If certain constraints are respected, this method outputs models of ZF when given models of ZF (and was first invented in order to prove the independence of the axiom of foundation from the other axioms of ZF) but it works even better with NF, since the construction preserves stratified formulæ.
Included are new demonstrations of the consistency of the entire set of axioms for Euclidean geometry, and of the independence of the axiom of parallels from the other Euclidean axioms.
Thus in set theory, the 'discovery' of a new axiom about real numbers, such as the axiom of definable determinacy, is typically the end process of a long period of working with the candidate axiom and examining its consequences.
This follows from Zorn's lemma, an equivalent formulation of the Axiom of Choice.
For an account of the Axiom of Replacement, and in particular von Neumann's work with it, see Kanamori 2012.
Sergeyev begins by adding a new axiom to the axiom of real numbers, which he calls the infinite unit axiom.
Some consequences of the axiom of definable determinateness.
The Zermelo-Fraenkel axiom system and the special role of the axiom of choice and its various equivalents.
Further work by Skolem and Fraenkel led to the formalization of the Separation axiom in terms of formulas of first-order, instead of the informal notion of property, as well as to the introduction of the axiom of Replacement, which is also formulated as an axiom schema for first-order formulas (see next section).
This axiom is presented as the axiom of negative introspection, or as the axiom of wisdom (since the agents possess Socratic wisdom, knowing that they do not know).
One gets the axiom of choice from the axioms of cardinalities and extensional codings; the details are technical.
