Über formal unentscheidbare Sätze der Principia Mathematica undver wandter Systeme I. Offprint from: Monatshefte für Mathematik und Physik 38, 1931. [Bound with:] Über die Vollständigkeit der Axiome des logischen Funktionenkalküls. Offprint from: Monatshefte für Mathematik und Physik 37, 1930. Leipzig: Akademische Verlagsgesellschaft, 1931, 1930. [Bound with:] VON WRIGHT, Georg Henrik. Typed letter signed ‘Georg Henrik von Wright’ in Swedish on Academy of Finland letterhead.

Leipzig: Akademische Verlagsgesellschaft, 1931; 1930.

First edition, extremely rare author’s presentation offprint, of Gödel’s famous incompleteness theorem, “one of the major contributions to modern scientific thought” (Nagel & Newman). “Every system of arithmetic contains arithmetical propositions, by which is meant propositions concerned solely with relations between whole numbers, which can neither be proved nor be disproved within the system. This epoch-making discovery by Kurt Gödel, a young Austrian mathematician, was announced by him to the Vienna Academy of Sciences in 1930 and was published, with a detailed proof, in a paper in the Monatshefte für Mathematik und Physik, Volume 38, pp. 173-198” (R. B. Braithwaite in Gödel/Meltzer, p. 1). “This theorem is an important limiting result regarding the power of formal axiomatics, but has also been of immense importance in other areas, such as the theory of computability” (Zach, p. 917). Gödel “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” (Britannica). The offprint of Gödel’s incompleteness theorem is here accompanied by an author’s presentation offprint of his earlier completeness theorem for first-order logic. “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. This, however, was nothing compared with what Gödel published in 1931—namely, the incompleteness theorem: ‘Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I’ (‘On Formally Undecidable Propositions of Principia Mathematica and Related Systems’)” (Britannica). Gödel intended to write a second part to the 1931 paper, but this was never published. OCLC lists two copies of the 1931 offprint (both in Canada), and none of the 1930 offprint. ABPC/RBH list two copies of each offprint, the most recent being those sold at Christie’s, London, 19 November 2014, which realised £104,500 ($167,000) and £35,000 ($55,930), respectively.

Provenance: The history of the present volume is explained in the accompanying letter from von Wright to von Plato, which reads, in translation:

11 Oct. 2000

Dear Jan,

The two essays were in the estate of Eino Kaila. In all probability, he had them directly from the Author. I hope that you appreciate having them. I had them bound together and hand them now, on the day of your inaugural lecture as Swedish professor of philosophy, to you with my wishes for the best of luck.

Your devoted,

Georg Henrik von Wright.

The Finnish philosopher Eino Kaila (1890-1958) worked in the early 1930s in Vienna and became associated to the Vienna Circle. He introduced its ideas to Finnish philosophical debate in Der Logistische Neupositivismus (1930, The new logical-positivism) and Inhimillinen tieto (1939, Human knowledge), an overview of the epistemological theory of logical empiricism. Kaila knew personally several members of the Circle and took part in its sessions, as did Gödel.

After Kaila’s death, the offprints were acquired by Georg Henrik von Wright (1916-2003), the famous Finnish philosopher (partly of Scottish ancestry) who had studied under Kaila at the University of Helsinki. Von Wright was also a relative of Kaila: his mother was the cousin of Kaila’s wife Anna. Von Wright, who made major contributions to logic and the philosophy of science, and latterly in ethics and the humanities, was deeply influenced by Ludwig Wittgenstein, succeeded him as professor at Cambridge University from 1948-52, and was later executor of Wittgenstein’s estate. Von Wright was the first holder of the Swedish-language Chair of Philosophy at the University of Helsinki (he was a member of the Swedish-speaking minority in Finland), a post he held from 1946 until his retirement in 1961, when he was appointed to the 12-member Academy of Finland. He is one of the very few philosophers to whom a volume is dedicated in the Library of Living Philosophers series. Its current editor, Randall E. Auxier, has written: ‘There is no Nobel Prize in philosophy, but being selected for inclusion in the Library of Living Philosophers is, along with the Gifford Lectures, perhaps the highest honor a philosopher can receive.’

