Technique of Inner Models

We prove that if then there is a Transitive Model
such that

where is The Continuum Hypothesis:

Sufficiently Strong
Absoluteness of Truth in a Model

Proof

Let be the Constructible Hierarchy.
We want to prove that

Any Transitive Model satisfies Axiom of Extensionality and Axiom of Foundation.
Also is Absolute, and satisfies Axiom of Infinity,
we also get Axiom of Infinity for all .
The operations and are Absolute Operation
for a Sufficiently Strong .
So we only need to prove

Find in .
Assume .
Take some such that .
Consider:

Form

The union is exactly the same.
Now we consider the Power-set axiom
This is more complicated, because it is not Absolute.
(But if it was absolute, it would be hopeless to find a powerset in a countable model)
Note that is Absolute.
Thus if is a powerset of then .
Also clearly as is Transitive.
Thus our candidate is .
If then satisfies the conditions of the powerset axiom.
Define

By Axiom of Replacement this is a set,
and it is a set of ordinals, so it has to have an upper bound
so there is some such that ,
so .
Let

then and thus .

Now consider Axiom of Separation.
We need the set

where are parameters.
Take formula

Then

This only works if is Absolute between and .
Apply Lévy Reflection Theorem to find
such that is Absolute between and .
Thus

and so separation holds.

Axiom of Replacement is on Example Sheet 2.

The only one left is the Axiom of Choice
We can provide a well-order of :
Fix some some on of Order Type .
Assume that is a wellorder of ,
define lexifcographically a wellorder of

and then write for

Make this into an end-extension of by

Thus

is a well-order of .
However, this was a recursive definition so its Absolute so:

and in particular Axiom of Choice holds in .
Axiom of Constructibility
Gödel’s Incompleteness Theorems
Gödel’s Condensation Lemma

Claim

For every , , there is some such that .

THIS PROVES CH