nLab material set theory

Overview

Material set theory (also called membership-based set theory) is a style of set theory with a primitive global membership relation “$\in$” where sets are characterized only by “$\in$” and propositional equality of sets. (The terminology ‘material’, or at least ‘materialistic’, goes back at least to Friedman 1997.) Material set theory contrasts with structural set theory (cf. material versus structural set theory), as well as with set theories which are neither structural nor material. Material set theories are simply sorted set theory.

Related concepts

• cumulative hierarchy

• unsorted set theory

• two-sorted set theory

• set-theoretic multiverse

• material versus structural set theory

References

• Håkon Robbestad Gylterud, Elisabeth Stenholm, Univalent material set theory (arXiv:2312.13024)

Relation to structural set theory is discussed in

• Michael Shulman, Comparing material and structural set theories (arXiv:1808.05204)

See also

• Harvey Friedman (1997). Foundational Completeness. FOM, 1997 November.