Set theories {Zermelo-Fraenkel set theory} {ZF set theory} can be axiomatic systems. Zermelo set theory has no paradoxes but is not consistent.
The empty set exists {axiom of empty set}. The empty set exists, so at least one set exists {axiom of existence}. Sets with same elements are equal {axiom of extension} {axiom of extensionality}. For two sets, another set exists that contains all and only elements of the two sets {axiom of union}. For two sets, another set exists that has the two sets as only elements {axiom of pairing}. For a set, another set exists whose elements are the subsets of the original set {axiom of powers} {axiom of power set}. A set exists that has the empty set as an element and, if an element is in the set, the set that contains only that element is an element in the set {axiom of infinity}.
Non-empty sets contain at least one element, and the non-empty set and the set of any element are disjoint sets {axiom of regularity}. For any set and any mapping, a subset of the set exists that has as elements the domain of the mapping {axiom of separation}. For any set and any mapping, a set exists that has as elements the range of the mapping over the original set's elements (as domain of mapping) {axiom of replacement} {axiom of specification}. For any set, a mapping exists that chooses one element of each subset (axiom of choice).
Mathematical Sciences>Set Theory>Axioms
Outline of Knowledge Database Home Page
Description of Outline of Knowledge Database
Date Modified: 2022.0224