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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.
Non-well-founded set theories are variants of axiomatic set theory that allow sets to be elements of themselves and otherwise violate the rule of well-foundedness. In non-well-founded set theories, the foundation axiom of ZFC is replaced by axioms implying its negation.
Set theory. Alternative set theory; Axiomatic set theory; Kripke–Platek set theory with urelements; Morse–Kelley set theory; Naive set theory; New Foundations; Positive set theory; Zermelo–Fraenkel set theory; Zermelo set theory; Set (mathematics) Simple theorems in the algebra of sets; Subset; Θ (set theory) Tree (descriptive set theory ...
If any set is postulated to exist, such as in the axiom of infinity, then the axiom of empty set is redundant because it is equal to the subset {}.Furthermore, the existence of a member in the universe of discourse, i.e., ∃x(x=x), is implied in certain formulations [1] of first-order logic, in which case the axiom of empty set follows from the axiom of Δ 0-separation, and is thus redundant.
Naive set theory; New Foundations; Pocket set theory; Positive set theory; S (Boolos 1989) Scott–Potter set theory; Tarski–Grothendieck set theory; Von Neumann–Bernays–Gödel set theory; Zermelo–Fraenkel set theory; Zermelo set theory; Set (mathematics) Set-builder notation; Set-theoretic topology; Simple theorems in the algebra of ...
Pocket set theory (PST) is an alternative set theory in which there are only two infinite cardinal numbers, ℵ 0 (aleph-naught, the cardinality of the set of all natural numbers) and c (the cardinality of the continuum). The theory was first suggested by Rudy Rucker in his Infinity and the Mind. [1]
In set theory, a semiset is a proper class that is a subclass of a set. In the typical foundations of Zermelo–Fraenkel set theory, semisets are impossible due to the axiom schema of specification. The theory of semisets was proposed and developed by Czech mathematicians Petr Vopěnka and Petr Hájek (1972).
Stronger axioms of infinity exist, such as that the set of natural numbers is a strongly Cantorian set, or NFUM = NFU + Infinity + Large Ordinals + Small Ordinals which is equivalent to Morse–Kelley set theory plus a predicate on proper classes which is a κ-complete nonprincipal ultrafilter on the proper class ordinal κ.