General set theory (GST) isGeorge Boolos's (1998) name for a fragment of theaxiomatic set theoryZ. GST is sufficient for all mathematics not requiringinfinite sets, and is the weakest known set theory whosetheorems include thePeano axioms.

The ontology of GST is identical to that ofZFC, and hence is thoroughly canonical. GST features a singleprimitiveontological notion, that ofset, and a single ontological assumption, namely that all individuals in theuniverse of discourse (hence allmathematical objects) are sets. There is a singleprimitivebinary relation,set membership; that seta is a member of setb is writtena ∈ b (usually read "a is anelement ofb").

The symbolic axioms below are from Boolos (1998: 196), and govern how sets behave and interact. As withZ, the background logic for GST isfirst order logic withidentity. Indeed, GST is the fragment of Z obtained by omitting the axiomsUnion,Power Set, Elementary Sets (essentiallyPairing) andInfinity 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 setsx andy are the same set if they have the same members.

${\displaystyle \mathrm{\forall }x\mathrm{\forall }y[\mathrm{\forall }z[z\in x\leftrightarrow z\in y]\to x=y].}$

The converse of this axiom follows from the substitution property of equality.

2)Axiom Schema of Specification (orSeparation orRestricted Comprehension): Ifz is a set and${\displaystyle \varphi }$ is any property which may be satisfied by all, some, or no elements ofz, then there exists a subsety ofz containing just those elementsx inz which satisfy the property${\displaystyle \varphi }$. Therestriction toz is necessary to avoidRussell's paradox and its variants. More formally, let${\displaystyle \varphi (x)}$ be any formula in the language of GST in whichx may occur freely andy does not. Then all instances of the following schema are axioms:

${\displaystyle \mathrm{\forall }z\mathrm{\exists }y\mathrm{\forall }x[x\in y\leftrightarrow (x\in z\wedge \varphi (x))].}$

3)Axiom of Adjunction: Ifx andy are sets, then there exists a setw, theadjunction ofx andy, whose members are justy and the members ofx.^[1]

${\displaystyle \mathrm{\forall }x\mathrm{\forall }y\mathrm{\exists }w\mathrm{\forall }z[z\in w\leftrightarrow (z\in x\vee z=y)].}$

Adjunction refers to an elementary operation on two sets, and has no bearing on the use of that term elsewhere in mathematics, including incategory theory.

ST is GST with the axiom schema of specification replaced by theaxiom of empty set:

${\displaystyle \mathrm{\exists }x\mathrm{\forall }y[y\notin x].}$

Note that Specification is an axiom schema. The theory given by these axioms is notfinitely axiomatizable. Montague (1961) showed thatZFC is not finitely axiomatizable, and his argument carries over to GST. Hence any axiomatization of GST must include at least oneaxiom schema. With its simple axioms, GST is also immune to the three great antinomies ofnaïve set theory:Russell's,Burali-Forti's, andCantor's.

GST is Interpretable inrelation algebra because no part of any GST axiom lies in the scope of more than threequantifiers. This is thenecessary and sufficient condition given in Tarski and Givant (1987).

Setting φ(x) inSeparation tox≠x, and assuming that thedomain is nonempty, assures the existence of theempty set.Adjunction implies that ifx is a set, then so is${\displaystyle S(x)=x\cup \{x\}}$. GivenAdjunction, the usual construction of thesuccessor ordinals from theempty set can proceed, one in which thenatural numbers are defined as${\displaystyle \varnothing ,S(\varnothing ),S(S(\varnothing )),\dots ,}$. SeePeano's axioms. GST is mutually interpretable withPeano arithmetic (thus it has the same proof-theoretic strength as PA).

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 theoriesZFC andNBG, STinterpretsRobinson arithmetic (Q), so that ST inherits the nontrivial metamathematics of Q. For example, ST isessentially undecidable because Q is, and every consistent theory whose theorems include the ST axioms is also essentially undecidable.^[2][3] This includes GST and every axiomatic set theory worth thinking about, assuming these are consistent. In fact, theundecidability of ST implies the undecidability offirst-order logic with a singlebinary predicate letter.^[4]

Q is also incomplete in the sense ofGödel's incompleteness theorem. Any axiomatizable theory, such as ST and GST, whose theorems include the Q axioms is likewise incomplete. Moreover, theconsistency of GST cannot be proved within GST itself, unless GST is in fact inconsistent.

Given any modelM of ZFC, the collection ofhereditarily finite sets inM will satisfy the GST axioms. Therefore, GST cannot prove the existence of even a countableinfinite set, that is, of a set whose cardinality is${\displaystyle {\mathrm{\aleph }}_0}$. Even if GST did afford a countably infinite set, GST could not prove the existence of a set whosecardinality is${\displaystyle {\mathrm{\aleph }}_1}$, because GST lacks theaxiom of power set. Hence GST cannot groundanalysis andgeometry, and is too weak to serve as afoundation for mathematics.

Boolos was interested in GST only as a fragment ofZ that is just powerful enough to interpretPeano arithmetic. He never lingered over GST, only mentioning it briefly in several papers discussing the systems ofFrege'sGrundlagen andGrundgesetze, and how they could be modified to eliminateRussell's paradox. The systemAξ'[δ₀] in Tarski and Givant (1987: 223) is essentially GST with anaxiom schema of induction replacingSpecification, and with the existence of anempty set explicitly assumed.

GST is called STZ in Burgess (2005), p. 223.^[5] Burgess's theory ST^[6] is GST withEmpty Set replacing theaxiom schema of specification. That the letters "ST" also appear in "GST" is a coincidence.

• George Boolos (1999)Logic, Logic, and Logic. Harvard Univ. Press.
• Burgess, John, 2005.Fixing Frege. Princeton Univ. Press.
• Collins, George E., and Daniel, J. D. (1970)."On the interpretability of arithmetic in set theory".Notre Dame Journal of Formal Logic,11 (4): 477–483.
• Richard Montague (1961) "Semantical closure and non-finite axiomatizability" inInfinistic Methods. Warsaw: 45-69.
• Alfred Tarski,Andrzej Mostowski, andRaphael Robinson (1953)Undecidable Theories. North Holland.
• Tarski, A., and Givant, Steven (1987)A Formalization of Set Theory without Variables. Providence RI: AMS Colloquium Publications, v. 41.

• Stanford Encyclopedia of Philosophy:Set Theory—by Thomas Jech.

• Systems of set theory
• Z notation

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