Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I ("On Formally Undecidable Propositions of Principia Mathematica and Related Systems I") is a paper in
mathematical logicMathematical logic is a subfield of mathematics with close connections to computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
by
Kurt GödelKurt Gödel was an Austrian-American logician, mathematician and philosopher. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when many, such as Bertrand Russell, A. N...
. Dated November 17, 1930, it was originally published in German in the 1931 volume of
Monatshefte für Mathematik. Several English translations have appeared in print, and the paper has been included in two collections of classic mathematical logic papers.
Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I ("On Formally Undecidable Propositions of Principia Mathematica and Related Systems I") is a paper in
mathematical logicMathematical logic is a subfield of mathematics with close connections to computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
by
Kurt GödelKurt Gödel was an Austrian-American logician, mathematician and philosopher. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when many, such as Bertrand Russell, A. N...
. Dated November 17, 1930, it was originally published in German in the 1931 volume of
Monatshefte für Mathematik. Several English translations have appeared in print, and the paper has been included in two collections of classic mathematical logic papers. The paper is famous for the theorems it contains, which have many implications for
consistency proofIn logic, a consistent theory is one that does not contain a contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if it has a model; this is the sense used in traditional Aristotelian logic,...
s in mathematics, and for the techniques that Gödel invented to prove these theorems.
Outline and key results
The main results established are Gödel's first and second
incompleteness theoremsIn mathematical logic, Gödel's incompleteness theorems, proved by Kurt Gödel in 1931, are two theorems stating inherent limitations of all but the most trivial formal systems for arithmetic of mathematical interest. The theorems are of considerable importance to the philosophy of mathematics...
, which have had an enormous impact on the field of
mathematical logicMathematical logic is a subfield of mathematics with close connections to computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
. These appear as theorems VI and XI, respectively, in the paper.
In order to prove these results, Gödel introduced in this paper a method now known as Gödel numbering. In this method, each sentence and formal proof in first-order arithmetic is assigned a particular natural number. Gödel shows that many properties of these proofs can be defined within any theory of arithmetic that is strong enough to define the
primitive recursive functionThe primitive recursive functions are defined using primitive recursion and composition as central operations and are a strict subset of the recursive functions...
s. (The contemporary terminology for
recursive functionRecursive function may refer to:* Recursion : a procedure or subroutine, implemented in a programming language, whose implementation references itself...
s and
primitive recursive functionThe primitive recursive functions are defined using primitive recursion and composition as central operations and are a strict subset of the recursive functions...
s had not yet been established when the paper was published; Gödel used the word
rekursiv ("recursive") for what are now known as primitive recursive functions.) The method of Gödel numbering has since become common in mathematical logic.
Because the method of Gödel numbering was novel, and to avoid any ambiguity, Gödel presented a list of 45 explicit formal definitions of primitive recursive functions and relations used to manipulate and test Gödel numbers. He used these to give an explicit definition of a formula Bew(
x) that is true if and only if
x is the Gödel number of a sentence φ and there exists a natural number that is the Gödel number of a proof of φ (The German word for proof is
Beweis).
A second new technique invented by Gödel in this paper was the use of self-referential sentences. Gödel showed that the classical
paradoxA paradox is a statement or group of statements that leads to a contradiction or a situation which defies intuition. The term is also used for an apparent contradiction that actually expresses a non-dual truth...
es of self-reference, such as "
This statement is falseIn philosophy and logic, the liar paradox, known to the ancients as the pseudomenon, encompasses paradoxical statements such as "This sentence is false." or "The next sentence is false. The previous sentence is true." These statements are paradoxical because there is no way to assign them a...
," can be recast as self-referential formal sentences of arithmetic. Informally,
the sentence employed to prove Gödel's first incompleteness theorem says "This statement is not provable." The fact that such self-reference can be expressed within arithmetic was not known until Gödel's paper appeared; independent work of
Alfred TarskiAlfred Tarski was a Polish logician and mathematician...
on his
indefinability theoremTarski's undefinability theorem, stated and proved by Alfred Tarski in 1936, is an important limitative result in mathematical logic, the foundations of mathematics, and in formal semantics...
was conducted around the same time but not published until 1936.
