1. ## Kuratowski Induction

Show that for any $\displaystyle E \subset A$, the following are equivalent:

a). $\displaystyle E \in K$ for every Kuratowski-Inductive $\displaystyle K \subset \mathcal{P}(A)$.
b). $\displaystyle E$ is finite.
I've given this a shot, but I think it's a bit jumbled. I decided to try induction:

The first Kuratowski-Inductive subset of $\displaystyle \mathcal{P}(A)$ is when $\displaystyle K=\{\phi \}$. This is finite since $\displaystyle |K|=1$.

Assume each of the next $\displaystyle n-1$ sets formed by unions with elements $\displaystyle x_1, \ldots , x_{n-1} \in A$ are finite.
ie. $\displaystyle \{ \phi \} \bigcup \{x_1\} \bigcup \ldots \bigcup \{x_{n-1} \}=\{\phi, \ x_1, \ldots, \ x_{n-1} \}$.

If we add another $\displaystyle x_n \in A$ we get:

$\displaystyle \{\phi, x_1, \ldots , \ x_{n-1} \} \bigcup \{x_n \}=\{\phi, x_1, \ldots , \ x_{n} \}$.

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.

2. Originally Posted by Showcase_22
Kuratowski-Inductive subset
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.