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A humorous variant of Gödel's ontological proof is mentioned in Quentin Canterel's novel The Jolly Coroner. [26] [page needed] The proof is also mentioned in the TV series Hand of God. [specify] Jeffrey Kegler's 2007 novel The God Proof depicts the (fictional) rediscovery of Gödel's lost notebook about the ontological proof. [27]
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.
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 ...
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.
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.
Bernays included a full proof of the incompleteness theorems in the second volume of Grundlagen der Mathematik , along with additional results of Ackermann on the ε-substitution method and Gentzen's consistency proof of arithmetic. This was the first full published proof of the second incompleteness theorem.
Gödel's original proof of the theorem proceeded by reducing the problem to a special case for formulas in a certain syntactic form, and then handling this form with an ad hoc argument. In modern logic texts, Gödel's completeness theorem is usually proved with Henkin's proof, rather than with Gödel's
In mathematical logic, Rosser's trick is a method for proving a variant of Gödel's incompleteness theorems not relying on the assumption that the theory being considered is ω-consistent (Smorynski 1977, p. 840; Mendelson 1977, p. 160).