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.
== Ontology == 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").
== Axioms == The
symbolic axioms below are from Boolos (1998: 196), and govern how sets behave and interact.
The natural language versions of the axioms are intended to aid the intuition. The background
logic is first order logic with identity. 1) Axiom of Extensionality: The sets x and
y are the same set if they have the same members.
∀ x
∀ y
[ ∀
z [
z ∈
x ↔
z ∈
y ]
→ x
= y
] .
{\displaystyle \forall x\forall y[\forall z[z\in x\leftrightarrow z\in y]\rightarrow
x=y].} 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
ϕ
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
ϕ
. The restriction to z is necessary to avoid
Russell's paradox and its variants. More formally, let
ϕ (
x )
be any formula in the language of GST in which
x is free and y is not. Then all instances of the following schema are axioms:
∀ z
∃ y
∀ x
[ x
∈ y
↔ (
x ∈
z ∧
ϕ (
x )
) ]
.
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.
∀ x
∀ y
∃ w
∀ z
[ z
∈ w
↔ (
z ∈
x ∨
z =
y )
] .
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.
== Discussion == GST is the fragment of Z obtained by omitting
the axioms Union, Power Set, Elementary Sets (essentially Pairing) and Infinity, then taking
Adjunction, a theorem of Z, as an axiom. The result is a first order theory.
Setting φ(x) in Separation to x≠x, and assuming that the domain is nonempty, assures
the existence of the empty set. Adjunction implies that if x is a set, then so is
S (
x )
= x
∪ {
x }
{\displaystyle S(x)=x\cup \{x\}} . Given Adjunction, the usual construction
of the successor ordinals from the empty set can proceed, one in which the natural numbers
are defined as
∅ ,
S (
∅ )
,
S (
∅ )
) ,
… ,
(see Peano's axioms). More generally, given any model M of ZFC, the collection of hereditarily
finite sets in M will satisfy the GST axioms. Therefore, GST cannot prove the existence
of even a countable infinite set, that is, of a set whose cardinality is ℵ0. Even if
GST did afford a countably infinite set, GST could not prove the existence of a set whose
cardinality is
ℵ
1
{\displaystyle \aleph _{1}} , because GST lacks the axiom of power set.
Hence GST cannot ground analysis and geometry, and is too weak to serve as a foundation for
mathematics. Boolos was interested in GST only as a fragment
of Z that is just powerful enough to interpret Peano arithmetic. He never lingered over GST,
only mentioning it briefly in several papers discussing the systems of Frege's Grundlagen
and Grundgesetze, and how they could be modified to eliminate Russell's paradox. The system
Aξ'[δ0] in Tarski and Givant (1987: 223) is essentially GST with an axiom schema of
induction replacing Specification, and with the existence of an empty set explicitly assumed.
GST is called STZ in Burgess (2005), p. 223. Burgess's theory ST is GST with Empty Set
replacing the axiom schema of specification. That the letters "ST" also appear in "GST"
is a coincidence.
== Metamathematics == The most remarkable fact about ST (and hence
GST), is that these tiny fragments of set theory give rise to such rich metamathematics.
While ST is a small fragment of the well-known canonical set theories ZFC and NBG, ST interprets
Robinson arithmetic (Q), so that ST inherits the nontrivial metamathematics of Q. For example,
ST is essentially undecidable because Q is, and every consistent theory whose theorems
include the ST axioms is also essentially undecidable. This includes GST and every axiomatic
set theory worth thinking about, assuming these are consistent. In fact, the undecidability
of ST implies the undecidability of first-order logic with a single binary predicate letter.Q
is also incomplete in the sense of Gödel's incompleteness theorem. Any axiomatizable
theory, such as ST and GST, whose theorems include the Q axioms is likewise incomplete.
Moreover, the consistency of GST cannot be proved within GST itself, unless GST is in
fact inconsistent. GST is:
Mutually interpretable with Peano arithmetic (thus it has the same proof-theoretic strength
as PA); Immune to the three great antinomies of naïve
set theory: Russell's, Burali-Forti's, and Cantor's;
Not finitely axiomatizable. Montague (1961) showed that ZFC is not finitely axiomatizable,
and his argument carries over to GST. Hence any axiomatization of GST must either include
at least one axiom schema such as Separation; Interpretable in relation algebra because
no part of any GST axiom lies in the scope of more than three quantifiers. This is the
necessary and sufficient condition given in Tarski and Givant (1987).
== Footnotes