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If a system of equations is inconsistent, then the equations cannot be true together leading to contradictory information, such as the false statements 2 = 1, or + = and + = (which implies 5 = 6). Both types of equation system, inconsistent and consistent, can be any of overdetermined (having more equations than unknowns), underdetermined ...
Gödel's original statement and proof of the incompleteness theorem requires the assumption that the system is not just consistent but ω-consistent. A system is ω-consistent if it is not ω-inconsistent, and is ω-inconsistent if there is a predicate P such that for every specific natural number m the system proves ~P(m), and yet the system ...
Such a theory is consistent if and only if it does not prove a particular sentence, called the Gödel sentence of the theory, which is a formalized statement of the claim that the theory is indeed consistent. Thus the consistency of a sufficiently strong, recursively enumerable, consistent theory of arithmetic can never be proven in that system ...
A good example is the relative consistency of absolute geometry with respect to the theory of the real number system. Lines and points are undefined terms (also called primitive notions) in absolute geometry, but assigned meanings in the theory of real numbers in a way that is consistent with both axiom systems. [citation needed]
Since, in a system without a global clock, defining the last operation among writes is difficult, some restrictions can be applied on the values that can be returned by a read operation. The goal of data-centric consistency models is to provide a consistent view on a data store where processes may carry out concurrent updates.
If a formal system is syntactically complete, a corresponding formal theory is called complete if it is a consistent theory. Gödel's incompleteness theorem shows that any computable system that is sufficiently powerful, such as Peano arithmetic, cannot be both consistent and syntactically complete.
Eventual consistency is a weak guarantee – most stronger models, like linearizability, are trivially eventually consistent. Eventually-consistent services are often classified as providing BASE semantics (basically-available, soft-state, eventual consistency), in contrast to traditional ACID (atomicity, consistency, isolation, durability).
In the case of the systems of polynomial equations, it may happen that an overdetermined system has a solution, but that no one equation is a consequence of the others and that, when removing any equation, the new system has more solutions. For example, () =, () = has the single solution =, but each equation by itself has two solutions.