Zermelo-Fraenkel set theory

 

Zermelo–Fraenkel set theory, with the axiom of choice, commonly abbreviated ZFC, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics.

Introduction

ZFC consists of a single primitive ontological notion, that of set, and a single ontological assumption, namely that all individuals in the universe of discourse (i.e., 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" or "a is in b"). ZFC is a first-order theory; hence the background logic is first-order logic. These axioms govern how sets behave and interact.

In 1908, Ernst Zermelo proposed the first axiomatic set theory, Zermelo set theory. This axiomatic theory did not allow the construction of the ordinal numbers; while most of "ordinary mathematics" can be developed without ever using ordinals, ordinals are an essential tool in most set-theoretic investigations. Moreover, one of Zermelo's axioms invoked a concept, that of a "definite" property, whose operational meaning was ambiguous. In 1922, Abraham Fraenkel and Thoralf Skolem independently proposed defining a "definite" property as any property that could be formulated in first-order logic, such that all atomic formulae involve set membership or equality. From their work emerged the axiom schema of replacement. Appending this axiom, as well as the axiom of regularity, to Zermelo set theory yields the theory denoted by ZF.

Adding the axiom of choice (AC) to ZF yields ZFC. When a mathematical result requires the axiom of choice, this is sometimes stated explicitly. The reason for singling out AC in this manner is that AC is inherently nonconstructive; it posits the existence of a set (the choice set), without specifying just how that set is to be constructed. Hence results proved using AC may involve sets that, although they can be proved to exist (at least if one is not committed to a constructivist ontology), can never be constructed explicitly. For instance, the axiom of choice implies the existence of a well-order on any set. While we cannot construct a well-order for the set of real numbers R, AC guarantees the existence of such an order.

ZFC has an infinite number of axioms because the Replacement axiom is actually an axiom schema. It is known that both ZFC and ZF set theory cannot be axiomatized by a finite set of axioms; this was first demonstrated by Richard Montague (1961). On the other hand, Von Neumann–Bernays–Gödel set theory (NBG) can be finitely axiomatized. The ontology of NBG includes classes as well as sets; a set is a class that is a member of another class. NBG and ZFC are equivalent set theories, in the sense that any theorem about sets (i.e., not mentioning classes in any way) which can be proved in one theory can be proved in the other.

Because of Gödel's second incompleteness theorem, the consistency of ZFC cannot be proved within ZFC itself (unless it is actually inconsistent). Thus, to the extent that ZFC is identified with ordinary mathematics, the consistency of ZFC cannot be demonstrated in ordinary mathematics. The consistency of ZFC does follow from the existence of a weakly inaccessible cardinal, something whose existence is not provable in ZFC (unless ZFC is inconsistent). Nevertheless, almost no one fears that ZFC harbors an unsuspected contradiction; if ZFC were inconsistent, it is widely believed that that fact would have been uncovered by now. This much is certain — ZFC does not fall as easily as naive set theory did to its three great paradoxes: Russell's paradox, the Burali-Forti paradox, and Cantor's paradox.

Drawbacks of ZFC that have been discussed in the literature include:

• It is stronger than what is required for nearly all of everyday mathematics (Saunders MacLane and Solomon Feferman have each made this point);
• Compared to some other axiomatizations of set theory, ZFC is comparatively weak. For example, it does not admit the existence of a universal set (as in New Foundations) or class (as in NBG), under pain of Russell's paradox;
• Saunders MacLane (a founder of category theory) and others have argued that all axiomatic set theories do not do justice to the way mathematics works in practice. This view asserts that mathematics is not about collections of abstract objects and their properties, but about structure and mappings that preserve structure.

Axioms

There are many equivalent formulations of the ZFC axioms; for a rich but somewhat dated discussion of this fact, see Fraenkel et al (1973). The following particular axiom set is that of Kunen (1980). English descriptions have been added for clarity.

1) Axiom of extensionality: Two sets are the same if they have the same elements.

The converse of this axiom is a consequence of the substitution property of equality.

