Emil Leon Post

Emil Leon Post
Born	February 11, 1897 Augustów, then Russian Empire , today Poland
Died	April 21, 1954, New York City, U.S.
Fields	Mathematics
Alma mater	Columbia University
Known for	Formulation 1, Post correspondence problem, completeness-proof of Principia's propositional calculus

Emil Leon Post, Ph.D., (February 11, 1897, Augustów – April 21, 1954, New York City) was a mathematician and logician.

Early work

Post was born into a Polish-Jewish family that immigrated to America when he was a child. After completing his Ph.D. in mathematics at Columbia University, he did a post doctorate at Princeton University. While at Princeton, he came very close to discovering the incompleteness of Principia Mathematica, which Kurt Gödel proved in 1931. Post then became a high school mathematics teacher in New York City. In 1936, he was appointed to the mathematics department at the City College of the College of the City of New York, where he remained until his death.

In his Columbia University doctoral thesis, Post proved, among other things, that the propositional calculus of Principia Mathematica was complete: all tautologies are theorems, given the Principia axioms and the rules of substitution and modus ponens. Post also devised truth tables independently of Wittgenstein and C.S. Peirce and put them to good mathematical use. Jean Van Heijenoort's (1966) well-known source book on mathematical logic reprinted Post's classic article setting out these results.

Recursion theory

In 1936, Post developed, independently of Alan Turing, a mathematical model of computation that was essentially equivalent to the Turing machine model. Intending this as the first of a series of models of equivalent power but increasing complexity, he titled his paper Formulation 1. (This model is sometimes called "Post's machine" or a Post-Turing machine, but is not to be confused with Post's tag machines or other special kinds of Post canonical system, a computational model using string rewriting and developed by Post in the 1920s but first published in 1943).

The unsolvability of his Post correspondence problem turned out to be exactly what was needed to obtain unsolvability results in the theory of formal languages.

In an influential address to the American Mathematical Society in 1944, he raised the question of the existence of an uncomputable recursively enumerable set whose Turing degree is less than that of the halting problem. This question, which became known as Post's Problem, stimulated much research. It was solved in the affirmative in the 1950s by the introduction of the powerful priority method in recursion theory.

Selected papers

• 1936, "Finite Combinatory Processes - Formulation 1," Journal of Symbolic Logic 1: 103-105.
• 1943, "Formal Reductions of the General Combinatorial Decision Problem," American Journal of Mathematics 65: 197-215.
• 1944, "Recursively enumerable sets of positive integers and their decision problems," Bulletin of the American Mathematical Society 50: 284-316. Introduces the important concept of many-one reduction.

See also

• Post's problem
• Post's theorem
• Post's inversion formula
• Arithmetical hierarchy
• Post's lattice
• Functional completeness

References

• Emil Leon Post Papers 1927-1991 - American Philosophical Society, Philadelphia, Pennsylvania.
• Davis, Martin (1993). The Undecidable (Ed.), pp. 288–406. Dover. ISBN 0-486-43228-9. Reprints several papers by Post.
• Davis, Martin (1994). "Emil L. Post: His Life and Work" in Davis, M., ed., Solvability, Provability, Definability: The Collected Works of Emil L. Post. Birkhäuser: xi—xxviii. A biographical essay.

External links

• O'Connor, John J.; Robertson, Edmund F., "Emil Leon Post", MacTutor History of Mathematics archive .
• Emil Leon Post at the Mathematics Genealogy Project
• An interview with Martin Davis - Notices of the AMS, May 2008. Much material on Emil Post from his first-hand recollections.