Wilhelm Friedrich Ackermann (; ; 29 March 1896 – 24 December 1962) was a German
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
and
logician
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
best known for his work in
mathematical logic
Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of for ...
and the
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. All primitive recursive functions are total ...
, an important example in the
theory of computation
In theoretical computer science and mathematics, the theory of computation is the branch that deals with what problems can be solved on a model of computation, using an algorithm, how efficiently they can be solved or to what degree (e.g., a ...
.
Biography
Ackermann was born in
Herscheid
Herscheid is a municipality in the southern Märkischer Kreis, in North Rhine-Westphalia, Germany.
Geography
Herscheid is located in the ''Ebbegebirge'' (''"Ebbe Mountains"''), a part of the Sauerland mountains. Altitudes in the municipality exte ...
,
Germany
Germany,, officially the Federal Republic of Germany, is a country in Central Europe. It is the second most populous country in Europe after Russia, and the most populous member state of the European Union. Germany is situated betwe ...
, and was awarded a Ph.D. by the
University of Göttingen
The University of Göttingen, officially the Georg August University of Göttingen, (german: Georg-August-Universität Göttingen, known informally as Georgia Augusta) is a public research university in the city of Göttingen, Germany. Founded ...
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
Peano induction (although it did use e.g. induction over the length of proofs). This was one of two major works in
proof theory
Proof theory is a major branchAccording to Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. Jon Barwise, Barwise (1978) consists of four correspo ...
in the 1920s and the only one following
Hilbert's school of thought.
From 1929 until 1948, he taught at the Arnoldinum Gymnasium in
Burgsteinfurt
Steinfurt (; Westphalian: ''Stemmert'') is a city in North Rhine-Westphalia, Germany. It is the capital of the district of Steinfurt. From roughly 1100-1806, it was the capital of the County of Steinfurt.
Geography
Steinfurt is situated north- ...
, and then at
Lüdenscheid
Lüdenscheid () is a city in the Märkischer Kreis district, in North Rhine-Westphalia, Germany. It is located in the Sauerland region.
Geography
Lüdenscheid is located on the saddle of the watershed between the Lenne and Volme rivers which b ...
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
The University of Münster (german: Westfälische Wilhelms-Universität Münster, WWU) is a public university, public research university located in the city of Münster, North Rhine-Westphalia in Germany.
With more than 43,000 students and over ...
.
In 1928, Ackermann helped
David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
turn his 1917 – 22 lectures on introductory
mathematical logic
Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of for ...
into a text, ''
Principles of Mathematical Logic''. This text contained the first exposition ever of
first-order logic
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
, and posed the problem of its
completeness and
decidability (
Entscheidungsproblem
In mathematics and computer science, the ' (, ) is a challenge posed by David Hilbert and Wilhelm Ackermann in 1928. The problem asks for an algorithm that considers, as input, a statement and answers "Yes" or "No" according to whether the statem ...
). Ackermann went on to construct
consistency proof
In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent i ...
s for
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
(1937),
full arithmetic (1940),
type-free logic (1952), and a new
axiomatization of
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
(1956).
In turn, Hilbert's support vanished when Ackermann got married:
Later in life, Ackerman continued working as a high school teacher. Still, he kept continually engaged in the field of research and published many contributions to the foundations of mathematics until the end of his life. He died in
Lüdenscheid
Lüdenscheid () is a city in the Märkischer Kreis district, in North Rhine-Westphalia, Germany. It is located in the Sauerland region.
Geography
Lüdenscheid is located on the saddle of the watershed between the Lenne and Volme rivers which b ...
,
Germany
Germany,, officially the Federal Republic of Germany, is a country in Central Europe. It is the second most populous country in Europe after Russia, and the most populous member state of the European Union. Germany is situated betwe ...
in December 1962.
See also
*
Ackermann's bijection
*
Ackermann coding In mathematics and computer science, the BIT predicate or Ackermann coding, sometimes written BIT(''i'', ''j''), is a predicate that tests whether the ''j''th bit of the number ''i'' is 1, when ''i'' is written in binary.
History
The BIT pred ...
*
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. All primitive recursive functions are total ...
*
Ackermann ordinal
In 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.
Unfortunately there is ...
*
Ackermann set theory
In mathematics and logic, Ackermann set theory (AST) is an axiomatic set theory proposed by Wilhelm Ackermann in 1956.
The language
AST is formulated in first-order logic. The formal language, language L_ of AST contains one binary relation \in ...
*
Hilbert–Ackermann system
*
Entscheidungsproblem
In mathematics and computer science, the ' (, ) is a challenge posed by David Hilbert and Wilhelm Ackermann in 1928. The problem asks for an algorithm that considers, as input, a statement and answers "Yes" or "No" according to whether the statem ...
*
Ordinal notation
In mathematical logic and set theory, an ordinal notation is a partial function mapping the set of all finite sequences of symbols, themselves members of a finite alphabet, to a countable set of ordinals. A Gödel numbering is a function mapping t ...
*
Inverse 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. All primitive recursive functions are total ...
Bibliography
*1928. "On Hilbert's construction of the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s" in
Jean van Heijenoort
Jean Louis Maxime van Heijenoort (; July 23, 1912 – March 29, 1986) was a historian of mathematical logic. He was also a personal secretary to Leon Trotsky from 1932 to 1939, and an American Trotskyist until 1947.
Life
Van Heijenoort was born ...
, ed., 1967. ''From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931''. Harvard Univ. Press: 493–507.
*1940. "Zur Widerspruchsfreiheit der Zahlentheorie", ''Mathematische Annalen'', vol. 117, pp 162–194.
*1950 (1928). (with
David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
) ''
Principles of Mathematical Logic''. Chelsea. Translation of 1938 German edition.
*1954. ''Solvable cases of the
decision problem
In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question of the input values. An example of a decision problem is deciding by means of an algorithm whethe ...
''. North Holland.
References
External links
*
*
Erich Friedman's page on Ackermannat
Stetson University
Stetson University is a private university with four colleges and schools located across the I–4 corridor in Central Florida with the primary undergraduate campus in DeLand. The university was founded in 1883 and was later established in 1887 ...
Hermes, ''In memoriam WILHELM ACKERMANN 1896-1962''(PDF, 945 KB)
Author profilein the database
zbMATH
zbMATH Open, formerly Zentralblatt MATH, is a major reviewing service providing reviews and abstracts for articles in pure mathematics, pure and applied mathematics, produced by the Berlin office of FIZ Karlsruhe – Leibniz Institute for Informa ...
{{DEFAULTSORT:Ackermann, Wilhelm
1896 births
1962 deaths
Computability theorists
People from Lüdenscheid
People from the Province of Westphalia
University of Göttingen alumni
University of Münster faculty
German Lutherans
20th-century German mathematicians
German logicians
German philosophers
German male writers
20th-century Lutherans
Members of the Göttingen Academy of Sciences and Humanities