Perlis's approach was to develop a series of theories towards a
complete commonsense set theory. He first proposed an axiom scheme of
set formation for a naive set theory which he named 
(Perlis
1987):
Here 
is any formula and Ind is a predicate symbol with the
intended extension ``individuals.'' This theory lacks further axioms,
like an axiom of extensionality, which can be easily added. However,
Ind can sometimes be critically rich, i.e., if 
is the same
with Ind itself, then y may be too large to be an individual.
(This is the case of Cantor's Axiom of Abstraction.) Therefore, a
theory for a hierarchical extension for Ind is required. To support
the cumulative hierarchy, Perlis extended this theory to a new one
called 
using Ackermann's Scheme (Ackermann 1956) which is a
formal principle of this hierarchy:
where 
can be interpreted as ``x can be built up as a
collection from previously obtained entities.'' 
is
consistent with respect to ZF. Unfortunately, it is hierarchical and
hence not able to deal with self-referring sets.
Perlis finally proposed 
which is a synthesis of the universal
reflection theory of Gilmore-Kripke (Gilmore 1974), which forms
entities regardless of their origins and self-referential aspects, and
the hierarchical theory of Ackermann (1956). GK set theory has the
following axiom scheme where each well-formed formula 
has
a reification (name) 
with variables free as in
and distinguished variable x
where y does not appear in 
(cf. Note 16). There is also a
definitional equivalence (denoted by 
) axiom:
GK is consistent with respect to ZF (Perlis 1985). Perlis then proposed the following axioms to augment GK:
These axioms provide extensional constructions, i.e., collections
determined only by their members. Thus, while GK provides the
representation of circularity, these axioms support the cumulative
construction mechanism. This theory can deal with problems like
non-profit organization membership described earlier (Perlis 1988).
But Perlis could not prove the consistency of 
yet because
this requires linking two notions of membership of the two theories.