General set theory

General set theory (GST) is George Boolos's (1998) name for a fragment of the axiomatic set theory Z. GST is sufficient for all mathematics not requiring infinite sets, and is the weakest known set theory whose theorems include the Peano axioms. The ontology of GST is identical to that of ZFC, and hence is thoroughly canonical. GST features a single primitive ontological notion, that of set, and a single ontological assumption, namely that all individuals in the universe of discourse (hence all mathematical objects) are sets. There is a single primitive binary relation, set membership; that set a is a member of set b is written a ∈ b (usually read "a is an element of b"). The symbolic axioms below are from Boolos (1998: 196), and govern how sets behave and interact. As with Z, the background logic for GST is first order logic with identity. Indeed, GST is the fragment of Z obtained by omitting the axioms Union, Power Set, Elementary Sets (essentially Pairing) and Infinity and then taking a theorem of Z, Adjunction, as an axiom. The natural language versions of the axioms are intended to aid the intuition.

1. Axiom of Extensionality: The sets x and y are the same set if they have the same members. The converse of this axiom follows from the substitution property of equality.
2. Axiom Schema of Specification (or Separation or Restricted Comprehension): If z is a set and is any property which may be satisfied by all, some, or no elements of z, then there exists a subset y of z containing just those elements x in z which satisfy the property . The restriction to z is necessary to avoid Russell's paradox and its variants. More formally, let be any formula in the language of GST in which x may occur freely and y does not. Then all instances of the following schema are axioms:
3. Axiom of Adjunction: If x and y are sets, then there exists a set w, the adjunction of x and y, whose members are just y and the members of x. Adjunction refers to an elementary operation on two sets, and has no bearing on the use of that term elsewhere in mathematics, including in category theory.