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Consider the problem of Linearly Constrained Convex Quadratic Programming. Under reasonable assumptions (the problem is feasible, the system of constraints is regular at every point, and the quadratic objective is strongly convex), the active-set method terminates after finitely many steps, and yields a global solution to the problem.
The Zermelo–Fraenkel set theory, a result of the axiomatic method applied to set theory, allowed the "proper" formulation of set-theory problems and helped avoid the paradoxes of naïve set theory. One such problem was the continuum hypothesis. Zermelo–Fraenkel set theory, with the historically controversial axiom of choice included, is ...
Kurt Gödel proved in 1940 that the negation of the continuum hypothesis, i.e., the existence of a set with intermediate cardinality, could not be proved in standard set theory. [2] The second half of the independence of the continuum hypothesis – i.e., unprovability of the nonexistence of an intermediate-sized set – was proved in 1963 by ...
In non-well-founded set theories, the foundation axiom of ZFC is replaced by axioms implying its negation. The study of non-well-founded sets was initiated by Dmitry Mirimanoff in a series of papers between 1917 and 1920, in which he formulated the distinction between well-founded and non-well-founded sets; he did not regard well-foundedness as ...
The concept of a stable model, or answer set, is used to define a declarative semantics for logic programs with negation as failure.This is one of several standard approaches to the meaning of negation in logic programming, along with program completion and the well-founded semantics.
In set theory, an infinite set is not considered to be created by some mathematical process such as "adding one element" that is then carried out "an infinite number of times". Instead, a particular infinite set (such as the set of all natural numbers) is said to already exist, "by fiat", as an assumption or an axiom. Given this infinite set ...
In set theory, is also used to indicate 'not in the set of': is the set of all members of U that are not members of A. Regardless how it is notated or symbolized , the negation ¬ P {\displaystyle \neg P} can be read as "it is not the case that P ", "not that P ", or usually more simply as "not P ".
In mathematics, Suslin's problem is a question about totally ordered sets posed by Mikhail Yakovlevich Suslin () and published posthumously.It has been shown to be independent of the standard axiomatic system of set theory known as ZFC; Solovay & Tennenbaum (1971) showed that the statement can neither be proven nor disproven from those axioms, assuming ZF is consistent.