In general, an object is complete if nothing needs to be added to it. This notion is made more specific in various fields.

## Logical completeness

In logic, completeness is the converse of soundness for formal systems. Logic is the study of the principles of valid demonstration and Inference. For contraposition in the field of traditional logic see Contraposition (traditional logic. In Mathematical logic, a Logical system has the soundness property If and only if its Inference rules prove only formulas that are In formal logic, a formal system (also called a logical system, a logistic system, or simply a logic Formal systems in mathematics consist A formal system has "completeness" when all tautologies are theorems whereas a formal system has "soundness" when all theorems are tautologies. In Propositional logic, a tautology (from the Greek word ταυτολογία is a Propositional formula that is true under any possible valuation In Mathematics, a theorem is a statement proven on the basis of previously accepted or established statements Kurt Gödel, Leon Henkin, and Post all published proofs of completeness. Kurt Gödel (kʊɐ̯t ˈgøːdl̩ (April 28 1906 – January 14 1978 was an Austrian American Logician, Mathematician and Philosopher Leon Henkin ( 19 April 1921 – 1 November[[ 006]] was a Logician at the University of California Berkeley. Emil Leon Post, PhD, ( February 11 1897, Augustów – April 21 1954, New York City) was a Mathematician (See History of the Church-Turing thesis. This article is an extension of the history of the Church-Turing thesis. ) A system is consistent if a proof never exists for both P and not P. The proof of Gödel's incompleteness theorem proves that no recursive system that is sufficiently powerful, such as the Peano axioms, can be both consistent and complete. In Mathematical logic, Gödel's incompleteness theorems, proved by Kurt Gödel in 1931 are two Theorems stating inherent limitations of all but the most In Mathematical logic, the Peano axioms, also known as the Dedekind-Peano axioms or the Peano postulates, are a set of Axioms for the Natural

• A language is expressively complete if it can express the subject matter for which it is intended.
• A formal system is complete with respect to a property iff every sentence that has the property is a theorem. In modern Philosophy, Mathematics, and Logic, a property is an Attribute of an object; thus a red object is said to have the property
• A formal system is functionally complete if it has adequate logical connectives to express all of the theorems of the language. In Logic, a set S of Logical connectives is functionally complete (also expressively adequate or simply adequate) if every possible Table of logic symbolsIn Logic, two sentences (either in a formal language or a natural language may be joined by means of a logical connective to form a compound sentence
• A formal system is strongly complete or complete in the strong sense iff no sentence which is not a theorem can become a theorem through the addition of a new basic rule to the deductive apparatus of the formal system (a rule of inference or an axiom) without the system becoming unsound. A deductive system (also called a deductive apparatus of a Formal system) consists of the Axioms (or Axiom schemata and Rules of inference In Logic, a rule of inference (also called a transformation rule) is a function from sets of formulae to formulae In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject First-order sentential calculus is strongly complete. This is a technical mathematical article about the area of mathematical logic variously known as "propositional calculus" or "propositional logic"
• A formal system is extremely complete or complete in the extreme sense iff every sentence is a theorem.
• A formal system is deductively complete iff there are no formulas constructed on the base of the system (the axioms) which are derivable by the rules of the system as theorems and which are not tautologies. In formal logic, a formal system (also called a logical system, a logistic system, or simply a logic Formal systems in mathematics consist In Mathematical logic, a well-formed formula (often abbreviated WFF, pronounced "wiff" or "wuff" is a Symbol or string of symbols (a In Mathematics, a proof is a convincing demonstration (within the accepted standards of the field that some Mathematical statement is necessarily true
• In one sense, a formal system S is syntactically complete or maximally complete iff for each formula A of the language of the system either A or ~A is a theorem of S. This is also called negation completeness. In another sense, a formal system is syntactically complete iff no unprovable schema can be added to it as an axiom schema without inconsistency. In Mathematical logic, an axiom schema generalizes the notion of Axiom. Truth-functional propositional logic is semantically, and syntactically complete. This is a technical mathematical article about the area of mathematical logic variously known as "propositional calculus" or "propositional logic" First-order predicate logic is semantically complete, but not syntactically or negation complete. First-order logic (FOL is a formal Deductive system used in mathematics philosophy linguistics and computer science [1]
• For a formal system S in formal language L, S is semantically complete iff every logically valid formula of L (every formula which is true under every interpretation of L) is a theorem of S. That is, $\models_{\mathrm L} A$ $\to$ $\ \vdash_{\mathrm S} A$. [1]

## Mathematical completeness

In mathematics, "complete" is a term that takes on specific meanings in specific situations, and not every situation in which a type of "completion" occurs is called a "completion". Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and See, for example, algebraically closed field or compactification. In Mathematics, a field F is said to be algebraically closed if every Polynomial in one Variable of degree at least 1 with Coefficients In Mathematics, compactification is the process or result of enlarging a Topological space to make it compact.

