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Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic. It was first proved by Kurt Gödel in 1929.

A first-order formula is called logically valid if it is true in every structure for its language. The completeness theorem shows that if a formula is logically valid then there is a finite deduction (a formal proof) of the formula. The deduction is a finite object that can be verified by hand or computer. This relationship between truth and provability establishes a close link between model theory and proof theory in mathematical logic.

An important consequence of the completeness theorem is that it is possible to enumerate the logical consequences of any effective first-order theory, by enumerating all the correct deductions using axioms from the theory.

Gödel's incompleteness theorem, referring to a different meaning of completeness, shows that if any sufficiently strong effective theory of arithmetic is consistent then there is a formula (depending on the theory) which can neither be proven nor disproven within the theory. Nevertheless the completeness theorem applies to these theories, showing that any logical consequence of such a theory is provable from the theory.

Reference:
http://en.wikipedia.org/wiki/Completeness_theorem

An infinitary logic is a logic that allows infinitely long statements and/or infinitely long proofs. Infinitary logics have different properties from those of standard first-order logic. In particular, infinitary logics often fail to be compact or complete. Notions of compactness and completeness that are equivalent in finitary logic sometimes are not so in infinitary logic. So for infinitary logics the notions of strong compactness and strong completeness are defined. In this article we shall be concerned with Hilbert-type infinitary logics, as these have been extensively studied and constitute the most straightforward extensions of finitary logic. These are not, however, the only infinitary logics around.

Considering whether a certain infinitary logic named Ω-logic is complete promises to throw light on the continuum hypothesis.

Reference:
http://en.wikipedia.org/wiki/Infinitary_logic

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끝!

In mathematics, higher-order logic is distinguished from first-order logic in a number of ways.

One of these is the type of variables appearing in quantifications; in first-order logic, roughly speaking, it is forbidden to quantify over predicates. See second-order logic for systems in which this is permitted.

Another way in which higher-order logic differs from first-order logic is in the constructions allowed in the underlying type theory. A higher-order predicate is a predicate that takes one or more other predicates as arguments. In general, a higher-order predicate of order n takes one or more (n − 1)th-order predicates as arguments, where n > 1. A similar remark holds for higher-order functions.

Higher-order logics are more expressive, but their properties, in particular with respect to model theory, make them less well-behaved for many applications. By a result of Gödel, classical higher-order logic does not admit a (recursively axiomatized) sound and complete proof calculus; however, such a proof calculus does exist which is sound and complete with respect to Henkin models.

Examples of higher order logics include the Church's Simple Theory of Types and calculus of constructions.

Reference:
http://en.wikipedia.org/wiki/Higher_order_logic

Finite model theory is a subfield of model theory that focuses on properties of logical languages, such as first-order logic, over finite structures, such as finite groups, graphs, databases, and most abstract machines. It focuses in particular on connections between logical languages and computation, and is closely associated with discrete mathematics, complexity theory, and database theory.

Many important results of first-order logic and classical model theory fail when restricted to finite structures, including the compactness theorem, the Craig interpolation lemma, the Los-Tarski preservation theorem, the Downward Löwenheim-Skolem theorem, and Gödel's completeness theorem. The essential problem is that in this context, first-order logic is not sufficiently expressive. By extending first-order logic with operators such as transitive closure and least fixed point, and by using fragments of second-order logic, we obtain new logics that have more interesting properties over finite structures.

One important subfield of finite model theory, descriptive complexity, connects the expressivity of various logical languages with the capabilities of various abstract machines. For example, if a language can be expressed in first-order logic with a least fixed point operator added, a Turing machine can determine in polynomial time (see P) whether a given string is in the language. Descriptive complexity allows results to be transferred between computational complexity and mathematical logic and gives additional evidence that the standard complexity classes are "natural." Neil Immerman states "In the history of mathematical logic most interest has concentrated on infinite structures....Yet, the objects computers have and hold are always finite. To study computation we need a theory of finite structures."

Another important result of finite model theory are the zero-one laws, which establish that many types of logical formulas hold for either almost all or almost no finite structures. For example, the proportion of graphs of size n that are connected approaches one as n approaches infinity, while the proportion that contain an isolated vertex approaches zero. In fact the same is true of any graph property that can be checked in polynomial time: it either approaches one or approaches zero. This has ramifications for randomized algorithms on large finite structures.

Finite model theory also studies finite restrictions of logic, such as first-order logic with only a fixed limit of k variables.

Reference:
http://en.wikipedia.org/wiki/Finite_model_theory

Algebra is a branch of mathematics concerning the study of structure, relation and quantity. The name is derived from the treatise written by the Persian mathematician, astronomer, astrologer and geographer, Muhammad bin Mūsā al-Khwārizmī titled Kitab al-Jabr wa-l-Muqabala (meaning "The Compendious Book on Calculation by Completion and Balancing"), which provided symbolic operations for the systematic solution of linear and quadratic equations.

Together with geometry, analysis, combinatorics, and number theory, algebra is one of the main branches of mathematics. Elementary algebra is often part of the curriculum in secondary education and provides an introduction to the basic ideas of algebra, including effects of adding and multiplying numbers, the concept of variables, definition of polynomials, along with factorization and determining their roots.

Algebra is much broader than elementary algebra and can be generalized. In addition to working directly with numbers, algebra covers working with symbols, variables, and set elements. Addition and multiplication are viewed as general operations, and their precise definitions lead to structures such as groups, rings and fields.

Reference:
http://en.wikipedia.org/wiki/Algebra

Secondary education is the stage of education following primary school. Secondary education is generally the final stage of compulsory education. The next stage of education is usually college or university. Secondary education is characterized by transition from the typically compulsory, comprehensive primary education for minors to the optional, selective tertiary, "post-secondary", or "higher" education (e.g., university, vocational school) for adults. Depending on the system, schools for this period or a part of it may be called secondary schools, high schools, gymnasia, lyceums, middle schools, colleges, vocational schools and preparatory schools, and the exact meaning of any of these varies between the systems.

The exact boundary between primary and secondary education varies from country to country and even within them, but is generally around the fifth to the tenth year of education. Secondary education occurs mainly during the teenage years. In the United States and Canada primary and secondary education together are sometimes referred to as K-12 education.

The purpose of secondary education can be to give common knowledge, to prepare for either higher education or vocational education, or to train directly to a profession.

Reference:
http://en.wikipedia.org/wiki/Secondary_education