In mathematical logic, an alternative set theory is any of the alternative mathematical approaches to the concept of set and any alternative to the de facto standard set theory described in axiomatic set theory by the axioms of Zermelo–Fraenkel set theory.

Alternative set theories

Alternative set theories include:^[1]

• Vopěnka's alternative set theory
• Von Neumann–Bernays–Gödel set theory
• Morse–Kelley set theory
• Tarski–Grothendieck set theory
• Ackermann set theory
• Type theory
• New Foundations
• Positive set theory
• Internal set theory
• Naive set theory
• S (set theory)
• Kripke–Platek set theory
• Scott–Potter set theory
• Constructive set theory
• Zermelo set theory
• General set theory