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On Computable Numbers, with an Application to the Entscheidungsproblem
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This groundbreaking research by Alan Turing, a pioneer in computer science, introduces the concept of what we now call the \'Turing Machine.\' This is a theoretical device that Turing used to explore the limits of what can be calculated or computed. In simpler terms, it's like an abstract 'robot' that follows very specific rules to manipulate symbols on a strip of tape, which represents data.The paper addresses a critical question in mathematics and logic known as the \'Entscheidungsproblem\' (German for 'decision problem'). This problem, posed by mathematician David Hilbert, asks whether there is a definitive method (an algorithm) that can be applied to any mathematical statement to determine its truth or falsehood. Turing's work essentially shows that there are indeed limits to what can be computed; there are some mathematical problems that no algorithm can solve.Through the Turing Machine concept, Turing sets the stage for the development of modern computers. He shows that such a machine, simple as it might seem, could, in theory, perform any computation that a modern computer can, given enough time and memory. This is why we refer to modern computers as \'Turing-complete\' systems.The implications of this work are profound, extending beyond mathematics into the realms of computer science, philosophy, and cognitive science, influencing how we understand not only computation but also the nature of human thought and consciousness.In summary, \'On Computable Numbers, with an Application to the Entscheidungsproblem\' is not just a cornerstone of computer science; it's a key piece in the puzzle of understanding the limits of computation and the potential of artificial intelligence.

Otf COMPUTABLE NUMBERS. 9. The extent of the computable numbers. No attempt has yet been made to show that the " computable " numbers include all numbers which would naturally be regarded as computable. Al I arguments which can be given are bound to be, fundamentally, appeals to intuition, and for this reason rather unsatisfactory mathematically. The real question at issue is " What are the possible processes which can be carried out in computing a number?" The arguments which I shall use are of three kinds.
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(a) A direct appeal to intuition. (6) A proof of the equivalence of two definitions (in case the new definition has a greater intuitive appeal). (c) Giving examples of large classes of numbers which are computable. Once it is granted that computable numbers are all c: computable"". several other propositions of the same character follow. In particular, it follows that, if there is a general process for determining whether a formula of the Hilbert function calculus is provable, then the determination can bo carried out by a machine. I. [Type (a)]. This argument is only an elaboration of the ideas of 1.
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Computing is normally done by writing certain symbols on paper. "We may suppose this paper is divided into squares like a child's arithmetic book. In elementary arithmetic the two-dimensional character of the paper is sometimes used. But such a use is always avoidable, and I think that it will be agreed that the two-dimensional character of paper is no essential I assume then that the computation is carried out on of computation. one-dimensional paper, i.e. on a tape divided into squares. I shall also suppose that the number of symbols which may be printed is finite. If we were to allow an infinity of symbols, then there would be symbols differing to an arbitrarily small extent j. The effect of this restriction of the number of symbols is not very serious. It is always possible to use sequences of symbols in the place of single symbols. Thus an Arabic numeral such as
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The symbol is defined as a set of points in this square, viz. the set occupied by printer's ink. If these sets are restricted to be measurable, we can define the "distance" between two symbols as the cost of transforming one symbol into the other if the cost of moving unit area of printer's ink unit distance is unity, and there is an infinite supply of ink at x = 2. y = 0. With this topology the symbols form a conditionally compact space. 249 250 A. M. TUBING [NOV. 12,
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