Truth and Turing : Systems of Logic based on Ordinals

We should like to link Turing’s construction in Systems of Logic based on Ordinals on progressions of theories, with some recent similar looking progressions of axiomatisations of truth sets. However we first set the scene by sketching the original paper. It is interesting for two fundamental reasons. Firstly he introduces, in a rather understated fashion, the notion of a variant of his original Turing Machine, which was to be the ‘omachine’ for ‘oracle-machine.’ The latter is the well known version of the basic machine, the ‘a-machine’ , introduced in his 1936 paper ‘On computable numbers ’. The o -machine is of course the standard Turing machine equipped with an oracle tape. In the paper Turing describes rather a program that is allowed input at a stage of the computation when a special instruction is reached to ask for such input from the oracle tape. He envisaged then that in this way ‘non-computable’ functions could be introduced by calling for values. In the paper, after introducing this idea, he then repeats the argument that the halting problem was undecidable by such machines. He called this the ‘circularity question’: whether a particular TM M would eventually loop on a particular input. (I shall use TM to abbreviate Turing machines (with or without oracle tapes.) Just as the o -machine became the standard model for a computer (in Turing’s terms) so the o -machine has become for us the standard model for relativised computability : the notion that a set A ⊆ N can be computed ‘relative to a set B ⊆ N’ is that membership questions as to whether n ∈ A or not can be ‘reduced’ to finitely many similar queries of the set B, where we imagine the oracle tape of the machine to have the characteristic function of B written out as a series of 0’s and 1’s. We write nowadays in this case ‘A ≤T B’ for this relation. Sets A,B of numbers equivalent under ≤T are then declared to be in the same ‘Turing degree’ of incomputability. Thus the whole theory of such algorithmic degrees can be effected, using this model. This however only occupies a page and a half. This is not what the paper is about. It is only a tool in his investigation of the second fundamental idea to emerge from the paper: the notion of an ‘ordinal progression’. One has to admire the sweep of the paper: merely 8 years after Gödel’s paper on the Incompleteness Phenomenon, and only 3 years after his own paper On Computable Numbers he attempted to grapple with the incompleteness phenomena of formal systems by systematically extending theories T = T0 ⊆ T1 ⊆ · · · by adding at each stage a consistency statement about the preceding theory. The assumption is that our acceptance of a theory T somehow also impels us to accept its consistency. Who would work in Peano Arithmetic (PA) if they believed Con(PA) was false? And of course it is the consistency statement ‘Con(PA)’ that Gödel showed was a statement unprovable in PA (assuming that it was itself consistent). Martin Davis refers to the paper in his introduction in a volume of collected sources