Recursive functionals and quantifiers of finite types revisited. V

Publisher Summary In the eighteen years since my last visit to Oslo, I have cherished a very pleasant recollection of an evening spent in the home of Thoralf Skolem. I am glad that the younger Norwegian logicians are keeping up the tradition so brilliantly started by him investigators from all parts have been pursuing higher and generalized recursion theory while I have been looking away. Also I am pleased to observe how many able In the theory developed in my original papers (1959 and 1963) on the subject I a m revisiting, there is a partial recursive function ϕ(σ 2 ,a) such that ϕ(λτ 1 θ,a,τ 1 ),a) to avoid this anomaly was not then pursued in depth is a partial recursive function of a for no completely defined (1963 LVI p. 110). A suggestion there (p.111) for extending the theory Platek in 1966 (to which my attention was drawn by Gandy 1967 p. 239) used my 1963 p. 110 example to show that the first recursion theorem (Kleene I M 1952a p. 348) does not hold in my 1959, 1963 theory ( as Ialsodidin 1963 LXVIpp. 120, 125). Platek developed a theory avoiding these anomalies by using what the called" hereditarily consistent" functionals. The present treatment (which for me is a resurrection of my 1963 p. 111 suggestion) gives the first recursion theorem a central role thinking along these lines before encountering Gandy 1967 and Platek 1966 exact relationship of the present treatment to Platek's and other work since 1963 remains to be clarified.