  It is well known that ZFC, despite its usefulness as a foundational theory
for mathematics, has two unwanted features: it cannot be written down
explicitly due to its infinitely many axioms, and it has a countable model due
to the L\"{o}wenheim-Skolem theorem. This paper presents the axioms one has to
accept to get rid of these two features. For that matter, some twenty axioms
are formulated in a nonstandard first-order language with countable many
constants: to this collection of axioms is associated a universe of discourse
consisting of a class of objects, each of which is a set, and a class of
arrows, each of which is a function. The axioms of ZF are derived from this
finite axiom scheme, and it is shown that it does not have a countable
model--if it has a model at all, that is. Furthermore, the axioms of category
theory are proven to hold: the present universe may therefore serve as an
ontological basis for category theory. However, it has not been investigated
whether any of the soundness and completeness properties hold for the present
theory: the inevitable conclusion is therefore that only further research can
establish whether the present results indeed constitute an advancement in the
foundations of mathematics.