2) Axiom of regularity (also called the Axiom of foundation): Every non-empty set x contains some member y such that x and y are disjoint sets.

3) Axiom schema of specification (also called the Axiom schema of separation or the Axiom schema of restricted comprehension): If z is a set, and is any property which may characterize the elements x of z, then there is a subset y of z containing those 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 ZFC, and let be free variables appearing in . Then:

Specification is part of Z but redundant in ZF, because there are variants of ZF's axiom schema of replacement that turn Specification into a theorem.

4) Axiom of pairing: If x and y are sets then there exists a set containing both of them.

Pairing is part of Z but is redundant in ZF, because ZF's axiom schema of replacement (in combination with the axiom of infinity) turns Pairing into a theorem.

5) Axiom of union: For any set there is a set A containing every set that is a member of some member of .

6) Axiom schema of replacement: Informally, if the domain of a function f is a set, then the range of f is also a set, subject to a restriction to avoid paradoxes. Formally, let the formula and its free variables be as described in the axiom schema of specification. Then:

Here the quantifer means that only one exists, up to equality.

The next axiom employs the notation , where is some set. From axioms 1 through 6 above, the existence and uniqueness of and of the empty set can be proved. The latter fact makes redundant Kunen's axiom (not shown) asserting the existence of at least one set.

7) Axiom of infinity: There exists a set X such that the empty set is a member of X and whenever y is in X, so is S(y).

8) Axiom of power set: For any set x there is a set y that contains every subset of x.

Here is an abbreviation for .

9) Axiom of choice: For any set X there is a binary relation R which well-orders X. This means that R is a linear order on X and every nonempty subset of X has an element which is minimal under R.

Alternative forms of axioms 1-8 are often encountered. For example, the axiom of pairing (#4) is often changed to say that for any sets x and y there is a set containing exactly x and y. Similarly, the axioms of union, replacement, and power set are often written to say that the desired set contains only those sets which it must contain. An axiom is sometimes added which asserts that the empty set exists. For an example of some of these variations, see the list of axioms in Jech [2003].

The Axiom of choice has many equivalent statements; that is, there are many statements that can be proved equivalent to axiom 9 using axioms 1-8. The name "axiom of choice" refers to one such statement, namely that there exists a choice function for every set of nonempty sets. Since the existence of a choice function for finite sets is easily proved, this formulation is interesting because of what it asserts about certain infinite sets.

The list above includes two infinite axiom schemes. In his 1957 PhD thesis, Montague proved that no finite axiomatization of ZFC is possible, and thus that any axiomatization of ZFC must include at least one axiom scheme.

See also

• Axiomatic set theory
• Zermelo set theory
• Whitehead-Russell axioms
• Von Neumann–Bernays–Gödel set theory
• List of statements undecidable in ZFC
• Z notation

Bibliography

• Abian, Alexander, 1965. The Theory of Sets and Transfinite Arithmetic. W B Saunders.
• Keith Devlin, 1996 (1984). The Joy of Sets. Springer.
• Abraham Fraenkel, Yehoshua Bar-Hillel, and Azriel Levy, 1973 (1958). Foundations of Set Theory. North Holland.
• Hatcher, William, 1982 (1968). The Logical Foundations of Mathematics. Pergamon.
• Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN 3-540-44085-2.
• Kunen, Kenneth, 1980. Set Theory: An Introduction to Independence Proofs. Elsevier. ISBN 0-444-86839-9.
• Suppes, Patrick, 1972 (1960). Axiomatic Set Theory. Dover.
• Tourlakis, George, 2003. Lectures in Logic and Set Theory, Vol. 2. Cambridge Univ. Press.
• Jean van Heijenoort, 1967. From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931. Harvard Univ. Press. Includes annotated English translations of the classic articles by Zermelo, Frankel, and Skolem bearing on ZFC.

External links

• Stanford Encyclopedia of Philosophy: Set Theory -- by Thomas Jech.
• Metamath. An online project for mathematical proofs using an abstract language, which can be verified by a computer program.
• Zermelo Fraenkel Axioms on PlanetMath

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