- "Die Vollständigkeit der Axiome des logischen Funktionenkalküls."Offprint from: Aus den Monatsheften für Mathematik und Physik, Vol 37, part 1. Leipzig: Akademische Verlagsgesellschaft, 1930.
AND: "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I." Offprint from: Aus den Monatsheften für Mathematik und Physik, Vol 38, part 1. Leipzig: Akademische Verlagsgesellschaft, 1931.
- paper, ink
WITH: VON WRIGHT, Georg Henrik. Typed letter signed "George Henrik von Wright" in Swedish on Academy of Finland letterhead, stating that the two papers came from the estate of Finnish philosopher Eino Kaila, and that Wright, who was the first holder of the Swedish Chair of Philosophy and the University of Helsinki, was presenting these as a gift to his successor.
In 1928, German mathematician David Hilbert, in his address to the International Congress of Mathematicians in Bologna posed three famous challenges to the mathematical community:
1. To prove that all true mathematical statements could be proven, that is, the completeness of mathematics.
2. To prove that only true mathematical statements could be proven, that is, the consistency of mathematics.
3. To prove the decidability of mathematics, that is, the existence of a decision procedure to decide the truth or falsity of any given mathematical proposition.
The first two questions of completeness and consistency were famously answered two years later by the Austrian logician Kurt in his "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I.” In this revolutionary paper, Gödel introduced his Incompleteness theorem, which "showed that even powerful logical systems could not hope to encompass the full scope of mathematical truth." Gödel showed that, for any axiomatic system powerful enough to describe the natural numbers 1: If the system is consistent, then it cannot be complete, and 2: the consistency of the axioms cannot be proven within the system. Just a year earlier, Gödel had published “Die Vollständigkeit der Axiome des logischen Funktionenkalküls,” a concise version of his 1929 doctoral thesis, in which he gave the first proof of his Completeness theorem, showing that the axiomatic system of logic is complete.
The third problem posed by Hilbert became known as the Entscheindungsproblem [Decision Problem]. Alan Turing famously solved this problem in his landmark 1936 paper "On Computable Numbers, with an Application to the Entscheindungsproblem." The process of doing so led to his development of a universal computing machine.
Known in English as "On formally Undecidable Propositions in Principia Mathematica and Related Systems I," and "The Completeness of the Axioms of the Functional Calculus of Logic" the papers had an enormous impact on the fields of mathematics, computer science, and philosophy. Von Neumann said of them: “Kurt Gödels' achievement in modern logic is singular and monumental. Indeed it is more than a monument, it is a landmark which will remain visible far in space and time. The subject of logic has certainly completely changed its nature and possibilities with Gödel's achievement." (Halmos)
The Finnish philosopher Eino Kaila (1890-1958) worked in the early 1930s in Vienna and became associated to the Vienna Circle of philosophers. He was greatly interested in logic, and was presented with the two offprints of Gödel's two papers on Completeness and Incompleteness. After Kaila's death, the offprints were given to Georg Henrik von Wright, the famous philosopher who was successor to Ludwig Wittgenstein as professor at Cambridge University from 1948-52. Von Wright was the first holder of the Swedish Chair of philosophy at the University of Helsinki, and later a member of the Academy of Finland. He is one of the very few philosophers to whom a volume is dedicated in the Library of Living Philosophers series, a distinction often compared to the Nobel prize.