In the year 2000, von Wright had the two offprints bound together as the present volume, with the spine lettered ‘Gödel: ZWEI AUFSATZE’ (Gödel: Two Essays), and presented it to his successor as Swedish-language Chair of Philosophy at the University of Helsinki, Jan von Plato (b. 1951). Von Plato works on proof theory, and is the author of Structural Proof Theory (2001) and Proof Analysis: A Contribution to Hilbert's Last Problem (2011).

Following his graduation from the Gymnasium in Brno, Moravia, in 1924, Gödel (1906-78) went to Vienna to begin his studies at the University. Vienna was to be his home for the next fifteen years, and in 1929 he was also to become an Austrian citizen. Gödel’s principal teacher was the German mathematician Hans Hahn (1879-1934), who was interested in modern analysis and set-theoretic topology, as well as logic, the foundations of mathematics, and the philosophy of science. It was Hahn who introduced Gödel to the group of philosophers around Moritz Schlick (1882-1936); this group was later baptized as the ‘Vienna Circle’ and became identified with the philosophical doctrine called logical positivism. Gödel attended meetings of the Circle quite regularly in the period 1926-1928, but in the following years gradually moved away from it as his own developing philosophical views were opposed to those of the Circle. Nevertheless, the lectures on mathematical logic of one member of the Circle, Rudolph Carnap (1891-1970), were one of the main influences on Gödel in his choice of direction for creative work. The other was the Grundzüge der theoretischen Logik (1928) by David Hilbert (1862-1943) and Wilhelm Ackermann (1896-1962), which posed as an open problem the question whether a certain system of axioms for the first-order predicate calculus is complete. In other words, does it suffice for the derivation of every statement that is logically valid (in the sense of being correct under every possible interpretation of its basic terms and predicates)? Gödel arrived at a positive solution to the completeness problem and with that notable achievement commenced his research career. The work, which was to become his doctoral dissertation at the University of Vienna, was finished in the summer of 1929. The degree itself was granted in February 1930, and a revised version of the dissertation was published as ‘Über die Vollständigkeit der Axiome des logischen Funktionenkalküls.’ Although recognition of the fundamental significance of this work would be a gradual matter, at the time the results were already sufficiently distinctive to establish Gödel as a rising star.

Gödel’s solution of the completeness problem posed in Hilbert and Ackermann constituted his first major result … The question was whether validity in the first-order predicate calculus (or the restricted functional calculus, as it was then called) is equivalent to provability in a specific system of axioms and rules of inference. Gödel’s affirmative solution actually established more, implying one version of the ‘downward’ Löwenheim-Skolem theorem … Gödel also extended this result to denumerable sets of formulas which, if consistent with the system, have a denumerable model [this is now called the ‘compactness theorem’]. The paper largely follows the dissertation, with two significant exceptions, one being a deletion and one an addition. First of all, the very interesting informal section with which [the dissertation] began was omitted in [the published paper]. In that section Gödel had situated his work relative to the ideas of Hilbert and Brouwer, arguing against both in certain respects. Even more noteworthy is that he had already raised the possibility of incompleteness of mathematical axiom systems in the deleted introduction. Secondly, Gödel added … the completeness theorem [which] proved to be fundamental for the subject of model theory some years later” (Feferman et al, p. 17).

