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Russell Paradox – supremum
Russell Paradox
(Bertrand Russell, 1901) A paradox of set theory which necessitated a more careful axiomatization of set theory in the 1920’s and 1930’s: Naively, some sets are members of themselves and some are not. For instance, the set of all apples is not itself an apple, but the set of all sets does seem to be a set. So consider the set X of all sets that are not members of themselves. We may ask, is X a member of itself? If it is then it cannot be, because of the way in which X itself was defined, but if it isn’t then it must be, by the same reasoning. Contradiction. The Russell paradox is resolved in modern set theory by a foundation axiom or axiom of regularity, and by limiting the “size” of objects we call sets. For example, the “set of all sets” is considered not to be a set but a proper class.
Schroeder-Bernstein Theorem
If there exists an injection from a set X into a set Y, and also an injection from Y into X, then there exists a bijection from X to Y, and hence X and Y have the same cardinality.
semi-lattice
A set with a single binary operation that is idempotent, commutative, and associative.
Cf. lattice.
sentential calculus
See propositional calculus.
set
Naively, any well-defined collection considered as a single, abstract object. By “well-defined” is meant that it is always possible to determine for a given set when something is an element of the set and when not. In formal set theory, the term “set” is not defined, but is a primitive term whose meaning is informed purely by the axioms in which it appears.
Cf. ZF, ZFC.
set algebra
See algebra of sets.
set difference
The set difference of two sets A and B, denoted A - B or A \ B, is the set of elements that is contained in A but not in B. This is equivalent to the intersection between A and the complement of B.
set function
A function whose domain of definition is a collection of sets.
set ring
See ring of sets.
set theory
Naive set theory: The study of sets (i.e., well-defined collections of objects) which have a binary extensional relation (set membership) defined on them.
Abstract set theory: As naive set theory, but with all sets built using only elements which are themselves sets (beginning with the empty set, which has no members).
Formal set theory: Any of several axiom systems of abstract set theory in the language of first-order logic, such as Zermelo Fraenkel set theory, Gödel-Bernays set theory, Quine’s New Foundations, etc.
Sid’s Paradox
ARTICLE
A paradox of knowledge: is the distinction between matters of opinion and matters of fact a matter of opinion or a matter of fact? See the article for discussion.
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.
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