don't use
+  -- sup-x  : {n : Level } → ( OD → Ordinal ) →  Ordinal
+  -- sup-lb : {n : Level } → { ψ : OD →  Ordinal } → {z : Ordinal }  →  z o< sup-o ψ → z o< osuc (ψ (sup-x ψ))
+
postulate  odAxiom : ODAxiom
open ODAxiom odAxiom
```
```--- a/Ordinals.agda	Wed Jun 24 14:05:38 2020 +0900
+++ b/Ordinals.agda	Sun Jun 28 18:09:04 2020 +0900
@@ -20,6 +20,7 @@
¬x<0 :   { x  : ord  } → ¬ ( x o< o∅  )
<-osuc :  { x : ord  } → x o< osuc x
osuc-≡< :  { a x : ord  } → x o< osuc a  →  (x ≡ a ) ∨ (x o< a)
+     is-limit :  { x : ord  } → Dec ( ¬ ((y : ord) → x ≡ osuc y) )
TransFinite : { ψ : ord  → Set (suc n) }
→ ( (x : ord)  → ( (y : ord  ) → y o< x → ψ y ) → ψ x )
→  ∀ (x : ord)  → ψ x```