Thus, constructing the set of Dedekind cuts serves the purpose of embedding the original ordered set " S ", which might not have had the least-upper-bound property, within a ( usually larger ) linearly ordered set that does have this useful property.
42.
He is noted for many theorems including : H�lder's inequality, the Jordan H�lder theorem, the theorem stating that every linearly ordered group that satisfies an Archimedean property is isomorphic to a subgroup of the additive analysis, including the theories of partial differential equations and function spaces.
43.
Gentzen's proof is arguably finitistic, since the transfinite ordinal ? 0 can be encoded in terms of finite objects ( for example, as a Turing machine describing a suitable order on the integers, or more abstractly as consisting of the finite trees, suitably linearly ordered ).
44.
Thus, the question " is "'P "'a proper subset of "'NP "'" can be reformulated as " is existential second-order logic able to describe languages ( of finite linearly ordered structures with nontrivial signature ) that first-order logic with least fixed point cannot ? ".
45.
We wish to define here a subset " Z " of a linearly ordered topological space " X " such that no total order on " Z " generates the subspace topology on " Z ", so that the subspace topology will not be an order topology even though it is the subspace topology of a space whose topology is an order topology.
46.
:The answer is nonunique, since the toplogies on R are not linearly ordered, even assuming that the question is what is the coarsest topology refining the usual topology on R . One coarsest topology that refines the usual one is obtained by taking the usual topology, and then in addition declaring as open any set of the form (-& infin;, 0 ] & cup; U where U is open in the usual topology . talk ) 21 : 07, 21 March 2011 ( UTC)
47.
Other finite linearly ordered MV-algebras are obtained by restricting the universe and operations of the standard MV-algebra to the set of n equidistant real numbers between 0 and 1 ( both included ), that is, the set \ { 0, 1 / ( n-1 ), 2 / ( n-1 ), \ dots, 1 \ }, which is closed under the operations \ oplus and \ lnot of the standard MV-algebra; these algebras are usually denoted MV n.
