AxiomAn axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word ἀξίωμα (), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'. The precise definition varies across fields of study. In classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question.
Gödel numberingIn mathematical logic, a Gödel numbering is a function that assigns to each symbol and well-formed formula of some formal language a unique natural number, called its Gödel number. The concept was developed by Kurt Gödel for the proof of his incompleteness theorems. () A Gödel numbering can be interpreted as an encoding in which a number is assigned to each symbol of a mathematical notation, after which a sequence of natural numbers can then represent a sequence of symbols.
Semantics of logicIn logic, the semantics of logic or formal semantics is the study of the semantics, or interpretations, of formal and (idealizations of) natural languages usually trying to capture the pre-theoretic notion of entailment. The truth conditions of various sentences we may encounter in arguments will depend upon their meaning, and so logicians cannot completely avoid the need to provide some treatment of the meaning of these sentences.
SoundnessIn logic or, more precisely, deductive reasoning, an argument is sound if it is both valid in form and its premises are true. Soundness also has a related meaning in mathematical logic, wherein logical systems are sound if and only if every formula that can be proved in the system is logically valid with respect to the semantics of the system. In deductive reasoning, a sound argument is an argument that is valid and all of its premises are true (and as a consequence its conclusion is true as well).
George BooleGeorge Boole (buːl; 2 November 1815 – 8 December 1864) was a largely self-taught English mathematician, philosopher, and logician, most of whose short career was spent as the first professor of mathematics at Queen's College, Cork in Ireland. He worked in the fields of differential equations and algebraic logic, and is best known as the author of The Laws of Thought (1854) which contains Boolean algebra. Boolean logic is credited with laying the foundations for the Information Age. Boole was the son of a shoemaker.
Computably enumerable setIn computability theory, a set S of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable, listable, provable or Turing-recognizable if: There is an algorithm such that the set of input numbers for which the algorithm halts is exactly S. Or, equivalently, There is an algorithm that enumerates the members of S. That means that its output is simply a list of all the members of S: s1, s2, s3, ... . If S is infinite, this algorithm will run forever.
Symbol (formal)A logical symbol is a fundamental concept in logic, tokens of which may be marks or a configuration of marks which form a particular pattern. Although the term "symbol" in common use refers at some times to the idea being symbolized, and at other times to the marks on a piece of paper or chalkboard which are being used to express that idea; in the formal languages studied in mathematics and logic, the term "symbol" refers to the idea, and the marks are considered to be a token instance of the symbol.
Richard's paradoxIn logic, Richard's paradox is a semantical antinomy of set theory and natural language first described by the French mathematician Jules Richard in 1905. The paradox is ordinarily used to motivate the importance of distinguishing carefully between mathematics and metamathematics. Kurt Gödel specifically cites Richard's antinomy as a semantical analogue to his syntactical incompleteness result in the introductory section of "On Formally Undecidable Propositions in Principia Mathematica and Related Systems I".
Formalism (philosophy of mathematics)In the philosophy of mathematics, formalism is the view that holds that statements of mathematics and logic can be considered to be statements about the consequences of the manipulation of strings (alphanumeric sequences of symbols, usually as equations) using established manipulation rules. A central idea of formalism "is that mathematics is not a body of propositions representing an abstract sector of reality, but is much more akin to a game, bringing with it no more commitment to an ontology of objects or properties than ludo or chess.
Axiom schemaIn mathematical logic, an axiom schema (plural: axiom schemata or axiom schemas) generalizes the notion of axiom. An axiom schema is a formula in the metalanguage of an axiomatic system, in which one or more schematic variables appear. These variables, which are metalinguistic constructs, stand for any term or subformula of the system, which may or may not be required to satisfy certain conditions. Often, such conditions require that certain variables be free, or that certain variables not appear in the subformula or term.
