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In programming language theory and proof theory, the Curry–Howard correspondence is the direct relationship between computer programs and mathematical proofs.It is also known as the Curry–Howard isomorphism or equivalence, or the proofs-as-programs and propositions-or formulae-as-types interpretation.
According to J. Mackenzie, the first edition of the book "deserves to become the standard textbook in its field", which he reiterated for the second edition. [10] [11] Reviewers particularly noted the book's utility as either a supplement to standard logic textbooks or as a primary text for courses on non-classical logic.
It is to be regretted that this first comprehensive and thorough-going presentation of a mathematical logic and the derivation of mathematics from it [is] so greatly lacking in formal precision in the foundations (contained in 1– 21 of Principia [i.e., sections 1– 5 (propositional logic), 8–14 (predicate logic with identity/equality), 20 ...
P. Oxy. 29, one of the oldest surviving fragments of Euclid's Elements, a textbook used for millennia to teach proof-writing techniques. The diagram accompanies Book II, Proposition 5. [1] A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the
In modern logic texts, Gödel's completeness theorem is usually proved with Henkin's proof, rather than with Gödel's original proof. Henkin's proof directly constructs a term model for any consistent first-order theory. James Margetson (2004) developed a computerized formal proof using the Isabelle theorem prover. [4] Other proofs are also known.
In mathematics and logic, a direct proof is a way of showing the truth or falsehood of a given statement by a straightforward combination of established facts, usually axioms, existing lemmas and theorems, without making any further assumptions. [1]
In proof theory and mathematical logic, sequent calculus is a family of formal systems sharing a certain style of inference and certain formal properties. The first sequent calculi systems, LK and LJ, were introduced in 1934/1935 by Gerhard Gentzen [1] as a tool for studying natural deduction in first-order logic (in classical and intuitionistic versions, respectively).
Several English translations have appeared in print, and the paper has been included in two collections of classic mathematical logic papers. The paper contains Gödel's incompleteness theorems, now fundamental results in logic that have many implications for consistency proofs in mathematics. The paper is also known for introducing new ...