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existential quantifier – infinity
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 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...
Hempel’s Ravens Paradox
ARTICLE
A paradox of inductive logic described by the philosopher C.G. Hempel. See the article for a complete exposition.
homomorphism
A function f from one algebra to another is called a homomorphism if it preserves operations on elements, that is, if for any a, b in the domain, f(ab) = f(a)f(b).
Cf. group homomorphism, ring homomorphism.
hyperset
A set which is not well-founded, i.e., which involves self-membership or, equivalently, an infinite descending membership chain. Example: the Quine atom x = {x}.
identity function
A function that maps each domain element to itself. Also called the identity map.
identity map
See identity function.
image
Given a function f with domain X, the image under f of a subset A of X, denoted f(A), is the subset of the range consisting of those elements to which elements of A are mapped by f.
inaccessible cardinal
A cardinal k is called weakly inaccessible if and only if it is a regular limit cardinal. A cardinal k is called strongly inaccessible if and only if k > w and for every cardinal l less than k, 2l is less than k. If GCH is assumed, then weakly inaccessibles are strongly inaccessible. Ordinary set theory (ZFC) cannot prove that weakly inaccessibles exist.
Cf. König’s Lemma.
independent axiom
In a formal mathematical theory, a formula or statement in the theory is said to be independent if it cannot be derived (proved, deduced) from the other axioms of the theory.
inf
Abbreviation of infimum.
infimum
The infimum of any subset of a linear order (linearly ordered set) is the greatest lower bound of the subset. In particular, the infimum of any set of numbers is the largest number in the set which is less than or equal to every other number in the set. In a complete linear order the infimum of any bounded set always exists.
Cf. supremum, least upper bound axiom.
infinity
Infinity is a concept understood in different ways depending upon the context in which the word is used. In particular, infinity is not a number in the ordinary sense. The so-called extended real numbers include either a positive or a negative infinity (but not both). When this is done, the algebraic forms "infinity plus negative infinity", "infinity times zero", and "infinity divided by infinity" are undefined. Infinite ordinals may be countable or uncountable. Whether actually infinite totalities may be admitted, used, or analyzed remains a contentious issue in the philosophy of mathematics. See the minitext for a thorough treatment.
Related MiniText: Infinity -- You Can't Get There From Here...
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