Independent axiomatizability of sets of sentences
Some of the parameters investigated here are familiar: (i) the consequence relation (classical, intuitionistic, etc.), (ii) the cardinality of the sets, and whether they are (equivalent to) (iii) recursive(ly enumerable) sets. Less familiar is attention to (iv) the logical operators by which the sentences are built up and, for the equivalence between two sets X and Y, to (v) whether elements in, say Y are built up from the sentences in X themselves or merely from their propositional letters (top of p. 287). Striking results establish sensitivity to all these parameters. Some of the results are in Reznikoff's earlier published and unpublished work, others correct it. Attention is drawn to ranges, of parameters above, not studied here, for example (p. 273), infinitary propositional operators in (iv) for intuitionistic consequence, and to open problems likely to be overlooked even by a careful reader; for example (p. 292) whether, for classical consequence, every recursive first-order theory has an independent r.e. (and hence recursive) axiomatization. Since many arguments have an algebraic flavour the author notes interpretations in terms familiar from algebra. (But since universal algebra is involved there is not yet any compelling reason to shift the emphasis away from traditional logical formulations.) This line of thought is pursued by \textit{J. Grygiel} in work on so-called absolute independence [Stud. Logica 48, 77-84 (1989)]. The paper shows convincingly that the notion of independent axiomatization passes one test for membership in the household of knowledge: it has a number of memorable properties that are candidates for connecting it with other things one knows or wants to know. A further, also demanding test requires the discovery of such things. The author emphasizes in his introduction a complementary point: the (cl)aims in the early foundational literature for (the suggested consequences of) the notion above are in conflict with experience; as in the case of consistency as sufficient for correctness or recursiveness as even a first step towards decidability. Viewed this way independent axiomatizability, compared to recursive enumerability, lacks a counterpart of \textit{G. Higman}'s result on subgroups of finitely presented groups [Proc. R. Soc. Lond., Ser. A 262, 455-475 (1961; Zbl 0104.021)].
scientific article; zbMATH DE number 4125379
scientific article