I think the old version is not correct. If $f : A \rightarrow B$ is not surjective, then there is more than one relation $\phi : A \looparrowright B$ such that $\forall a \in A. \phi(a,f(a))$, because we are free to choose whether $\phi(a,b)$ holds for all $b \notin \im f$.

Robin Adams

]]>added singleton subset function operator

Anonymous

]]>Fixed typo in Axiom of fibers. The function $g$ needs codomain $A$.

I wonder, though whether one could posit the existence of an injection $i:f^{-1}(b) \hookrightarrow A$, not just a function, and then the universal property involving $g$ could be replaced by a weak universal property (i.e. existence but not uniqueness).

]]>fixed the axiom of fibers as it seemed to be missing the universal property for fibers

Anonymous

]]>Are Axioms 3 and 4 a bit too weak? I could imagine the fibre axiom permitting something too small, and the product axiom something too big. Is this remedied by the other axioms?

Edit: oh, I see why 3 should be ok. Still thinking about 4, though.

]]>all the ZFC axioms have structural counterparts, which means that one could come up with a structural set theory which is equivalent in strength to ZFC.

Anonymous

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