Wilhelm Ackermann
Encyclopedia
Wilhelm Friedrich Ackermann (29 March 1896, Herscheid
municipality, Germany
– 24 December 1962, Lüdenscheid
, Germany
) was a German
mathematician
best known for the Ackermann function
, an important example in the theory of computation
.
Ackermann was awarded the Ph.D. by the University of Göttingen in 1925 for his thesis Begründung des "tertium non datur" mittels der Hilbertschen Theorie der Widerspruchsfreiheit, which was a consistency proof of arithmetic apparently without full Peano induction
(although it did use e.g. induction over the length of proofs). From 1929 until 1948, he taught at the Arnoldinum Gymnasium in Burgsteinfurt
, and then at Lüdenscheid
until 1961. He was also a corresponding member of the Akademie der Wissenschaften (Academy of Sciences) in Göttingen, and was an honorary professor at the University of Münster
.
In 1928, Ackermann helped David Hilbert
turn his 1917-22 lectures on introductory mathematical logic
into a text, Principles of Mathematical Logic. This text contained the first exposition ever of first-order logic
, and posed the problem of its completeness
and decidability (Entscheidungsproblem
). Ackermann went on to construct consistency proof
s for set theory
(1937), full arithmetic (1940), type-free logic
(1952), and a new axiomatization of set theory
(1956).
Although Ackermann did not choose a university career and rather continued as a high school teacher, he was continually engaged in research and published many contributions to the foundations of mathematics until the end of his life.
Herscheid
Herscheid is a municipality in the southern Märkischer Kreis, in North Rhine-Westphalia, Germany.-Geography:Herscheid is located in the Ebbegebirge , a part of the Sauerland mountains. Altitudes in the municipality extend from 250m above sea level in the valley of the Schwarze Ahe up to the highest...
municipality, Germany
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
– 24 December 1962, Lüdenscheid
Lüdenscheid
Lüdenscheid is a town in the Märkischer Kreis district, in North Rhine-Westphalia, Germany. It is located in the Sauerland region. Lüdenscheid is seat of the administration of the Märkischer Kreis district...
, Germany
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
) was a German
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
best known for the Ackermann function
Ackermann function
In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function that is not primitive recursive...
, an important example in the theory of computation
Theory of computation
In theoretical computer science, the theory of computation is the branch that deals with whether and how efficiently problems can be solved on a model of computation, using an algorithm...
.
Ackermann was awarded the Ph.D. by the University of Göttingen in 1925 for his thesis Begründung des "tertium non datur" mittels der Hilbertschen Theorie der Widerspruchsfreiheit, which was a consistency proof of arithmetic apparently without full Peano induction
Peano axioms
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano...
(although it did use e.g. induction over the length of proofs). From 1929 until 1948, he taught at the Arnoldinum Gymnasium in Burgsteinfurt
Steinfurt
Steinfurt is a town in North Rhine-Westphalia, Germany. It is the capital of the district of Steinfurt.-Geography:Steinfurt is situated north-west of Münster, North Rhine-Westphalia. Its name came into being in 1975 when the two – up to then independent – parts of the city – Borghorst and...
, and then at Lüdenscheid
Lüdenscheid
Lüdenscheid is a town in the Märkischer Kreis district, in North Rhine-Westphalia, Germany. It is located in the Sauerland region. Lüdenscheid is seat of the administration of the Märkischer Kreis district...
until 1961. He was also a corresponding member of the Akademie der Wissenschaften (Academy of Sciences) in Göttingen, and was an honorary professor at the University of Münster
University of Münster
The University of Münster is a public university located in the city of Münster, North Rhine-Westphalia in Germany. The WWU is part of the Deutsche Forschungsgemeinschaft, a society of Germany's leading research universities...
.
In 1928, Ackermann helped David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
turn his 1917-22 lectures on introductory mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical 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...
into a text, Principles of Mathematical Logic. This text contained the first exposition ever of first-order logic
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...
, and posed the problem of its completeness
Gödel's completeness theorem
Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic. It was first proved by Kurt Gödel in 1929....
and decidability (Entscheidungsproblem
Entscheidungsproblem
In mathematics, the is a challenge posed by David Hilbert in 1928. The asks for an algorithm that will take as input a description of a formal language and a mathematical statement in the language and produce as output either "True" or "False" according to whether the statement is true or false...
). Ackermann went on to construct consistency proof
Consistency proof
In 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 and only if it has a model, i.e. there exists an interpretation under which all...
s for set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
(1937), full arithmetic (1940), type-free logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
(1952), and a new axiomatization of set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
(1956).
Although Ackermann did not choose a university career and rather continued as a high school teacher, he was continually engaged in research and published many contributions to the foundations of mathematics until the end of his life.
See also
- Ackermann codingAckermann codingAckermann coding is the encoding of finite sets as natural numbers as devised by Wilhelm Ackermann in his 1940 paper "Die Widerspruchsfreiheit der allgemeinen Mengenlehren"....
- Ackermann ordinalAckermann ordinalIn mathematics, the Ackermann ordinal is a certain large countable ordinal, named after Wilhelm Ackermann. The term "Ackermann ordinal" is also occasionally used for the small Veblen ordinal, a somewhat larger ordinal....
- Ackermann set theoryAckermann set theoryAckermann set theory is a version of axiomatic set theory proposed by Wilhelm Ackermann in 1956.- The language:Ackermann set theory is formulated in first-order logic. The language L_A consists of one binary relation \in and one constant V . We will write x \in y for \in...
- Inverse Ackermann function
External links
- Erich Friedman's page on Ackermann at Stetson UniversityStetson UniversityStetson University is a private university with four colleges and schools located across the I-4 corridor in Central Florida. The primary undergraduate campus is located in DeLand, Florida, USA. In the 2012 U.S...
- Hermes, In memoriam WILHELM ACKERMANN 1896-1962 (pdf 945 KB)