I've given this a shot, but I think it's a bit jumbled. I decided to try induction:Show that for any , the following are equivalent:
a). for every Kuratowski-Inductive .
b). is finite.
The first Kuratowski-Inductive subset of is when . This is finite since .
Assume each of the next sets formed by unions with elements are finite.
If we add another we get:
This is finite since there are n+1 elements in this set.
Is this right?
Also, do I need to prove the converse? My notes tell me that the converse is by definition.