Absoluteness Of Recursive Operations

Let be a Theory and .
Let , , be Absolute Operations for Transitive Models of
and define

proves that is uniquely defined Definition by recursion.
Let be an attempt if it satisfies the above on some Initial Segment.
Suppose some Theory is such that

• any two attempts , agree everywhere
• for any , there is an attempt such that
  Then is an Absolute Operation for Transitive Models of .

Proof

Note that is defined by a formula Formula Hierarchy
which has to be Absolute (in a Transitive Model)
so is an Absolute Operation.