The usual set theory with extensionality, empty set, pairing, union, power set, infinity and full separation, before adjoining replacement. Modern presentations often include foundation; the convention should be specified when comparing theories. Class-theoretic variants add classes and class comprehension, rather than treating proper classes as sets.
Every proper class is in bijection with the universe. Over class-theoretic Zermelo set theory, with class-parameter separation, this is equivalent to replacement plus global choice. For replacement, a proper functional image of a set would give a surjection from that set onto the universe, contradicted by diagonal separation. The proper class of ordinals supplies the global well-order. Conversely rank followed by uniformly chosen rank-segment well-orders is set-like, and every proper class in that order has order type equal to the entire ordinal class.
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Zermelo set theory, often referred to as Zermelo's axiomatic set theory, is an early foundational system for set theory developed by the German mathematician Ernst Zermelo in the early 20th century, primarily around 1908. This system provides a framework for understanding sets and their properties while addressing certain paradoxes that arise in naive set theory, such as Russell's paradox.