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Witness (mathematics)
In mathematical logic, a witness is a specific value to be substituted for variable of an existential statement of the form such that is true. Examples For example, a theory ''T'' of arithmetic is said to be inconsistent if there exists a proof in ''T'' of the formula "0 = 1". The formula I(''T''), which says that ''T'' is inconsistent, is thus an existential formula. A witness for the inconsistency of ''T'' is a particular proof of "0 = 1" in ''T''. Boolos, Burgess, and Jeffrey (2002:81) define the notion of a witness with the example, in which ''S'' is an ''n''-place relation on natural numbers, ''R'' is an ''(n+1)''-place recursive relation, and ↔ indicates logical equivalence In logic and mathematics, statements p and q are said to be logically equivalent if they have the same truth value in every model. The logical equivalence of p and q is sometimes expressed as p \equiv q, p :: q, \textsfpq, or p \iff q, depending ... (if and only if): :: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Mathematical Logic
Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and Mathematical analysis, analysis. In the early 20th century it was shaped by David Hilbert's Hilbert's program, program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |