## A note on the Entscheidungsproblem (pp. 40-41) and Correction to A note on the Entscheidungsproblem (pp. 101-102) and Review of A. M. Turing. On Computable numbers, with an application to the Entscheidungsproblem (pp. 42-43) (Church) WITH Finite combinatory processes-formulation I (pp. 103-105) (Post) WITH Computability and lambda-definability (pp. 153-164) and The Ø-Function in lambda-K-Conversion (pp. 164-166) (Turing) in Journal of Symbolic Logic, Volume 1, 1936 AND Volume 2, 1937

1937. 1st Edition. FIRST EDITION OF A COLLECTION OF LANDMARK PAPERS, here housed together in Volumes 1 and 2 of The Journal of Symbolic Logic. Collectively, the volumes contain some of the most important contributions to the fields of mathematical and computational mathematics ever written. The first and second papers are seminal works by Alonzo Church on Hilbert's Entscheidungsproblem problem; collectively, the papers are often referred to as Church's Theorem. Here Church was the first to demonstrate that David Hilbert's Entscheidungsproblem (positing the question as to whether any given mathematical statement could, in principle, be found true or false) was untrue. "Church showed that it is impossible to decide algorithmically whether statements in arithmetic are true or false. It follows that a general solution to the problem does not exist; equivalently, first-order logic is undecidable" (Princeton Companion to Mathematics, 816). Church solved the problem by devising and employing lambda-definability, or the 'lambda-calculus.' A Function of positive integers is said to be lambda-definable if the values of the function can be calculated by a process of repeated substitution. He "had earlier shown the 'existence of an unsolvable problem of elementary number theory,' but [this paper] was the first to put his findings into the exact form of an answer to Hilbert's problem. Church's paper bears on the question of what is computable, a problem addressed more directly by Alan Turing in his paper 'On computable numbers' published a few months later. The notion of an 'effective' or 'mechanical' computation in logic and mathematics because known as the Church-Turing thesis (Hook & Norman: Origins of Cyberspace, 250). In the third paper, Church's review of Turing's paper 'On computable numbers,' Church coined the term 'Turing Machine.' Here Church acknowledges Turing's method of solving Hilbert's problem as more satisfying, stating that "computability by a Turing machine... has the advantage of making the identification with effectiveness in the ordinary sense evident immediately" (Church). In the fourth paper, "Finite combinatory processes-Formulation I," (also seminal) the Polish-American mathematician Emil Post describes "a model of extreme simplicity which he conjectured in 'logically equivalent to recursiveness,' and which was later proved to be so... Post's model of a computation, essentially a logic-automaton, differs from the Turing-machine model in a further 'atomization' of the acts a human 'computer' would perform during a computation" (Wikipedia). "Post's paper was intended to fill a conceptual gap in Church's paper on 'An unsolvable problem...'. Church had answered in the negative Hilbert's Entscheidungsproblem, but failed to provide the assertion that any such definitive method could be expressed as a formula in Church's lambda-calculus. Post proposed that a definite method would be one written in the form of instructions to a mind-less worker operating on an infinite line of 'boxes' (equivalent to the Turing machines 'tape')" (Origins of Cyberspace, 356). "Post suggests a computation scheme by which a 'worker' can solve all problems in symbolic logic by performing only machinelike 'primitive acts.' Remarkably, the instructions given to the 'worker' in Post's paper and [those of Turing] top a Universal Turing Machine were identical (A Computer Perscpective, 125). Item #374

CONDITION & DETAILS: [No place]: The Association for Symbolic Logic. Royal 8vo. (10 x 7 inches; 250 x 175mm). Ex-libris: Bibliotheque Universite de Moncton; Volume I bearing only a very faint stamp on the front blank flyleaf and a binder's stamp at the foot of the front pastedown; no markings on the spine; Volume II bearing a very faint stamp on the second blank flyleaf and at the head of the title page; small binder's stamp at the foot of the front pastedown; no markings on the spine. Volumes 1 and 2 in full. Volume 1: [6], 218, [4]. Volume 2: [iv], 188, [4]. Recently rebound in black cloth; gilt-lettered at the spine. Tightly and very solidly bound. The title page of Volume I is supplied in facsimile. All else, near fine condition throughout.

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Price:
$2,500.00
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