 Mac Lane set theory, introduced by , is Zermelo set theory with the axiom of separation restricted to first-order formulas in which every quantifier is bounded,
Mac Lane set theory is similar in strength to topos theory with a natural number object, or to the system in Principia mathematica.It is strong enough to carry out almost all ordinary mathematics not directly connected with set theory or logic.The introduction states that the very existence of the discipline of set theory "seems to be threatened by certain contradictions or "antinomies", that can be derived from its principles – principles necessarily governing our thinking, it seems – and to which no entirely satisfactory solution has yet been found".