In footnote 48a, Gödel stated that a planned second part of the paper would establish a link between consistency proofs and type theory, but Gödel did not publish a second part of the paper before his death. His 1958 paper in
Dialectica did, however, show how type theory can be used to give a consistency proof for arithmetic.
Published English translations
During his lifetime three English translations of Gödel's paper were printed, but the process was not without difficulty. The first English translation was by Bernard Meltzer; it was published in 1963 as a standalone work by Basic Books and has since been reprinted by Dover and reprinted by Hawking (
God Created the Integers, Running Press, 2005:1097ff). The Meltzer version was adversely reviewed by
Raymond SmullyanRaymond Merrill Smullyan is an American mathematician, concert pianist, logician, philosopher, and magician.Born in Far Rockaway, New York, his first career was stage magic. He then earned a BSc from the University of Chicago in 1955 and his Ph.D. from Princeton University in 1959...
(1966). According to Dawson's biography of Gödel (Dawson 1997:216),
"Fortunately, the Meltzer translation was soon supplanted by a better one prepared by Elliott Mendelson for Martin Davis's anthology The Undecidable; but it too was not brought to Gödel's attention until almost the last minute, and the new translation was still not wholly to his liking ... when informed that there was not time enough to consider substituting another text, he declared that Mendelson's translation was 'on the whole very good' and agreed to its publication.3 [3 Afterward he would regret his compliance, for the published volume was marred throughout by sloppy typography and numerous misprints.]
The translation by Elliott Mendelson appears in the collection
The Undecidable (
DavisMartin David Davis, is an American mathematician, known for his work on Hilbert's tenth problem . He received his Ph.D. from Princeton University in 1950, where his adviser was Alonzo Church . He is Professor Emeritus at New York University. He is the co-inventor of the Davis-Putnam and the DPLL...
1965:5ff). This translation also received a harsh review by Bauer-Medelburg (1965), who in addition to giving a detailed list of the typographical errors also described what he believed to be serious errors in the translation.
A translation by
Jean van HeijenoortJean Louis Maxime van Heijenoort was a pioneer historian of mathematical logic...
appears in the collection
From Frege to Gödel: A source book in Mathematical Logic (van Heijenoort 1967). A review by
Alonzo ChurchAlonzo Church was an American mathematician and logician who made major contributions to mathematical logic and the foundations of theoretical computer science. He is best known for the lambda calculus, Church–Turing thesis, Frege–Church ontology, and the Church–Rosser theorem.- Life :Alonzo...
(1972) described this as "the most careful translation that has been made" but also gave some specific criticisms of it. Dawson (1997:216) notes:
"The translation Gödel favored was that by Jean van Heijenoort ... In the preface to the volume van Heijenoort noted that Gödel was one of four authors who had personally read and approved the translations of his works."
This approval process was laborious. Gödel introduced changes to his text of 1931, and negotiations between the men were "protracted": "Privately van Heijenoort declared that Gödel was the most doggedly fastidious individual he had ever known." Between them they "exchanged a total of seventy letters and met twice in Gödel's office in order to resolve questions concerning subtleties in the meanings and usage of German and English words." (Dawson 1997:216-217).
Although not a translation of the original paper, a very useful 4th version exists that "cover[s] ground quite similar to that covered by Godel's original 1931 paper on undecidability" (Davis 1952:39), as well as Gödel's own extensions of and commentary on the topic. This appears as
On Undecidable Propositions of Formal Mathematical Systems (Davis 1965:39ff) and represents the lectures as transcribed by Stephen Kleene and
J. Barkley RosserJohn Barkley Rosser Sr. was an American logician, a student of Alonzo Church, and known for his part in the Church-Rosser theorem, in lambda calculus. He also developed what is now called the Rosser sieve, in number theory. He was later Director of the Army Mathematics Research Center at the...
while Gödel delivered them at the Institute for Advanced Study in Princeton N.J. in 1934. Two pages of errata and additional corrections by Gödel were added by Davis to this version. This version is also notable because in it Gödel first describes the Herbrand suggestion that gave rise to the (general, i.e. Herbrand-Gödel) form of
recursionRecursion, in mathematics and computer science, is a method of defining functions in which the function being defined is applied within its own definition. The term is also used more generally to describe a process of repeating objects in a self-similar way...
.
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