Information:  - Mathematics (from Greek  "máthma", knowledge, study, learning) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics.  - Science is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe.  - Physics (from , from "phúsis" "nature") is the natural science that involves the study of matter and its motion and behavior through space and time, along with related concepts such as energy and force. One of the most fundamental scientific disciplines, the main goal of physics is to understand how the universe behaves.  - William Craig ( November 13 , 1918 -- January 13 , 2016 ) was an American academic and philosopher , who taught at the University of California , Berkeley , in Berkeley , California . His research interests included mathematical logic , and the philosophy of science , and he is best known for the Craig interpolation theorem .  - Theoretical computer science, or TCS, is a division or subset of general computer science and mathematics that focuses on more abstract or mathematical aspects of computing and includes the theory of computation.  - Nevada (Spanish for "snowy"; see pronunciations) is a state in the Western, Mountain West, and Southwestern regions of the United States of America. Nevada is the 7th most extensive, the 34th most populous, and the 9th least densely populated of the 50 United States. Nearly three-quarters of Nevada's people live in Clark County, which contains the Las VegasParadise metropolitan area where three of the state's four largest incorporated cities are located. Nevada's capital is Carson City. Nevada is officially known as the "Silver State" because of the importance of silver to its history and economy. It is also known as the "Battle Born State", because it achieved statehood during the Civil War (the words "Battle Born" also appear on the state flag); as the "Sage-brush State", for the native plant of the same name; and as the "Sage-hen State". Nevada borders Oregon to the northwest, Idaho to the northeast, California to the west, Arizona to the southeast and Utah to the east.  - First-order logic  also known as first-order predicate calculus and predicate logic  is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects and allows the use of sentences that contain variables, so that rather than propositions such as "Socrates is a man" one can have expressions in the form "there exists X such that X is Socrates and X is a man" where "there exists" is a quantifier and "X" is a variable. This distinguishes it from propositional logic, which does not use quantifiers.  - Logic in computer science covers the overlap between the field of logic and that of computer science. The topic can essentially be divided into three main areas:  - In mathematical logic, Craig's interpolation theorem is a result about the relationship between different logical theories. Roughly stated, the theorem says that if a formula  implies a formula , and the two have at least one atomic variable symbol in common, then there is a third formula , called an interpolant, such that every nonlogical symbol in  occurs both in  and ,  implies , and  implies . The theorem was first proved for first-order logic by William Craig in 1957. Variants of the theorem hold for other logics, such as propositional logic. A stronger form of Craig's theorem for first-order logic was proved by Roger Lyndon in 1959; the overall result is sometimes called the CraigLyndon theorem.  - Philosophy of science is a branch of philosophy concerned with the foundations, methods, and implications of science. The central questions of this study concern what qualifies as science, the reliability of scientific theories, and the ultimate purpose of science. This discipline overlaps with metaphysics, ontology, and epistemology, for example, when it explores the relationship between science and truth.  - The University of California (UC) is a public university system in the U.S. state of California. Under the California Master Plan for Higher Education, the University of California is a part of the state's three-system public higher education plan, which also include the California State University system and the California Community Colleges System.  - Metaphysics is a branch of philosophy investigating the fundamental nature of being and the world that encompasses it. Metaphysics attempts to answer two basic questions:  - Los Angeles (Spanish for "The Angels"), officially the City of Los Angeles and often known by its initials L.A., is the second-most populous city in the United States (after New York City), the most populous city in California and the county seat of Los Angeles County. Situated in Southern California, Los Angeles is known for its mediterranean climate, ethnic diversity, sprawling metropolis, and as a major center of the American entertainment industry. Los Angeles lies in a large coastal basin surrounded on three sides by mountains reaching up to and over .  - A public university is a university that is predominantly funded by public means through a national or subnational government, as opposed to private universities. Whether a national university is considered public varies from one country (or region) to another, largely depending on the specific education landscape.   - Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects.  - Philosophy (from Greek , "philosophia", literally "love of wisdom") is the study of general and fundamental problems concerning matters such as existence, knowledge, values, reason, mind, and language. The term was probably coined by Pythagoras (c. 570  c. 495 BC). Philosophical methods include questioning, critical discussion, rational argument and systematic presentation. Classic philosophical questions include: Is it possible to know anything and to prove it? What is most real? However, philosophers might also pose more practical and concrete questions such as: Is there a best way to live? Is it better to be just or unjust (if one can get away with it)? Do humans have free will?  - Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. It bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science. The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems. Mathematical logic is often divided into the fields of set theory, model theory, recursion theory, and proof theory. These areas share basic results on logic, particularly first-order logic, and definability. In computer science (particularly in the ACM Classification) mathematical logic encompasses additional topics not detailed in this article; see Logic in computer science for those.  - A formal system or logical calculus is any well-defined system of abstract thought based on the model of mathematics. A formal system need not be mathematical as such; for example, Spinoza's "Ethics" imitates the form of Euclid's "Elements".. Spinoza employed Euclidiean elements such as "axioms" or "primitive truths", rules of inferences etc. so that a calculus can be build using these. For nature of such primitive truths, one can consult Tarski's "Concept of truth for a formalized language".   - The Pacific Ocean is the largest of the Earth's oceanic divisions. It extends from the Arctic Ocean in the north to the Southern Ocean (or, depending on definition, to Antarctica) in the south and is bounded by Asia and Australia in the west and the Americas in the east.  - In mathematics, model theory is the study of classes of mathematical structures (e.g. groups, fields, graphs, universes of set theory) from the perspective of mathematical logic. The objects of study are models of theories in a formal language. A set of sentences in a formal language is called a theory; a model of a theory is a structure (e.g. an interpretation) that satisfies the sentences of that theory.  - Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathematics. In this latter sense, the distinction between foundations of mathematics and philosophy of mathematics turns out to be quite vague. Foundations of mathematics can be conceived as the study of the basic mathematical concepts (number, geometrical figure, set, function, etc.) and how they form hierarchies of more complex structures and concepts, especially the fundamentally important structures that form the language of mathematics (formulas, theories and their models giving a meaning to formulas, definitions, proofs, algorithms, etc.) also called metamathematical concepts, with an eye to the philosophical aspects and the unity of mathematics. The search for foundations of mathematics is a central question of the philosophy of mathematics; the abstract nature of mathematical objects presents special philosophical challenges.  - The City of New York, often called New York City or simply New York, is the most populous city in the United States. With an estimated 2015 population of 8,550,405 distributed over a land area of just , New York City is also the most densely populated major city in the United States. Located at the southern tip of the state of New York, the city is the center of the New York metropolitan area, one of the most populous urban agglomerations in the world. A global power city, New York City exerts a significant impact upon commerce, finance, media, art, fashion, research, technology, education, and entertainment, its fast pace defining the term "New York minute". Home to the headquarters of the United Nations, New York is an important center for international diplomacy and has been described as the cultural and financial capital of the world.  - Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories. Emphasis on metamathematics (and perhaps the creation of the term itself) owes itself to David Hilbert's attempt to secure the foundations of mathematics in the early part of the 20th Century. Metamathematics provides "a rigorous mathematical technique for investigating a great variety of foundation problems for mathematics and logic" (Kleene 1952, p. 59). An important feature of metamathematics is its emphasis on differentiating between reasoning from inside a system and from outside a system. An informal illustration of this is categorizing the proposition "2+2=4" as belonging to mathematics while categorizing the proposition "'2+2=4' is valid" as belonging to metamathematics.  - The University of California, Berkeley, (also referred to as Berkeley, UC Berkeley, and Cal) is a public research university located in Berkeley, California. Founded in 1868, Berkeley is the oldest of the ten research universities affiliated with the University of California system, and is often cited as the top public university in the United States and around the world.  - Ontology is the philosophical study of the nature of being, becoming, existence or reality as well as the basic categories of being and their relations. Traditionally listed as a part of the major branch of philosophy known as metaphysics, ontology often deals with questions concerning what entities exist or may be said to exist and how such entities may be grouped, related within a hierarchy, and subdivided according to similarities and differences. Although ontology as a philosophical enterprise is highly theoretical, it also has practical application in information science and technology, such as ontology engineering.  - Epistemology is the branch of philosophy concerned with the theory of knowledge.  - Proof theory is a major branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of the logical system. As such, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.  - California is the most populous state in the United States and the third most extensive by area. Located on the western (Pacific Ocean) coast of the U.S., California is bordered by the other U.S. states of Oregon, Nevada, and Arizona and shares an international border with the Mexican state of Baja California. The state capital is Sacramento. Los Angeles is California's most populous city, and the country's second largest after New York City. The state also has the nation's most populous county, Los Angeles County, and its largest county by area, San Bernardino County.    'william craig ' is related to which object entity through the relation of 'field of work'?  Choices: - asia  - classification  - computing  - economy  - education  - engineering  - epistemology  - ethics  - fashion  - finance  - history  - information  - language  - logic  - mathematical logic  - mathematics  - metaphysics  - model theory  - nature  - ontology  - philosophy  - physics  - plant  - proof theory  - research  - science  - set theory  - spanish  - technology  - theoretical computer science  - united states of america
Answer:
proof theory