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The concept of proof is formalized in the field of mathematical logic. [ 12] A formal proof is written in a formal language instead of natural language. A formal proof is a sequence of formulas in a formal language, starting with an assumption, and with each subsequent formula a logical consequence of the preceding ones.
Proof theory is a major branch [1] of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as lists, boxed lists, or trees, which are constructed ...
Sir Andrew John Wiles. Wiles's proof of Fermat's Last Theorem is a proof by British mathematician Andrew Wiles of a special case of the modularity theorem for elliptic curves. Together with Ribet's theorem, it provides a proof for Fermat's Last Theorem. Both Fermat's Last Theorem and the modularity theorem were believed to be impossible to ...
By Gödel's completeness result, it must hence have a natural deduction proof (shown right). 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 . The completeness theorem applies to any first-order theory: If T is ...
In programming language theory and proof theory, the Curry–Howard correspondence (also known as the Curry–Howard isomorphism or equivalence, or the proofs-as-programs and propositions- or formulae-as-types interpretation) is the direct relationship between computer programs and mathematical proofs. It is a generalization of a syntactic ...
3-540-63698-6. Proofs from THE BOOK is a book of mathematical proofs by Martin Aigner and Günter M. Ziegler. The book is dedicated to the mathematician Paul Erdős, who often referred to "The Book" in which God keeps the most elegant proof of each mathematical theorem. During a lecture in 1985, Erdős said, "You don't have to believe in God ...
Download as PDF; Printable version; Appearance. ... This is a list of mathematical theories. ... Proof theory; Quantum theory; Queue theory;
Gödel left a fourteen-point outline of his philosophical beliefs in his papers. [1] Points relevant to the ontological proof include: 4. There are other worlds and rational beings of a different and higher kind. 5. The world in which we live is not the only one in which we shall live or have lived. 13.