Sets

13. AXIOMATIC DEVELOPMENT OF SET THEORY

In an axiomatic development of a branch of mathematics, one begins with:

1. Undefined terms

2. Undefined relations

3. Axioms relating the undefined terms and undefined relations. Then, one develops theorems based upon the axioms and definitions Example 14:1:

In an axiomatic development of Plane Euclidean geometry

1. “Points” and “lines” are undefined terms

2. “Points on a line” or, equivalent, “line contain a point” is an undefined relation

3. Two of the axioms are:

Axiom 1: Two different points are on one and only one line

Axiom 2: Two different lines cannot contain more than one point in common

In an axiomatic development of set theory:

1. “Element’ and “set” are undefined terms

2. “Element belongs to a set” is undefined relation

3. Two of the axioms are

Axiom of Extension: Two sets A and B are equal if and only if every element in A belongs to B and every element in B belongs to A.

Axiom of Specification: Let P(x) be any statement and let A be any set. Then there exists a set:

B= {aa ∈ A, P (a) is true}

Here, P(x) is a sentence in one variable for which P(a) is true or false for any a∈A. for example P(x) could be the sentence “x² = 4” or “x is a member of the United Nations”