The ten years 1929-1939 were a period of intense work for Gödel which resulted in his major achievements in mathematical logic. In 1930 he began to pursue Hilbert’s programme for establishing the consistency of formal axiom systems for mathematics by finitary means. “According to Hilbert, there is a central ‘finitary’ core of mathematics that is unquestionably reliable. Its subject matter is strings of characters on a finite alphabet or, equivalently, natural numbers. There are, of course, infinitely many strings and natural numbers, but Hilbert did not regard them as a ‘complete and closed’ totality. The domains are merely ‘potentially infinite’, in the sense that there is no upper bound on the size of strings that can be considered — given any string, one can always produce a larger one. As indicated, unrestricted quantifiers are banned from finitary mathematics; every quantifier must be restricted to a finite domain. To be sure, mathematics goes well beyond the finitary and, unlike the intuitionists and constructivists, Hilbert is not out to restrict available methodology. The idea is that the non-finitary parts of mathematics be regarded as meaningless, akin to the ideal ‘points at infinity’ sometimes introduced into geometry. The purpose of non-finitary systems is to streamline inferences leading to finitary conclusions. With a view like this, we need some assurance that employing the non-finitary methods will not lead to results that are refuted on finitary grounds; that is, we need a guarantee that the non-finitary system is consistent with finitary mathematics. To achieve this, the Hilbert programme called for the discourse of each branch of mathematics to be cast in a rigorously specified deductive system. These deductive systems are to be studied syntactically, with the aim of establishing their consistency. For this metamathematics, only finitary methods are to be employed. Thus, if the programme were successful, finitary mathematics would establish that deductive systems are consistent, and can be used to derive finitary results with full assurance that the latter are correct” (Shapiro, pp. 647-8).

Gödel started by working on the consistency problem for analysis, which he sought to reduce to that for arithmetic, but his plan led him to an obstacle related to the well-known paradoxes of truth and definability in ordinary language. While Gödel saw that these paradoxes did not apply to the precisely specified languages of the formal systems he was considering, he realized that analogous non-paradoxical arguments could be carried out by substituting the notion of provability for that of truth. Pursuing this realization, he was led to the following unexpected conclusions. Any formal system S in which a certain amount of theoretical arithmetic can be developed and which satisfies some minimal consistency conditions is incomplete: one can construct an elementary arithmetical statement A such that neither A nor its negation is provable in S. In fact, the statement so constructed is true, since it expresses its own unprovability in S via a representation of the syntax of S in arithmetic (the technical device used for this construction is now called ‘Gödel numbering’). Furthermore, one can construct a statement C which expresses the consistency of S in arithmetic, and C is not provable in S if S is consistent. It follows that, if the body of finitary combinatorial reasoning that Hilbert required for execution of his consistency program could all be formally developed in a single consistent system S, then the program could not be carried out for S or any stronger (consistent) system. The incompleteness results were published as [‘Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I’]; the stunning conclusions and the novel features of his argument quickly drew wide attention and brought Gödel recognition as a leading thinker in the field … Gödel’s incompleteness work became his Habilitationsschrift (a kind of higher dissertation) at the University of Vienna in 1932. In his report on it, Hahn lauded Gödel’s work as epochal, constituting an achievement of the first order” (Feferman et al, pp. 6-7). On 23 October 1930, Hahn presented an abstract of Gödel’s paper to the Vienna Academy of Sciences; the full paper was received for publication by the Monatshefte, which Hahn edited, on 17 November 1930 and published early in 1931.

Gödel succinctly summarizes his paper in the first paragraph (translation from Gödel/Meltzer): “The development of mathematics in the direction of greater exactness has – as is well-known – led to large tracts of it being formalized, so that proofs can be carried out by following a few mechanical rules. The most comprehensive formal systems yet set up are, on the one hand, the system of Principia Mathematica and, on the other hand, the axiom system for set theory of Zermelo-Fraenkel (later extended by J. v. Neumann). These two systems are so extensive that all methods of proof used in mathematics today have been formalized in them, i.e., reduced to a few axioms and rules of inference. It may therefore be surmised that these axioms and ruled of inference are also sufficient to decide all mathematical questions which can in any way at all be expressed formally in the systems concerned. It is shown below that this is not the case, and that in both the systems mentioned there are in fact relatively simple problems in the theory of ordinary whole numbers which cannot be decided from the axioms. This situation is not due in some way to the special nature of the systems set up, but holds for a very extensive class of formal systems, including, in particular, all those arising from the addition of a finite number of axioms to the two systems mentioned, provided that thereby no false propositions … become provable.”

“One of the first to recognise the potential significance of Gödel’s incompleteness results and to encourage their full development was John von Neumann. Only three years older than Gödel, the Hungarian-born von Neumann was already well known in mathematical circles for his brilliant and extremely diverse work in set theory, proof theory, analysis and mathematical physics” (Feferman et al, p. 6). Von Neumann said: “Kurt Gödel’s 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, p. 383).

