Gödel's incompleteness theoremsGödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The theorems are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible. The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.
Logical atomismLogical atomism is a philosophical view that originated in the early 20th century with the development of analytic philosophy. It holds that the world consists of ultimate logical "facts" (or "atoms") that cannot be broken down any further, each of which can be understood independently of other facts. Its principal exponent was the British philosopher Bertrand Russell. It is also widely held that the early works of his Austrian-born pupil and colleague, Ludwig Wittgenstein, defend a version of logical atomism, though he went on to reject it in his later Philosophical Investigations.
Philosophical InvestigationsPhilosophical Investigations (Philosophische Untersuchungen) is a work by the philosopher Ludwig Wittgenstein, published posthumously in 1953. Philosophical Investigations is divided into two parts, consisting of what Wittgenstein calls, in the preface, Bemerkungen, translated by Anscombe as "remarks". A survey among American university and college teachers ranked the Investigations as the most important book of 20th-century philosophy.
FactA fact is a true datum about one or more aspects of a circumstance. Standard reference works are often used to check facts. Scientific facts are verified by repeatable careful observation or measurement by experiments or other means. For example, "This sentence contains words." accurately describes a linguistic fact, and "The sun is a star" accurately describes an astronomical fact. Further, "Abraham Lincoln was the 16th President of the United States" and "Abraham Lincoln was assassinated" both accurately describe historical facts.
Truth functionIn logic, a truth function is a function that accepts truth values as input and produces a unique truth value as output. In other words: The input and output of a truth function are all truth values; a truth function will always output exactly one truth value; and inputting the same truth value(s) will always output the same truth value.
Atomic sentenceIn logic and analytic philosophy, an atomic sentence is a type of declarative sentence which is either true or false (may also be referred to as a proposition, statement or truthbearer) and which cannot be broken down into other simpler sentences. For example, "The dog ran" is an atomic sentence in natural language, whereas "The dog ran and the cat hid" is a molecular sentence in natural language. From a logical analysis point of view, the truth or falsity of sentences in general is determined by only two things: the logical form of the sentence and the truth or falsity of its simple sentences.
Frank Ramsey (mathematician)Frank Plumpton Ramsey (ˈræmzi; 22 February 1903 – 19 January 1930) was a British philosopher, mathematician, and economist who made major contributions to all three fields before his death at the age of 26. He was a close friend of Ludwig Wittgenstein and, as an undergraduate, translated Wittgenstein's Tractatus Logico-Philosophicus into English. He was also influential in persuading Wittgenstein to return to philosophy and Cambridge. Like Wittgenstein, he was a member of the Cambridge Apostles, the secret intellectual society, from 1921.
MetaphilosophyMetaphilosophy, sometimes called the philosophy of philosophy, is "the investigation of the nature of philosophy". Its subject matter includes the aims of philosophy, the boundaries of philosophy, and its methods. Thus, while philosophy characteristically inquires into the nature of being, the reality of objects, the possibility of knowledge, the nature of truth, and so on, metaphilosophy is the self-reflective inquiry into the nature, aims, and methods of the activity that makes these kinds of inquiries, by asking what is philosophy itself, what sorts of questions it should ask, how it might pose and answer them, and what it can achieve in doing so.
