Closing the Circle: An Analysis of Emil Post's Early Work

In 1931 Kurt Godel published his incompleteness results, and some years later Church and Turing showed that the decision problem for certain systems of symbolic logic has a negative solution. However, already in 1921 the young logician Emil Post worked on similar problems which resulted in what he called an “anticipation” of these results. For several reasons though he did not submit these results to a journal until 1941. This failure ‘to be the first’, did not discourage him: his contributions to mathematical logic and its foundations should not be underestimated. It is the purpose of this article to show that an interest in the early work of Emil Post should be motivated not only by this historical fact, but also by the fact that Post's approach and method differs substantially from those offered by Godel, Turing and Church. In this paper it will be shown how this method evolved in his early work and how it finally led him to his results.

[1]  M. Minsky.  Recursive Unsolvability of Post's Problem of "Tag" and other Topics in Theory of Turing Machines , 1961 .

[2]  Emil L. Post,et al.  Solvability, provability, definability: the collected works of Emil L. Post , 1994 .

[3]  Emil L. Post.  Introduction to a General Theory of Elementary Propositions , 1921 .

[4]  Emil L. Post.  Finite combinatory processes—formulation , 1936, Journal of Symbolic Logic.

[5]  Rolf Herken,et al.  Retrieved July 19 2004 Universal Turing Machine , 2011 .

[6]  J. Lucas.  Minds, Machines and Gödel , 1961, Philosophy.

[7]  Emil L. Post.  A variant of a recursively unsolvable problem , 1946 .

[8]  Emil L. Post.  Recursively enumerable sets of positive integers and their decision problems , 1944 .

[9]  Roger Penrose,et al.  Shadows of the mind - a search for the missing science of consciousness , 1994 .

[10]  Emil L. Post.  Recursive Unsolvability of a problem of Thue , 1947, Journal of Symbolic Logic.

[11]  A. Turing.  On Computable Numbers, with an Application to the Entscheidungsproblem. , 1937 .

[12]  Martin Davis,et al.  Influences of mathematical logic on computer science , 1988 .

[13]  Noam Chomsky,et al.  On Certain Formal Properties of Grammars , 1959, Inf. Control..

[14]  A. Church.  The calculi of lambda-conversion , 1941 .

[15]  A. Church.  An Unsolvable Problem of Elementary Number Theory , 1936 .

[16]  Martin Davis,et al.  The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions , 2004 .

[17]  Emil Post and His Anticipation of Gödel and Turing , 2004 .

[18]  K. Gödel.  Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I , 1931 .

[19]  Emil L. Post.  Formal Reductions of the General Combinatorial Decision Problem , 1943 .

[20]  Martin D. Davis,et al.  Why Gödel Didn't Have Church's Thesis , 1982, Inf. Control..