“The immediate effect of Gödel’s theorem was that the assumptions of Hilbert’s program were challenged. Hilbert assumed quite explicitly that arithmetic was complete in the sense that it would settle all questions that could be formulated in its language—it was an open problem he was confident could be given a positive solution … up to 1930 it was widely assumed that arithmetic, analysis, and indeed set theory could be completely axiomatized, and that once the right axiomatizations were found, every sentence of the theory under consideration could be either proved or disproved in the object-language theory itself. Gödel’s theorem showed that this was not so …

“Gödel’s results had a profound influence on the further development of the foundations of mathematics. One was that it pointed the way to a reconceptualization of the view of axiomatic foundations. Whereas a prevalent assumption prior to Gödel—and not only in the Hilbert school—was that incompleteness was at best an aberrant phenomenon, the incompleteness theorem showed that it was, in fact, the norm. It now seemed that many of the open questions of foundations, such as the continuum problem, might be further examples of incompleteness. Indeed, he succeeded not long after in showing that the axiom of choice and the continuum hypothesis are not refutable in Zermelo–Fraenkel set theory: [Paul] Cohen later showed that they were also not provable. The incompleteness theorem also played an important role in the negative solution to the decision problem for first-order logic by [Alonzo] Church. The incompleteness phenomenon not only applies to provability, but … also to the notion of computability and its limits.

“Perhaps more than any other recent result of mathematics, Gödel’s theorems have ignited the imagination of non-mathematicians. They inspired Douglas Hofstadter’s best- seller Gödel, Escher, Bach (1979), which compares phenomena of self-reference in mathematics, visual art, and music. They also figure prominently in the work of popular writers such as Rudy Rucker. Although they have sometimes been misused, as when self-described postmodern writers claim that the incompleteness theorems show that there are truths that can never be known, the theorems have also had an important influence on serious philosophy. John Lucas, in his paper ‘Minds, machines, and Gödel’ (1961) and more recently Roger Penrose in Shadows of the mind (1994) have given arguments against mechanism (the view that the mind is, or can be faithfully modeled by a digital computer) based on Gödel’s results. It has also been of great importance in the philosophy of mathematics: for instance, Gödel himself saw them as an argument for Platonism” (Zach, pp. 923-5).

After the publication of the incompleteness theorem, Gödel became an internationally known intellectual figure. He travelled to the United States several times and lectured extensively at Princeton University in New Jersey, where he met Albert Einstein. This was the beginning of a close friendship that would last until Einstein’s death in 1955. When war broke out in 1939, he fled Europe taking his wife to Princeton where, with Einstein’s help, he took up a position at the newly formed Institute for Advanced Studies. He spent the remainder of his life working and teaching there, retiring in 1976.

Feferman et al (eds.), Kurt Gödel: Collected Works: Volume I, 1986. Gödel, On formally undecidable propositions of Principia Mathematica and related Systems, Meltzer (tr.), 1962. Halmos, ‘The Legend of John von Neumann,’ American Mathematical Monthly 80 (1973), pp. 382-394. Nagel & Newman, Gödel ‘s Proof, 1958. Shapiro, ‘Metamathematics and computability,’ in Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, I. Grattan-Guinness (ed.), 1994. Zach, ‘Kurt Gödel, Paper on the Incompleteness Theorems (1931),’ pp. 917-925 in Landmark Writings in Western Mathematics 1640-1940, I. Grattan-Guinness (ed.), 2005.

8vo (227 x 153 mm). [1931:] pp. 173-198. [1930:] pp. 349-360. Original tan printed wrappers, front wrappers each with printed presentation statement in German ‘Überreicht vom Verfasser.’ A few light pencil notations in the 1931 offprint, presumably in the hand of Eino Kaila. Light diagonal crease to upper corner of the 1930 offprint, some faint crinkling and some small spots to front wrapper of second offprint, but otherwise fine. Bound together in brown cloth.

Item #4559

Price: $140,000.00