Lévy Reflection Theorem

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