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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.
Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. Zermelo–Fraenkel set theory with the axiom of choice included is abbreviated ZFC, where C stands for "choice", [1] and ZF refers to the ...
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 ...
To avoid illegal set formation, one must only use predicates of ZFC to define subsets. [13] Another example of the syntactic approach is the Vopěnka's alternative set theory, [14] which tries to find set-theory axioms more compatible with the nonstandard analysis than the axioms of ZF.
Vopěnka's "Alternative Set Theory" builds on some ideas of the theory of semisets, but also introduces more radical changes: for example, all sets are "formally" finite, which means that sets in AST satisfy the law of mathematical induction for set-formulas (more precisely: the part of AST that consists of axioms related to sets only is equivalent to the Zermelo–Fraenkel (or ZF) set theory ...
A structure may be implemented within a set theory ZFC, or another set theory such as NBG, NFU, ETCS. [1] Alternatively, a structure may be treated in the framework of first-order logic, second-order logic, higher-order logic, a type theory, homotopy type theory etc. [details 2]
Tarski–Grothendieck set theory (TG, named after mathematicians Alfred Tarski and Alexander Grothendieck) is an axiomatic set theory.It is a non-conservative extension of Zermelo–Fraenkel set theory (ZFC) and is distinguished from other axiomatic set theories by the inclusion of Tarski's axiom, which states that for each set there is a "Tarski universe" it belongs to (see below).
In mathematical logic, positive set theory is the name for a class of alternative set theories in which the axiom of comprehension holds for at least the positive formulas (the smallest class of formulas containing atomic membership and equality formulas and closed under conjunction, disjunction, existential and universal quantification).