If is a Hierarchy and any formula then
for any there is some
such that is Absolute between and .
Proof
Fix , and .
Let be the set of subformulas of (it is finite).
For each and let
And as is finite let
Further define ordinals and
Finally set .
Note that is a limit so
So any has some such that so
has a witness in .
We are done by Tarski-Vaught Test.
Proposition
Suppose above we take , so .
Let be the statement ” is the smallest uncountable Ordinal”
Proof