Note that completeness is a property of the metric and not of the topology, meaning that a complete metric space can be homeomorphic to a non-complete one.

His axioms, however, do not guarantee that the circles actually intersect, because they do not assert the geometrical property of continuity, which in Cartesian terms is equivalent to the completeness property of the real numbers.

The completeness theorem is a central property of first-order logic that does not hold for all logics.

Second-order logic, for example, does not have a completeness theorem for its standard semantics ( but does have the completeness property for Henkin semantics ), and the same is true of all higher-order logics.

* Computational completeness: This property says that we can define or implement any kind of computable function for the ODB, using DML of database system.

The converse of the soundness property is the semantic completeness property.

In analysis, the supremum or least upper bound of a set S of real numbers is denoted by sup S and is defined to be the smallest real number that is greater than or equal to every number in S. An important property of the real numbers is completeness: every nonempty subset of the set of real numbers that is bounded above has a supremum that is also a real number.

The proof is similar to the preceding statement ; the finite intersection property takes the role played by completeness.

Similarly, the completeness property cannot be expected to carry over, because the reals are the unique complete ordered field up to isomorphism.

An axiomatic system for which every model is isomorphic to another is called categorial ( sometimes categorical ), and the property of categoriality ( categoricity ) ensures the completeness of a system.

In statistics, completeness is a property of a statistic in relation to a model for a set of observed data.

The only real number axiom that does not follow easily from the definitions is the completeness of ≤, i. e. the least upper bound property.

For example, the completeness of a linear order, which is used to characterize the real numbers as a complete ordered field, is a non-first-order property.

Hence every completeness property has its dual, obtained by inverting the order-dependent definitions in the given statement.

Directed completeness alone is quite a basic property that occurs often in other order theoretic investigations, using for instance algebraic posets and the Scott topology.

The existence of least and greatest elements is a special completeness property of a partial order.

The completeness property is useful when language B is used to study or test a program written in language A, possibly by extracting key parts of the code: if this study or test proves that the program terminates in B, then it also terminates in A.

In practice, Turing completeness means that rules followed in sequence on arbitrary data can produce the result of any calculation.

A whole fish, chicken, or pig means luck and completeness in Chinese wedding culture.

Complete means that the set of functions satisfies the following completeness relation:

Formal rigour is the introduction of high degrees of completeness by means of a formal language where such proofs can be codified using set theories such as ZFC ( see automated theorem proving ).

* Both MA and AM remain unchanged if their definitions are changed to require perfect completeness, which means that Arthur accepts with probability 1 ( instead of 2 / 3 ) when x is in the language.

Also, as a 1978 Risk Assessment Review Group Report to the NRC pointed out, it is " conceptually impossible to be complete in a mathematical sense in the construction of event-trees and fault-trees … This inherent limitation means that any calculation using this methodology is always subject to revision and to doubt as to its completeness.

** Gödel's completeness theorem for first-order logic: every consistent set of first-order sentences has a completion.

Then, since every real is the limit of some Cauchy sequence of rationals, the completeness of the norm extends the linearity to the whole real line.

From 1950 onward, census forms were mailed to every address on record with the United States Post Office, including the Armed Services Postal System, in an effort to enhance completeness of the data collected.

A deductive system is called complete if every logically valid formula is the conclusion of some formal deduction, and the completeness theorem for a particular deductive system is the theorem that it is complete in this sense.

In 1928, Hilbert and Wilhelm Ackermann published Grundzüge der theoretischen Logik ( Principles of Mathematical Logic ), an introduction to first-order logic in which the problem of completeness was posed: Are the axioms of a formal system sufficient to derive every statement that is true in all models of the system?

It teaches that God's predestining decision is based on the knowledge of His own will rather than foreknowledge, concerning every particular person and event ; and, God continually acts with entire freedom, in order to bring about his will in completeness, but in such a way that the freedom of the creature is not violated, " but rather, established "

Turing completeness is significant in that every real-world design for a computing device can be simulated by a universal Turing machine.

In some cases, we can use FMP to prove Kripke completeness of a logic: every normal modal logic is complete wrt a class of modal algebras, and a finite modal algebra can be transformed into a Kripke frame.

Furthermore, an ordered field is complete if every non-empty subset of it that has an upper bound within the field has a least upper bound within the field, which should be compared to the ( slightly different ) order-theoretical notion of bounded completeness.

Note for completeness that condition ( 1 ) fails: any map t: B → A must map every two-cycle to the identity because the map has to be a group homomorphism, while the order of a two-cycle is 2 which can not be divided by the order of the elements in A other than the identity element, which is 3 as A is the alternating subgroup of, or namely the cyclic group of order 3.

Traceability refers to the completeness of the information about every step in a supply chain.

Arguably, one aspect of its completeness is that the concept of error is limited to events like lack of file space and requesting expansion of a string longer than the interpreter's working storage ; what would in many languages be described as illegal operations are dealt with in TRAC by defining a result ( often a null string ) for every possible combination of a function's argument strings.

; completeness: For every n-ary operator of A, there exists an n-ary operator of B such that

; completeness: For every term of language A and every terms of language B, if then there exists some such that.

; completeness: for every observable on terms of A, there exists an observable on terms of B such that for any term with observable, has observable.

; completeness: for every terms, if simulates then simulates.

One way to do so is to impose completeness on the revealed preference relation with regards to the situations, i. e. every possible situation must be taken into consideration by a consumer.

Chang's ( 1958, 1959 ) completeness theorem states that any MV-algebra equation holding in the standard MV-algebra over the interval will hold in every MV-algebra.