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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:

$(\forall a)(\exists x)(\forall y)(\exists z\in x)(\forall w)(w\in z\leftrightarrow (w\in y \wedge w\in a))$

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 $ZF-Power \ Set$.

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 $ZF-Power \ Set$.

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:

$(\forall a)(\exists x)(\forall y)(Ez\in x)(\forall w)(w\in z\leftrightarrow (w\in y \wedge w\in a))$

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 $ZF-Power \ Set$.

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:

$(\forall a)(\exists x)(\forall y)(\exists z\in x)(\forall w)(w\in z\leftrightarrow (w\in y \wedge w\in a))$

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 $ZF-Power \ Set$.