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 Turingcomputable 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.

The Abstract State Machine Thesis asserts that every classical algorithm is behaviorally equivalent to an abstract state machine. This thesis has been shown to follow from three natural postulates about algorithmic computation. Here, we prove that augmenting those postulates with an additional requirement regarding basic operations implies Church’s Thesis, namely, 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). In particular, this gives a natural axiomatization of Church’s Thesis, as Gödel and others suggested may be possible.

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Results going back to Turing and Gödel provide us with limitations on our ability to algorithmically decide the truth or falsity of mathematical assertions in a number of important mathematical contexts. Here we adapt some of this earlier work to very simplified mathematical models of discrete deterministic physical systems involving a few moving bodies (twelve point masses) in potentially infinite one dimensional space. There are two kinds of such limiting results that must be carefully distinguished. Results of the first kind state the nonexistence of any algorithm for determining whether any statement among a given set of statements is true or false. Results of the second kind are much deeper and present much greater challenges. They point to specific statements A, where we can neither prove nor refute A using accepted principles of mathematical reasoning. We give a brief survey of these limiting results. These include limiting results of the first kind: from number theory, group theory, and topology, in mathematics, and from idealized computing devices in theoretical computer science. We present a new limiting result of the first kind for simplified physical systems. We conjecture some related limiting results of the second kind, for simplified physical systems.