The problem depends on what your previous definitions of 'finite' and 'Kuratowski-inductive' are.

And it would help to know what previous theorems have been established.

What book are you working from? Suppes maybe? What exercise number?

What is your definition of 'K is Kuratowski-inductive'?

Do you mean [here 'u' is used for binary union]?:

if x subset of PK &

0 in x &

for all y in K, we have {y} in x &

for all b and c in x, we have buc in x

then, x = PK

I don't know whether that is what 'Kuratowski-inductive' means, but if it does, then K is finite iff K is Kuratowski-inductive.