Naive Set Theory

(from the book "Naïve Set Theory" by Paul R. Halmos, 1960)

Axioms of naive set theory

• axiom of extension

• axiom of unions

• axiom of specification

• axiom of pairing

• axiom of powers

• axiom of infinity

• axiom of substitution

• axiom of choice

The term "naïve" qualifies the presentation of set theory as being given informally, without rigorous compliance to a set of axioms.

Axioms of set theory (by Paul Halmos) 1. Axiom of extension: two sets are equal iff they have the same elements 2. Axiom of unions: for every collection of sets there exists a set that contains all the elements that belong to at least one of them 3. Axiom of specification: to every set S and to every condition P(a), there corresponds a set A whose elements are exactly those elements a ∈ S for which P(a) holds. 4. Axiom of pairing: for any two sets there is a set whom they both belong to 5. Axiom of powers: for each set there is a collection of sets that contains, as elements, all the subsets of the given set 6. Axiom of infinity: there is a set containing 0 and the successor of each of its elements 7. Axiom of substitution: if S(a,b) is a sentence such that ∀a ∈ A the set { b | S(a,b) } can be formed, then there exists a function f with domain A such that f(a) = { b | S(a,b) } for each a ∈ A (a reasonable operation over the element of a set yields a set) 8. Axiom of choice: the Cartesian product of a non-empty family of non-empty sets is non-empty