• In mathematical logic, a theory is complete, if it contains either $\scriptstyle S$ or $\scriptstyle \neg S$ for every sentence $\scriptstyle S$ in the language. Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical logic. In Mathematical logic, a theory is a set of sentences in a Formal language. In Mathematical logic, a first-order theory is complete, if for every sentence φ in its language it contains either φ itself or its negation This article is a technical mathematical article in the area of predicate logic A language is a dynamic set of visual auditory or tactile Symbols of Communication and the elements used to manipulate them
• In mathematical logic, a formal calculus for a logic L is strongly complete with respect to a certain semantics of L, if every statement P that follows semantically from a set of premises G can be derived syntactically from these premises within the calculus. Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical logic. In Mathematical logic, a proof calculus corresponds to a family of Formal systems that use a common style of formal inference for its Inference rules. In Mathematics, model theory is the study of (classes of mathematical structures such as groups, Fields graphs or even models Proof theory is a branch of Mathematical logic that represents proofs as formal Mathematical objects facilitating their analysis by mathematical techniques Formally, $\scriptstyle G\ \models\ P$ implies $\scriptstyle G\ \vdash\ P$. The calculus is complete, if the same holds for the empty set of premises $\scriptstyle G\,=\,\varnothing$ (i. e. , if all semantically true sentences of the logic can be proved). This article is a technical mathematical article in the area of predicate logic
• A metric space (or uniform space) is complete if every Cauchy sequence in it converges. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined In the Mathematical field of Topology, a uniform space is a set with a uniform structure. In Mathematics, a Cauchy sequence, named after Augustin Cauchy, is a Sequence whose elements become arbitrarily close to each other as the sequence In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" See complete space. In Mathematical analysis, a Metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has
• In functional analysis, a subset S of a topological vector space V is complete if its span is dense in V. For functional analysis as used in psychology see the Functional analysis (psychology article In Mathematics, a topological vector space is one of the basic structures investigated in Functional analysis. In the mathematical subfield of Linear algebra, the linear span, also called the linear hull, of a set of vectors in a Vector In Topology and related areas of Mathematics, a Subset A of a Topological space X is called dense (in X) if If V is separable, it follows that any vector in V can be written as a (possibly infinite) linear combination of vectors from S. In Mathematics a Topological space is called separable if it contains a countable dense subset that is there exists a sequence \{ x_n In Mathematics, linear combinations are a concept central to Linear algebra and related fields of mathematics In the particular case of Hilbert spaces (or more generally, inner product spaces), an orthonormal basis is a set that is both complete and orthonormal. This article assumes some familiarity with Analytic geometry and the concept of a limit. In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. In Mathematics, an orthonormal basis of an Inner product space V (i In Linear algebra, two vectors in an Inner product space are orthonormal if they are orthogonal and both of unit length
• A measure space is complete if every subset of every null set is measurable. In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with In Mathematics, a null set is a set that is negligible in some sense. See complete measure. In Mathematics, a complete measure (or more precisely a complete measure space) is a measure space in which every Subset of every Null
• In commutative algebra, a commutative ring can be completed at an ideal (in the topology defined by the powers of the ideal). Commutative algebra is the branch of Abstract algebra that studies Commutative rings their ideals, and modules over such rings See Completion (ring theory). In Commutative algebra, the term completion refers to several related Functors on Topological rings and modules
• More generally, any topological group can be completed at a decreasing sequence of open subgroups. In Mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the
• In statistics, a statistic is called complete if it does not allow an unbiased estimator of zero. Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. A statistic (singular is the result of applying a function (statistical Algorithm) to a set of data. See completeness (statistics). In Statistics, completeness is a property of a Statistic for which the statistic optimally obtains information about the unknown parameters characterizing the distribution
• In graph theory, a complete graph is an undirected graph in which every pair of vertices has exactly one edge connecting them. In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects In the mathematical field of Graph theory, a complete graph is a Simple graph in which every pair of distinct vertices is connected by an
• In category theory, a category C is complete if every diagram from a small category to C has a limit; it is cocomplete if every such functor has a colimit. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Mathematics, a complete category is a category in which all small limits exist In Category theory, a branch of mathematics a diagram is the categorical analogue of a Indexed family in Set theory. In Category theory, a branch of Mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts In Mathematics, a complete category is a category in which all small limits exist In Category theory, a branch of Mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts
• In order theory and related fields such as lattice and domain theory, completeness generally refers to the existence of certain suprema or infima of some partially ordered set. Order theory is a branch of Mathematics that studies various kinds of Binary relations that capture the intuitive notion of ordering providing a framework for saying In Mathematics, a lattice is a Partially ordered set (also called a poset) in which every pair of elements has a unique Supremum (the elements' Domain theory is a branch of Mathematics that studies special kinds of Partially ordered sets (posets commonly called domains. In the mathematical area of Order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered In Mathematics the infimum of a Subset of some set is the Greatest element, not necessarily in the subset that is less than or equal to all elements of In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement Notable special usages of the term include the concepts of complete Boolean algebra, complete lattice, and complete partial order (cpo). This article is about a type of Mathematical structure. For the notion from Computer science, see Complete Boolean algebra (computer science. In Mathematics, a complete lattice is a Partially ordered set in which all subsets have both a Supremum (join and an Infimum (meet In Mathematics, directed complete partial orders and complete partial orders are special classes of Partially ordered sets These orders called dcpo 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. In Mathematics, an ordered field is a field together with a Total ordering of its elements that agrees in a certain sense with the field operations In the mathematical field of Order theory, a Partially ordered set is bounded complete if all of its subsets which have some Upper bound also Up to isomorphism there is only one complete ordered field: the field of real numbers (but note that this complete ordered field, which is also a lattice, is not a complete lattice). In Mathematics, the phrase " up to xxxx" indicates that members of an Equivalence class are to be regarded as a single entity for some purpose In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics, the real numbers may be described informally in several different ways
• In algebraic geometry, an algebraic variety is complete if it satisfies an analog of compactness. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety See complete algebraic variety. In Mathematics, in particular in Algebraic geometry, a complete algebraic variety is an Algebraic variety X, such that for any variety

