Some strongly undecidable natural arithmetical problems, with an application to intuitionistic theories

Although Church and Turing presented their path-breaking undecidability results immediately after their explication of effective decidability in 1936, it has been generally felt that these results do not have any direct bearing on ordinary mathematics but only contribute to logic, metamathematics and the theory of computability. Therefore it was such a celebrated achievement when Yuri Matiyasevich in 1970 demonstrated that the problem of the solvability of Diophantine equations is undecidable. His work was building essentially on the earlier work by Julia Robinson, Martin Davis and Hilary Putnam (1961), who had showed that the problem of solvability of exponential Diophantine equations is undecidable. One should note, however, that although it was only Matiyasevich's result which finally solved Hilbert's tenth problem, already the earlier result had provided a perfectly natural problem of ordinary number theory which is undecidable.Nevertheless, both the set of Diophantine equations with solutions and the set of exponential Diophantine equations with solutions are still semi-decidable, that is, recursively enumerable (i.e., Σ10); if an equation in fact has a solution, this can be eventually verified. More generally, they are — as are their complements, the sets of equations with no solutions, which are Π10, — also Trial and Error decidable (Putnam [1965]), or decidable in the limit (Shoenfield [1959]), for every Δ20 set is (and conversely). This last-mentioned natural “liberalized” notion of decidability has begun more recently to play an essential role e.g., in so-called Formal Learning Theory (see e.g., Osherson, Stob, and Weinstein [1986], Kelly [1996]).

Can there be no nonrecursive functions?

In 1936 Alonzo Church proposed the following thesis: Every effectively computable number-theoretic function is general recursive. The classical mathematician can easily give examples of nonrecursive functions, e.g. by diagonalizing a list of all general recursive functions. But since no such function has been found which is effectively computable, there is as yet no classical evidence against Church's Thesis.The intuitionistic mathematician, following Brouwer, recognizes at least two notions of function: the free-choice sequence (or ordinary number-theoretic function, thought of as the ever-finite but ever-extendable sequence of its values) and the sharp arrow (or effectively definable function, all of whose values can be specified in advance).

XLVII.—Non-Associative Arithmetics

Introduction and SummaryThe systems of “partitive numbers” introduced in this paper differ from ordinary number systems in being subject to non-associative addition. They are intended primarily to serve as the indices of powers in algebraic systems having non-associative multiplication, or as the coefficients of multiples in systems with non-associative addition, but are defined more generally than is probably necessary for these purposes. They are essentially the same as root-trees (Setzbäume) with non-branching knots other than terminal knots ignored, with operations of addition and multiplication defined.Partitive numbers are of two kinds, partitioned cardinals and partitioned serials, defined respectively as the partition-types of repeatedly partitioned classes and series. For each kind, multiplication is binary (i.e. any ordered pair has a unique product) and associative. Addition is in general a free operation (i.e. the summands are not limited to two, and indeed, assuming the multiplicative axiom, may form an infinite class or series); but it is non-associative, which means that for example a + b + c (involving one operation of addition) is distinguished from (a + b) + c and a + (b + c) (involving two operations). A one-sided distributive law is obeyed:Partitioned cardinals are commutative in addition.