An axiom of materialization states that every set is isomorphic to a pure set of a certain form. They can be formulated either in structural set theory or material set theory, and in reasonably strong set theories they are often provable.
The axiom of well-founded materialization says that every set is isomorphic to a well-founded pure set, or equivalently (at least in a strong enough set theory) that every set can be embedded into a well-founded extensional graph. In material set theory this is also known as Coret’s Axiom B (Coret 64).
The axiom of ill-founded materialization says that every set can be embedded in an strongly extensional graph.
Since well-founded extensional graphs are strongly extensional, the well-founded axiom implies the ill-founded one.
The axiom of well-founded materialization follows from the axiom of choice: every well-ordered set is a well-founded extensional graph, so if every set is well-orderable then the axiom follows.
In material set theory, the axiom of well-founded materialization follows from the axiom of foundation, since then every set is already a well-founded pure set. Similarly, the axiom of ill-founded materialization follows from the axiom of anti-foundation.
In the presence of excluded middle and the axiom of replacement, every strongly extensional graph can be embedded into a well-founded extensional graph. Specifically, a graph is a coalgebra for the powerset functor, hence has a cone over the terminal coalgebra sequence for that endofunctor (which exists using the axiom of replacement, even though it does not converge to a terminal coalgebra). The kernels of the maps in this cone are the approximations to the bisimilarity of the graph, and hence converge to it at some ordinal $\alpha$ (namely the Hartogs number of the lattice of binary relations; this uses excluded middle). By strong extensionality, the bisimilarity is the identity, so the $\alpha^{\mathrm{th}}$ map in the cone is an injection into a well-founded set.
Thus, given excluded middle and replacement, the axiom of ill-founded materialization implies the well-founded one, hence the two are equivalent. In particular, if the axiom of foundation in ZF is replaced by the axiom of anti-foundation, then the axiom of well-founded materialization is also provable.
It is unclear whether the two axioms of materialization are distinct in weaker theories lacking excluded middle or replacement, such as IZF or Zermelo set theory with the axiom of foundation removed.
Either axiom of materialization implies that the category of sets is equivalent to its subcategory of the relevant kind of pure sets. Thus, for instance, Scott's trick can be used to prove isomorphism-invariant properties (see for instance this MO question).
J. Coret, Formules stratifides et axiome de fondation, Comptes Rendus hebdomadaires des seances de l’Academie des Sciences de Paris serie A, vol. 264 (1964)
Mike Shulman, Comparing material and structural set theories. Annals of Pure and Applied Logic 170(4), 2019, p465–504, arxiv
Last revised on April 10, 2019 at 23:55:48. See the history of this page for a list of all contributions to it.