  Frege's Grundgesetze was one of the 19th century forerunners to contemporary
set theory which was plagued by the Russell paradox. In recent years, it has
been shown that subsystems of the Grundgesetze formed by restricting the
comprehension schema are consistent. One aim of this paper is to ascertain how
much set theory can be developed within these consistent fragments of the
Grundgesetze, and our main theorem shows that there is a model of a fragment of
the Grundgesetze which defines a model of all the axioms of Zermelo-Fraenkel
set theory with the exception of the power set axiom. The proof of this result
appeals to G\"odel's constructible universe of sets, which G\"odel famously
used to show the relative consistency of the continuum hypothesis. More
specifically, our proofs appeal to Kripke and Platek's idea of the projectum
within the constructible universe as well as to a weak version of
uniformization (which does not involve knowledge of Jensen's fine structure
theory). The axioms of the Grundgesetze are examples of abstraction principles,
and the other primary aim of this paper is to articulate a sufficient condition
for the consistency of abstraction principles with limited amounts of
comprehension. As an application, we resolve an analogue of the joint
consistency problem in the predicative setting.
