Mislove, Moss, and Oles (1990) developed a partial set theory, ZFAP,
based on protosets, which is a generalization of HF---the set
of well-founded hereditarily finite sets (cf. Note 17). A protoset is
like a well-founded set except that it has some kind of
packaging which can hide some of its elements. There exists a
protoset 
which is empty except for packaging. From a finite
collection 
, one can construct the clear
protoset 
which has no packaging, and the
murky protoset 
which has some
elements, but also packaging. For example, a murky set like 
contains 2 and 3 as elements, but it might contain other elements,
too. We say that x is clarified by y, 
,
if one can obtain y from x by taking some packaging inside x and
replacing this by other protosets.
Partial set theory has a first order language L with three relation
symbols, 
(for actual membership), 
(for possible
membership), and set (for set existence). The theory consists of
two axioms and ZFA
, the relativization of all axioms of ZFA
(ZF + Aczel's AFA) to the relation set. The two axioms are (i)
Pict, which states that every partial set has a picture, a set
G which is a partial set graph (corresponding to the accessible
pointed graph of Aczel) and such that there is a decoration d of G
with the root decorated as x, and (ii) PSA, which states that every
such G has a unique decoration. Partial set theory ZFAP is the set
of all these axioms. ZFAP is a conservative extension of ZFA.