By Casnati G., Centina A.D.
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Extra info for A Characterization of Bielliptic Curves and Applications to Their Moduli Spaces
As it happens, this classification broadly follows the, chronological order of publication (except for the reviews), although there is much interleaving of items. All classification schemes have some degree of arbitrariness, and mine is no exception. For the most part, however, the scheme adopted here was dictated by the nature of the material itself. Those cases where that was less certain had mainly to do with articles on the borderline between technical work and philosophy. No attempt is made here to explain the contents of each category in any depth; that is the task of the introductory notes accompanying individual items throughout the first two volumes.
However, the article itself did not appear in print until the collection Davis 1965. Here Godel deals with ideas of absolute definability and demonstrability, independent of any formal languages and theories. The latter notion is only speculative, but for the former Godel suggests a definition of set-theoretic definability within set theory itself; this permits arbitrary ordinals as primitive terms, and so has since come to be called "ordinal definability". Later it was independently rediscovered by others, in particular Myhill and Scott 1971; the notion eventually turned out to be technically very useful.
They also provided the key theoretical concepts for modern general-purpose digital computers, as realized by Turing and von Neumann in the 1940s. (c) Speed-up theorems (1936a). This note contains the first statement (without proof) of what is now called a "speed-up" theorem. Godel had pointed out in his earlier work (1931, footnote 48a, and 19S2b) that passage to higher types allows one to prove statements undecidable in systems of lower levels, for example the consistency of those systems. Thus, by Godel's life and work 19 successively adding higher types one arrives at a sequence of ever stronger systems Sn.