  We analyze the effect of replacing several natural uses of definability in
set theory by the weaker model-theoretic notion of algebraicity. We find, for
example, that the class of hereditarily ordinal algebraic sets is the same as
the class of hereditarily ordinal definable sets, that is, HOA = HOD. Moreover,
we show that every (pointwise) algebraic model of ZF is actually pointwise
definable. Finally, we consider the implicitly constructible universe Imp---an
algebraic analogue of the constructible universe---which is obtained by
iteratively adding not only the sets that are definable over what has been
built so far, but also those that are algebraic (or equivalently, implicitly
definable) over the existing structure. While we know Imp can differ from L,
the subtler properties of this new inner model are just now coming to light.
Many questions remain open.
