Set Theory

4. For every set A and every given property, there is a set containing exactly the elements of A that have that property. A property is given by a formula φ of the first-order language of set theory.
6. is an ordinal that is not bijectable with any smaller ordinal.
7. is the branch of mathematical logic that studies sets, which can be informally described as collections of objects.
11. If two sets A and B have the same elements, then they are equal.
14. which asserts that every set can be well-ordered, i.e., it can be linearly ordered so that every non-empty subset has a minimal element.
15. Given any sets A and B, there exists a set, denoted by {A,B}, which contains A and B as its only elements. In particular, there exists the set {A} which has A as its only element.

1. is the study of the properties and structure of definable sets of real numbers and, more generally, of definable subsets of Rn and other Polish spaces
2. can also be viewed as sets, as any n-ary relation on the elements of a set A can be viewed as a set of n-tuples (a1,…,an) of elements of A.
3. Every non-empty set A contains an ∈-minimal element, that is, an element such that no element of A belongs to it.
5. For every given definable function with domain a set A, there is a set whose elements are all the values of the function.
8. There exists a set, denoted by ∅ and called the empty set, which has no elements.
9. For every set A there exists a set, denoted by P(A) and called the power set of A, whose elements are all the subsets of A.
10. For every set A of pairwise-disjoint non-empty sets, there exists a set that contains exactly one element from each set in A.
12. For every set A, there exists a set, denoted by ⋃A and called the union of A, whose elements are all the elements of the elements of A.
13. There exists an infinite set. In particular, there exists a set Z that contains ∅ and such that if A∈Z, then ⋃{A,{A}}∈Z.