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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:

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

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 $ZF-Power \ Set$.

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    $\begingroup$ what does $E$ mean? Possibly you mean $\exists$ $\endgroup$
    – Zuhair Al-Johar
    Commented Jan 27, 2020 at 21:53
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    $\begingroup$ I believe in presence of extensionality and some weak separation ($KP\omega$ should have enough) this axiom is equivalent to full power set. $\endgroup$
    – Wojowu
    Commented Jan 27, 2020 at 22:03
  • $\begingroup$ @ZuhairAl-Johar Yes, I edited $\endgroup$
    – Frode Alfson Bjørdal
    Commented Jan 27, 2020 at 22:04
  • $\begingroup$ @Wojowu It would be interesting if that could be whown $\endgroup$
    – Frode Alfson Bjørdal
    Commented Jan 27, 2020 at 22:06
  • $\begingroup$ @ I think if you add this to ZF - Power, then you recover full ZF. $\endgroup$
    – Zuhair Al-Johar
    Commented Jan 27, 2020 at 22:06

1 Answer 1

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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 $a$.

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  • $\begingroup$ @FrodeAlfsonBjørdal Friedman in the cited article seems to work in set theory without extensionality. Without it it makes no sense to write $a\cap y$, as there may not be a unique set satisfying its defining property. If you wish to also work in a theory without extensionality (which I'm not sure is the case), you should say so explicitly in your question. $\endgroup$
    – Wojowu
    Commented Jan 27, 2020 at 23:03
  • $\begingroup$ @Wojowu Extensionality is part of $KP\omega$ and $ZF-Power+Weak Power$, so I already made clear that I presuppose extensionality in the cases I expressed interest in. $\endgroup$
    – Frode Alfson Bjørdal
    Commented Jan 27, 2020 at 23:07
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    $\begingroup$ @FrodeAlfsonBjørdal In that case, what is unconvincing about this answer? $\endgroup$
    – Noah Schweber
    Commented Jan 27, 2020 at 23:07
  • $\begingroup$ @NoahSchweber Perhaps it is not unconvincing. It is getting late here, so let me see what I think tomorrow. $\endgroup$
    – Frode Alfson Bjørdal
    Commented Jan 27, 2020 at 23:09
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    $\begingroup$ @FrodeAlfsonBjørdal What problems? The sentence $(\forall a)(\exists x)(\forall y)(\exists z\in x)(z = y\cap a)$ is tautologically equivalent to the sentence $(\forall a)(\exists x)(\forall y) (y\cap a\in x)$. $\endgroup$
    – Alex Kruckman
    Commented Jan 28, 2020 at 17:03

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