Search results
Results From The WOW.Com Content Network
Gödel's ontological proof is a formal argument by the mathematician Kurt Gödel (1906–1978) for the existence of God. The argument is in a line of development that goes back to Anselm of Canterbury (1033–1109).
Gödel's discoveries in the foundations of mathematics led to the proof of his completeness theorem in 1929 as part of his dissertation to earn a doctorate at the University of Vienna, and the publication of Gödel's incompleteness theorems two years later, in 1931. The incompleteness theorems address limitations of formal axiomatic systems.
The application is a defense of Christianity stating that "If God does not exist, the Atheist loses little by believing in him and gains little by not believing. If God does exist, the Atheist gains eternal life by believing and loses an infinite good by not believing". [3] The atheist's wager has been proposed as a counterargument to Pascal's ...
Moreover, one may define a statement form Proof(x,y), which for every two numbers x and y is provable if and only if x is the Gödel number of a proof of the statement S and y = G(S). Proof(x,y) is in fact an arithmetical relation, just as "x + y = 6" is, though a much more complicated one.
Gödel's original proofs of the incompleteness theorems, like most mathematical proofs, were written in natural language intended for human readers. Computer-verified proofs of versions of the first incompleteness theorem were announced by Natarajan Shankar in 1986 using Nqthm ( Shankar 1994 ), by Russell O'Connor in 2003 using Coq ( O'Connor ...
Kurt Gödel (1925) The proof of Gödel's completeness theorem given by Kurt Gödel in his doctoral dissertation of 1929 (and a shorter version of the proof, published as an article in 1930, titled "The completeness of the axioms of the functional calculus of logic" (in German)) is not easy to read today; it uses concepts and formalisms that are no longer used and terminology that is often obscure.
Versions of the story can also be found in Logical Dilemmas: The Life and Work of Kurt Gödel (1997) By John W. Dawson; E: His Life, His Thought and His Influence on Our Culture (2006), edited by Donald Goldsmith and Marcia Bartusiak; Incompleteness: The Proof and Paradox of Kurt Gödel (2006) by Rebecca Goldstein; Godel: A Life Of Logic, The ...
Several English translations have appeared in print, and the paper has been included in two collections of classic mathematical logic papers. The paper contains Gödel's incompleteness theorems, now fundamental results in logic that have many implications for consistency proofs in mathematics. The paper is also known for introducing new ...