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singleton set – transitive set
singleton set
A set with exactly one element.
singular cardinal
A cardinal that is not regular.
space
Any abstract set with a structure defined on it, such as an order relation, metric, etc.
Cf. Euclidean space, Hilbert space, metric space, topological space.
stationary set
If a is an ordinal, a set S in a is called stationary if S has non-empty intersection with every closed unbounded subset of a.
subset
A set A is a subset of a set B if every element of A is also an element of B. If in addition B is a subset of A, then A = B, but if not then A may be said to be a proper subset of B.
Cf. superset.
sumset
Given a set A, the sumset of A, denoted by
is the set containing all of the elements of the elements of A, that is, it is the union of the elements of A.
sumset axiom
An axiom of set theory which states that if A is any set, then the sumset of A is also a set.
sup
Abbreviation of supremum.
superset
A set A is a superset of a set B if every element of B is an element of A.
Cf. subset.
supremum
The supremum of any subset of a linearly ordered set is the least upper bound of the subset. In particular, the supremum of any set of numbers is the smallest number in the set which is greater than or equal to every number in the set. In a complete linear order the supremum of any bounded set always exists.
Cf. infimum, least upper bound axiom.
surjection
A surjective function, i.e., a function that maps at least one element of its domain to each element of its range.
Cf. injection, bijection.
Suslin tree
Set Theory: For a an infinite cardinal, an a-Suslin tree is a tree T such that |T| = a, and every chain and every antichain of T has cardinality less than a.
Cf. Aronszajn tree.
symbolic logic
Logic reduced to syntax, i.e., which works only with uninterpreted symbols. The two most often used kinds of symbolic logic are the propositional calculus and the predicate calculus.
symmetric difference
The symmetric difference of two sets A and B is the set of those elements that are in either A or B but not both.
Cf. intersection, union.
symmetric relation
A relation ~ on a set X is symmetric if for any two elements x and y in X we have x ~ y if and only if y ~ x. The relation ~ is called asymmetric if for any two elements x and y we have that x ~ y implies it is not true that y ~ x, and it is called antisymmetric if whenever x ~ y and y ~ x then x = y. Note that a relation may be neither symmetric, asymmetric, nor antisymmetric.
Cf. reflexive relation, transitive relation.
Tarski Truth Theorem
Let T be a mathematical theory. Denote formulas of T by j, y, etc., and let T j be the statement that the sentence j is true in T Choose a canonical numbering of the formulas of T (e.g., Gφdel numbering) that assigns to each sentence of T a unique positive integer, and denote the positive integer associated with a sentence j by j. Then there is no formula y in T such that y(j) Tj.
In other words, truth in a theory T is not definable in T.
totally ordered set
A set with a total order defined on it.
total order
An order relation < on a set S is a total order exactly if, for any two elements x and y of S, either x < y, x = y, or y < x, but no two of the three. Sometimes also called a linear order. Every totally ordered set is order-isomorphic to an ordinal, and this ordinal is called its order type (or sometimes just type.)
Cf. partial order
transitive closure
If x is any set, the transitive closure of x is the set containing all of the elements of x, and all of the elements of the elements of x, and all of the elements of the elements of the elements of x, and so on. Formally, the transitive closure Tc(x) of x is defined by the recursive formulas
where w is the set of finite ordinals.
transitive relation
A relation ~ on a set X is called transitive if it is the case that for every x, y, and z in X, if x ~ y and y ~ z , then x ~ z. For example, the relation < (less-than) on the set of natural numbers is transitive.
Cf. reflexive relation, symmetric relation.
transitive set
A set is called transitive if it is equal to its transitive closure. That is, x is a transitive set if whenever y is an element of x and z is an element of y, then z is an element of x.
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