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empty set – Hempel’s Ravens Paradox
empty set
The unique set having no elements, generally denoted by a circle with a forward slash through it, or by an empty pair of braces.
Epimenides Paradox
See Liar Paradox.
epimorphism
A morphism f from X to Y is called an epimorphism when it is surjective, that is, when to each element y of Y there corresponds an x in X such that f(x) = y.
Cf. monomorphism.
equicontinuous
Let C[a,b] denote the space of all continuous functions on the real interval [a,b]. A subset S of C[0,1] is called equicontinuous at x in [a,b] if for any e greater than zero there is some d greater than zero such that for every function f in S we have:
equipotent
Two sets are equipotent if there exists a function between them that is bijective, that is, which is “one-to-one” and “onto.”
Cf. cardinal.
equivalence relation
An equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive. An equivalence relation on X gives rise to (and is determined by) a partition on X.
Cf. congruence relation.
Euclidean space
A space satisfying the axioms of Euclidean geometry. Specifically, a space in which the parallel postulate holds; equivalently, a space in which the distance between points is given by a straight line.
existential quantifier
See predicate calculus.
extended real numbers
The set of real numbers together with a “point(s) at infinity.” If both a positive and a negative infinity are added, then the resulting space is order equivalent and topologically equivalent (i.e., homeomorphic) to the unit interval [0, 1]. If only a single point at infinity is added, the resulting space is topologically equivalent to the unit circle.
fallacy
ARTICLE
An unjustified step in logic, or incorrect form of reasoning, leading to an invalid conclusion. See the article for a complete description.
filter
If X is a set (or class) and F is a family of subsets of X, then F is called a filter provided- F is closed under intersections, i.e., for any sets A, B in F their intersection is also in F, and
- F is closed under supersets (upwardly closed), i.e., if A is any set in F and B is any set in X containing A, then B is in F.
If X and F are as above, and if in addition for every subset Y of X either Y or its complement is in F, then F is called an ultrafilter (or maximal filter). Equivalently, an ultrafilter is a filter that is not the proper subset of any filter. A free ultrafilter is an ultrafilter which contains the complement of every finite set. If A is any subset of X, then the collection of supersets of A together with all finite intersections of supersets of A is called the filter generated by A. More generally, if U is any family of subsets of X that is closed unter finite intersections, then the family F of subsets which includes precisely U and all of the intersections of supersets of members of U is the filter generated by U, and U is called its filter base. If a filter F is generated by a singleton set, then it is called a principal filter.
finite intersection property
A collection of sets has the finite intersection property if every finite subcollection has non-empty intersection.
finite measure
See measure.
finite set
A set X is finite if there is a natural number n (possibly 0) such that X can be said to have exactly n elements. More formally, a set is finite if it is not bijective with any proper subset of itself.
flat pair
A pair of elements {a,b} with no order defined on the elements. Sometimes also called an unordered pair.
Cf. ordered pair.
functional calculus
See predicate calculus.
GCH
See: generalized continuum hypothesis.
generalized continuum hypothesis
The claim that there is no set of intermediate cardinality between any given set and its power set. The GCH is independent of (i.e., cannot be proved true or false by) the known axioms of set theory.
Cf. continuum hypothesis.
Related article: Gödel's Theorems
Related MiniText: Infinity -- You Can't Get There From Here...
Hausdorff space
A topological space is Hausdorff if every pair of distinct points have disjoint neighborhoods. Two disjoint compact sets in a Hausdorff space have disjoint neighborhoods. A compact subset of a Hausdorff space is closed.
Heine-Borel property
Every closed and bounded subset of the real numbers is compact.
Hempel’s Ravens Paradox
ARTICLE
A paradox of inductive logic described by the philosopher C.G. Hempel. See the article for a complete exposition.
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