Two forms of the genus Mytilus, M. edulis L. and M. galloprovincialis Lmk inhabit the rocky shores of Devon and Cornwall (Hepper, 1957). According to Gosling (1984), controversy has existed since 1860 as to whether M. galloprovincialis is a separate species or a subspecies of M. edulis. In a review of the systematic status of M. galloprovincialis, Gosling (1984) analysed the criteria used to identify and classify this form of Mytilus. Gosling did not reach any firm conclusion on the systematic status of these mytilids, but did suggest that the evidence favoured placing M. galloprovincialis as a variety, ecotype or even subspecies of M. edulis.

Kirby and Paris have exhibited combinatorial algorithms whose computations always terminate, but for which termination is not provable in elementary arithmetic. However, termination of these computations can be proved by adding an axiom first introduced by Goodstein in 1944. Our purpose is to investigate this axiom of Goodstein, and some of its variants, and to show that these are potentially adequate to prove termination of computations of a wide class of algorithms. We prove that many variations of Goodstein's axiom are equivalent, over elementary arithmetic, and contrast these results with those recently obtained for Kruskal's theorem.

In a recent article in this Journal (see [3]), J.P. Jones states and proves a theorem which purports to give an “absolute epistemological upper bound on the complexity of mathematical proofs” for recursively axiomatizable theories. However, Jones' statement of this result is misleading, and in fact defective, as can be seen by a close analysis of it. Such an analysis is the object of the present note.

The main point is that Jones' “epistemological bound” can in no way be considered a computational bound on the complexity of proofs. Not only is the “proof-theoretic interpretation of the number 243” contained in Jones' article objectionable but, more fundamentally, there is in a strong sense no way one can hope to recover anything like the full force suggested by Jones' original statement of the theorem.

We wish to insist that our comments concern only the difficulties surrounding Jones' Corollary 1 on p. 338 of his article and not his ingenious construction of universal Diophantine representations of r.e. sets presented in the same article.