``Pathologies'' in two syntactic categories of partial maps (Q1115866)

In connection with \textit{R. A. Di Paola} and \textit{A. Heller}'s project [J. Symb. Logic 52, 594-635 (1987; Zbl 0649.03032)] the categories S and S' of partial maps are considered. In S (S') these maps are taken modulo PA- provable equality (modulo equality being PA-provable for any fixed value of arguments). It is shown that S and S' are not dominical categories and hence the Di Paola-Heller's notion of totality does not correspond to the expected one. The author suggests to use the notion of Rosolini pointed p-categories. According to him all the results corresponding to the Di Paola-Heller's project are negative. Positive aspect of the paper: relevant examples of models of Rosolini's p-categories; syntactic classification of totality by category theoretic means; extension to S and S' (and all p-categories with Turing morphism) of many important properties of recursion categories; the relevant difference between p- categories with Turing morphism and dominical categories.