axiomatic set theory
version of set theory in which axioms are taken as uninterpreted rather than as formalizations of pre-existing truths; defined using a formal logic
New Foundations
axiomatic set theory permitting set comprehension by stratified formulae, hence with a universal set, but in which the singleton map 𝑥↦{𝑥} fails to exist
internal set theory
formalism that provides an axiomatic basis for nonstandard analysis where, rather than adding new elements to the reals, the axiomatic foundations are modified through syntactic enrichment
constructive set theory
axiomatic set theories based on the principles of mathematical constructivism
non-well-founded set theory
variants of axiomatic set theory that allow sets to be elements of themselves
S
system of mathematical set theory
general set theory
fragment of the axiomatic set theory Z