In The Consistency of Classical Set Theory Relative to a Set Theory with Intuitionistic Logic in THE JOURNAL OF SYMBOLIC LOGIC Volume 38, Number 2, June 1973 page 316 Harvey Friedman's axiom 8\* $Weak \ Power \ Set$ is:


$(\forall a)(\exists x)(\forall y)(\exists z\in x)(z=y\cap a)$


What do we know about weak power set? I am curious about what if anything weak power set can do when added to $KP\omega$ or $ZF-Power \ Set$.



Weak power set - what strength may it have?
The asserted set $x$ is just a set that contains all overlap sets between the set $a$ and any set, among its elements. Weak Power as written above is simply:


$(\forall a)(\exists x)(\forall y)(a \cap y\in x) $


Now in classical ZF all subsets of $a$ are overlaps with $a$, i.e. $z \subseteq a \to z \cap a=z$, so all of them would be included in the weak power of $a$ (just substitute each subset $z$ of $a$ instead of $y$ in the above formula), then by separation one can easily recover full $P(a)$ by separating on the weak power of $a$ using the property of being a subset of $a$.