## Computing

• In computational complexity theory, a problem P is complete for a complexity class C, under a given type of reduction, if P is in C, and every problem in C reduces to P using that reduction. Computational complexity theory, as a branch of the Theory of computation in Computer science, investigates the problems related to the amounts of resources In Computational complexity theory, a Computational problem is complete for a Complexity class when it is in a formal sense one of the "hardest" For example, each problem in the class NP-complete is complete for the class NP, under polynomial-time, many-one reduction. In Computational complexity theory, the Complexity class NP-complete (abbreviated NP-C or NPC) is a class of problems having two properties In Computational complexity theory, NP is one of the most fundamental Complexity classes The abbreviation NP refers to " N on-deterministic
• In computing, a data-entry field can autocomplete the entered data based on the prefix typed into the field; that capability is known as autocompletion. Computing is usually defined like the activity of using and developing Computer technology Computer hardware and software. Autocomplete is a feature provided by many Source code Text editors Word processors and Web browsers Autocomplete involves the program predicting
• In software testing, completeness has for goal the functional verification of call graph (between software item) and control graph (inside each software item).

## Economics, finance, and industry

• Complete market
• In auditing, completeness is one of the financial statement assertions that have to be ensured. In Economics, a complete market is one in which the complete set of possible gambles on future states-of-the-world can be constructed with existing Assets The most general definition of an audit is an evaluation of a person organization system process project or product For example, auditing classes of transactions. Rental expense which includes 12-month or 52-week payments should be all booked according to the terms agreed in the tenancy agreement.
• Oil or gas well completion, the process of making a well ready for production. In petroleum production completion is the process of making a well ready for production (or injection

## References

1. ^ a b Hunter, Geoffrey, Metalogic: An Introduction to the Metatheory of Standard First-Order Logic, University of California Pres, 1971

## completeness

### -noun

1. the state or condition of being complete
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