Zermelo-Fraenkel Axioms

The Zermelo-Fraenkel axioms are the basis for Zermelo-Fraenkel set theory. In the following (Jech 1997, p. 1),  stands for exists,  means for all,  stands for "is an element of,"  for the empty set,  for implies,  for AND,  for OR, and  for "is equivalent to."

1. Axiom of Extensionality: If  and  have the same elements, then .

(1)

2. Axiom of the Unordered Pair: For any  and  there exists a set  that contains exactly  and . (also called Axiom of Pairing)

(2)

3. Axiom of Subsets: If  is a property (with parameter ), then for any  and  there exists a set  that contains all those  that have the property . (also called Axiom of Separation or Axiom of Comprehension)

(3)

4. Axiom of the Sum Set: For any  there exists a set , the union of all elements of . (also called Axiom of Union)

(4)

5. Axiom of the Power Set: For any  there exists a set , the set of all subsets of .

(5)

6. Axiom of Infinity: There exists an infinite set.

(6)

7. Axiom of Replacement: If  is a function, then for any  there exists a set .

(7)

8. Axiom of Foundation: Every nonempty set has an -minimal element. (also called Axiom of Regularity)

(8)

9. Axiom of Choice: Every family of nonempty sets has a choice function.

(9)

The system of axioms 1-8 is called Zermelo-Fraenkel set theory, denoted "ZF." The system of axioms 1-8 minus the axiom of replacement (i.e., axioms 1-6 plus 8) is called Zermelo set theory, denoted "Z." The set of axioms 1-9 with the axiom of choice is usually denoted "ZFC."

Unfortunately, there seems to be some disagreement in the literature about just what axioms constitute "Zermelo set theory." Mendelson (1997) does not include the axioms of choice or foundation in Zermelo set theory, but does include the axiom of replacement. Enderton (1977) includes the axioms of choice and foundation, but does not include the axiom of replacement. Itô includes an Axiom of the empty set, which can be gotten from (6) and (3), via  and .

Abian (1969) proved consistency and independence of four of the Zermelo-Fraenkel axioms.

REFERENCES

Abian, A. "On the Independence of Set Theoretical Axioms." Amer. Math. Monthly 76, 787-790, 1969.

Devlin, K. The Joy of Sets: Fundamentals of Contemporary Set Theory, 2nd ed. New York: Springer-Verlag, 1993.

Enderton, H. B. Elements of Set Theory. New York: Academic Press, 1977.

Itô, K. (Ed.). "Zermelo-Fraenkel Set Theory." §33B in Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 1. Cambridge, MA: MIT Press, pp. 146-148, 1986.

Iyanaga, S. and Kawada, Y. (Eds.). "Zermelo-Fraenkel Set Theory." §35B in Encyclopedic Dictionary of Mathematics, Vol. 1. Cambridge, MA: MIT Press, pp. 134-135, 1980.

Jech, T. Set Theory, 2nd ed. New York: Springer-Verlag, 1997.Mendelson, E. Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall, 1997.

Zermelo, E. "Über Grenzzahlen und Mengenbereiche." Fund. Math. 16, 29-47, 1930.