Set Theory
Across
- 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.
Down
- 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.