# General set theory

**General set theory** (**GST**) is

## Ontology

The ontology of GST is identical to that of

*a*is a member of set

*b*is written

*a ∈ b*(usually read "

*a*is an element

*b*").

## Axioms

The symbolic axioms below are from Boolos (1998: 196), and govern how sets behave and interact. As with

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)

*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)

*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*.

^{[1]}

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

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

## Discussion

### Metamathematics

Note that Specification is an axiom schema. The theory given by these axioms is not

GST is Interpretable in

### Peano arithmetic

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 . Given *Adjunction*, the usual construction of the successor ordinals from the empty set can proceed, one in which the natural numbers are defined as . See

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

^{[4]}

Q is also incomplete in the sense of

### Infinite sets

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 . Even if GST did afford a countably infinite set, GST could not prove the existence of a set whose cardinality is , because GST lacks the

## History

Boolos was interested in GST only as a fragment of

*Grundlagen*and

*Grundgesetze*, and how they could be modified to eliminate Russell's paradox. The system

**Aξ'**[δ

_{0}] 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

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

## Footnotes

**^***Adjunction*is seldom mentioned in the literature. Exceptions are Burgess (2005)*passim*, and QIII in Tarski and Givant (1987: 223).**^**Burgess (2005), 2.2, p. 91.**^**Collins and Daniel (1970), in which ST is called**S**.**^**Tarski et al. (1953), p. 34.**^**The Empty Set axiom in STZ is redundant, because the existence of the empty set is derivable from the axiom schema of Specification.**^**Called S' in Tarski et al. (1953: 34).

## References

- 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" in
*Infinistic Methods*. Warsaw: 45-69. - Raphael 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.

## External links

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