The class of all recursive functions fails to possess a natural hierarchical structure, generated predicatively from "within". On the other hand, many (proof-theoretically significant) sub-recursive classes do. This paper attempts to measure the limit of predicative generation in this context, by classifying and characterizing those (predictably terminating) recursive functions which can be successively defined according to an autonomy condition of the form: allow recursions only over well-orderings which have already been "coded" at previous levels. The question is: how can a (...) recursion code a well-ordering? The answer lies in Girard's theory of dilators, but is reworked here in a quite different and simplified framework specific to our purpose. The "accessible" recursive functions thus generated turn out to be those provably recursive in $(\prod_{1}^{1}-CA)_{0}$. (shrink)

Edited proceedings of an interdisciplinary symposium on consciousness held at the University of Cambridge in January 1978. Includes a foreword by Freeman Dyson. Chapter authors: G. Vesey, R.L. Gregory, H.C. Longuet-Higgins, N.K. Humphrey, H.B. Barlow, D.M. MacKay, B.D. Josephson, M. Roth, V.S. Ramachandran, S. Padfield, and (editorial summary only) E. Noakes. A scanned pdf is available from this web site (philpapers.org), while alternative versions more suitable for copying text are available from https://www.repository.cam.ac.uk/handle/1810/245189. -/- Page numbering convention for the pdf version (...) viewed in a pdf viewer is as follows: 'go to page n' accesses the pair of scanned pages 2n and 2n+1. Applicable licence: CC Attribution-NonCommercial-ShareAlike 2.0. (shrink)

Here we prove that if T and T′ are strongly minimal theories, where T′ satisfies a certain property related to triviality and T does not, and T′ is model complete, then there is no computable embedding of Mod(T) into Mod(T′). Using this, we answer a question from [4], showing that there is no computable embedding of VS into ZS, where VS is the class of infinite vector spaces over Q, and ZS is the class of models of Th(Z, S). Similarly, (...) we show that there is no computable embedding of ACF into ZS, where ACF is the class of algebraically closed fields of characteristic 0. (shrink)

Introduction : last stands -- The world is weary of the past -- Reconstruction on a world scale -- The human zoo -- States of exception, exceptional states -- Postscript : toward Leviathan 3.0?