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For example, let $\phi$ be a sentence in $ZF$ and $ZFC\vdash \neg\phi$. Then, $\phi$ must not be provable in $ZF$, but we still don't know whether $ZF\vdash \neg\phi$. What should i call this sentence $\phi$ under $ZF$?

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You could say '$\phi$ is unprovable in ZF, if ZF is consistent.' Note that you don't really know whether or not $\phi$ is provable in ZF, because if ZF is inconsistent, then its certainly provable. –  goblin May 25 '13 at 16:55

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up vote 6 down vote accepted

I'd say that $\phi$ is consistently false with ZF or $\phi$ is unprovable from ZF. Because it is consistent to have $ZF+\lnot\phi$ (e.g. if we assume choice).

If you think it should be provably false from ZF itself, it is something to suggest as a conjecture. However if all you know is that it is consistent with ZF, this is all you can say.

For example $\phi$ could be the statement "There is no elementary embedding from the universe onto itself" (whatever it means), which is a statement we know is provably false from ZFC, but we do not know its provability from ZF (probably one of the biggest problems in choiceless set theory).

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$\phi$ is false, but it might not be disprovable in ZF.

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