Abstract
The common formulation of the Church-Turing thesis runs as follows:
Every computable partial function is computable by a Turing machine
Where by partial function I mean a function from a subset of natural numbers to natural numbers. As most textbooks relate, the thesis makes a connection between an intuitive notion (computable function) and a formal one (Turing machine). The claim is that the definition of a Turing machine captures the pre-analytic intuition that underlies the concept computation. Formulated in this way the Church-Turing thesis cannot be proved in the same sense that a mathematical proposition is provable. However, it can be refuted by an example of a function which is not Turing computable, but is nevertheless calculable by some procedure that is intuitively acceptable.
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Pitowsky, I. (2007). From Logic to Physics: How the Meaning of Computation Changed over Time. In: Cooper, S.B., Löwe, B., Sorbi, A. (eds) Computation and Logic in the Real World. CiE 2007. Lecture Notes in Computer Science, vol 4497. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73001-9_64
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DOI: https://doi.org/10.1007/978-3-540-73001-9_64
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