A natural axiomatization of computability and proof of church's thesis

Bulletin of Symbolic Logic 14 (3):299-350 (2008)
Abstract
Church's Thesis asserts that the only numeric functions that can be calculated by effective means are the recursive ones, which are the same, extensionally, as the Turing-computable numeric functions. The Abstract State Machine Theorem states that every classical algorithm is behaviorally equivalent to an abstract state machine. This theorem presupposes three natural postulates about algorithmic computation. Here, we show that augmenting those postulates with an additional requirement regarding basic operations gives a natural axiomatization of computability and a proof of Church's Thesis, as Gödel and others suggested may be possible. In a similar way, but with a different set of basic operations, one can prove Turing's Thesis, characterizing the effective string functions, and--in particular--the effectively-computable functions on string representations of numbers
Keywords effective computation   recursiveness   computable functions   Church's Thesis   Turing's Thesis   abstract state machines   algorithms   encodings
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    Michael Rescorla (2012). Copeland and Proudfoot on Computability. Studies in History and Philosophy of Science Part A 43 (1):199-202.
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