Kurt Gödel

Gödel also spelled Goedel, (born April 28, 1906, Brünn, Austria-Hungary [now Brno, Czech Rep.]—died Jan. 14, 1978, Princeton, N.J., U.S.), Austrian-born mathematician, logician, and philosopher who obtained what may be the most important mathematical result of the 20th century: his famous incompleteness theorem, which states that within any axiomatic mathematical system there are propositions that cannot be proved or disproved on the basis of the axioms within that system; thus, such a system cannot be simultaneously complete and consistent. This proof established Gödel as one of the greatest logicians since Aristotle, and its repercussions continue to be felt and debated today.  

In his doctoral thesis, “Über die Vollständigkeit des Logikkalküls” (“On the Completeness of the Calculus of Logic”), published in a slightly shortened form in 1930, Gödel proved one of the most important logical results of the century—indeed, of all time—namely, the completeness theorem, which established that classical first-order logic, or predicate calculus, is complete in the sense that all of the first-order logical truths can be proved in standard first-order proof systems. 

Gödel's incompleteness result is widely regarded as the most remarkable achievement of 20th century mathematics, although some mathematicians say it is logic, not math, and others call it the fundamental result of theoretical computer science, reformulated and extended by Church (1935) & Post & Turing (1936). This discipline did not yet officially exist back then but was effectively created through Gödel's work. It had enormous impact not only on computer science but also on philosophy and other fields.

Since Gödel's exhibition of the fundamental limits of proof and computation, and Konrad Zuse's subsequent construction of the first general-purpose programmable computer (1935-1941), there has been a lot of work on specialized algorithms solving problems taken from more or less general problem classes. Apparently, however, until recently one remarkable fact has escaped the attention of computer scientists: it is possible to use self-referential proof systems to build optimally efficient yet conceptually very simple universal problem solvers: the Gödel machines (2003).

Videos

Papers

Goedels_theorems.pdf
Kurt Gödel and the Foundations of Mathematics.pdf