Zermelo–Fraenkel set theory
Zermelo–Fraenkel set theory (abbreviated ZF) is a system of axioms used to describe set theory. When the axiom of choice is added to ZF, the system is called ZFC. It is the system of axioms used in set theory by most mathematicians today.
After Russell's paradox was found in the 1901, mathematicians wanted to find a way to describe set theory that did not have contradictions. Ernst Zermelo proposed a theory of set theory in 1908. In 1922, Abraham Fraenkel proposed a new version based on Zermelo's work.
Axioms
An axiom is a statement which is accepted without question, and which has no proof. ZF contains eight axioms.
- The axiom of extension says that two sets are equal if and only if they have the same elements. For example, the set [math]\displaystyle{ \{1, 3\} }[/math] and the set [math]\displaystyle{ \{3, 1\} }[/math] are equal.
- The axiom of foundation says that every set [math]\displaystyle{ S }[/math] (other than the empty set) contains an element that is disjoint (shares no members) with [math]\displaystyle{ S }[/math].
- The axiom of specification says that given a set [math]\displaystyle{ S }[/math], and a predicate [math]\displaystyle{ F }[/math] (a function that is either true or false), that a set exists that contains exactly those elements of [math]\displaystyle{ S }[/math] where [math]\displaystyle{ F }[/math] is true. For example, if [math]\displaystyle{ S = \{1, 2, 3, 5, 6\} }[/math], and [math]\displaystyle{ F }[/math] is "this is an even number", then the axiom says that the set [math]\displaystyle{ \{2, 6\} }[/math] exists.
- The axiom of pairing says that given two sets, there is a set whose members are exactly the two given sets. So, given the two sets [math]\displaystyle{ \{0, 3\} }[/math] and [math]\displaystyle{ \{2, 5\} }[/math], this axiom says that the set [math]\displaystyle{ \{\{0, 3\}, \{2, 5\}\} }[/math] exists.
- The axiom of union says that for any set, there exists a set that consists of just the elements of the elements of that set. For example, given the set [math]\displaystyle{ \{\{0, 3\}, \{2, 5\}\} }[/math], this axiom says that the set [math]\displaystyle{ \{0, 3, 2, 5\} }[/math] exists.
- The axiom of replacement says that for any set [math]\displaystyle{ S }[/math] and a function [math]\displaystyle{ F }[/math], that the set consisting of the results of calling [math]\displaystyle{ F }[/math] on all the members of [math]\displaystyle{ S }[/math] exists. For example, if [math]\displaystyle{ S = \{1, 2, 3, 5, 6\} }[/math] and [math]\displaystyle{ F }[/math] is "add ten to this number", then the axiom says that the set [math]\displaystyle{ \{11, 12, 13, 15, 16\} }[/math] exists.
- The axiom of infinity says that the set of all integers (as defined by the Von Neumann construction) exists. This is the set [math]\displaystyle{ \{0, 1, 2, 3, 4, ...\} }[/math]
- The axiom of power set says that the power set (the set of all subsets) of any set exists. For example, the power set of [math]\displaystyle{ \{2, 5\} }[/math] is [math]\displaystyle{ \{\{\}, \{2\}, \{5\}, \{2, 5\}\} }[/math]
Axiom of choice
The axiom of choice says that it is possible to take one object out of each of the elements of a set and make a new set. For example, given the set [math]\displaystyle{ \{\{0, 3\}, \{2, 5\}\} }[/math], the axiom of choice would show that a set such as [math]\displaystyle{ \{3, 5\} }[/math] exists. For finite sets, this axiom can be proved from the other axioms, but not for infinite sets.
References
- Bagaria, Joan (2014). "Zermelo-Fraenkel Set Theory | Axioms of ZF".
- Halmos, Paul (1974). Naive Set Theory. Springer. ISBN 0-387-90092-6.
- Kunen, Kenneth (1980). Set Theory: An Introduction to Independence Proofs. Elsevier. ISBN 0-444-86839-9.