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.