Axioms of Set Theory

In mathematics, each 'system' is defined by the axioms at the base of that system. Set Theory has a number of axioms that define it.

The axiom of existence

There exists a set with no elements.[1]
This axiom describes the empty set.

The axiom of extensionality

If every element of X is an element of Y and every element of Y is an element of X then X = Y.[1]
This axiom states that if sets X and Y have exactly the same elements, then they are the same set. This is also the definition of equality of sets.

The axiom schema of comprehension

Let P(x) be a property of x. For any set A there is a set B such that x ∈ B if and only if x ∈ A and P(x).[1]
This axiom states that if a property (P(x)) of elements (x) of a set (A) can be identified, then a subset (B) of the original set can be constructed that contains only the elements of A that have the property. For example, one property of integers is that an integer is either even or not even. Given the existence of the set of integers, and the property of even, a set containing all even integers can be constructed.

The axiom of pair

For any A and B, there is a set C such that x ∈ C if and only if x = A or x = B.[1]
This axiom says that unordered pairs can be created.

The axiom of union

For any set S, there exists a set U such that x ∈ U if and only if x ∈ A for some A ∈ S.[1]

This axioms says that the union of any two or more sets can be formed. It is understood here that set S is a set containing other sets.

The axiom of power sets

For any set S there exists a set P such that X ∈ P if and only if X ⊆ P.[1]

This axioms states that a power set can be created for any set. A power set is a set that contains all subsets of a set.

The axiom of choice

Let C be a collection of nonempty sets. Then we can choose a member from each set in that collection. In other words, there exists a function f defined on C with the property that, for each set S in the collection, f(S) is a member of S.[2]

This axioms states that a member can be selected from each of a series of infinite sets. This axiom is sometimes called Zermelo's axiom of choice.

The axiom of infinity

There is a set of which the null set is a member, and such that if any set is a member, the union of it and its unit set is also a member. [3] page 186

This axiom guarantees the existenc of at least one infinite set, the set of natural numbers.

The axiom schema of replacement

Any function whose domain is a set has a range which is also a set. [3] page 316

This axiom guarantees that if the input to a function is a set, then the output to the function is a set.

Axiom of foundation

Every non-empty set A contains an element that is not the set A itself.

This axiom states that a set may not be a member of itself. it is also called the axiom of regularity. In logic notation: For every set x where x is not the empty set, there exists element y that is a member of set x such that the intersection of y and x is the empty set.

References

  1. Jech, Thomas. Set Theory. 3rd edition. pp 3-4. Springer. April 1, 2006. Last Accessed 7/8/2018. http://plato.stanford.edu/entries/set-theory. Buy the book
  2. Karel Hrbacek and Thomas Jech. Introduction to Set Theory. 3rd Edition. CRC Press. June 22, 1999. Buy the book
  3. Eric Schechter. Axiom of Choice. Last Accessed 7/8/2018. http://www.math.vanderbilt.edu/~schectex/ccc/choice.html.
  4. Simon Blackburn. The Oxford Dictionary of Philosophy. 2nd Edition. pp 186,316. Last Accessed 7/8/2018. http://www.answers.com/topic/axiom-of-replacement. Buy the book

Cite this article as:

McAdams, David E. Axioms of Set Theory. 7/3/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/a/axiomsofsettheory.html.

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