''Principles of Mathematical Logic'' is the 1950 American translation of the 1938 second edition of David Hilbert's and

Wilhelm Ackermann Wilhelm Friedrich Ackermann (; ; 29 March 1896 – 24 December 1962) was a German mathematician and logician best known for his work in mathematical logic and the Ackermann function, an important example in the theory of computation. Biography ...

's classic text ''Grundzüge der theoretischen Logik'', on elementary

mathematical logic Mathematical logic is the study of 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 formal ...

. The 1928 first edition thereof is considered the first elementary text clearly grounded in the formalism now known as

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 ...

(FOL). Hilbert and Ackermann also formalized FOL in a way that subsequently achieved canonical status. FOL is now a core formalism of mathematical logic, and is presupposed by contemporary treatments of Peano arithmetic and nearly all treatments of axiomatic set theory. The 1928 edition included a clear statement of the

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 state ...

(

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 wheth ...

) for FOL, and also asked whether that logic was complete (i.e., whether all semantic truths of FOL were theorems derivable from the FOL axioms and rules). The former problem was answered in the negative first by

Alonzo Church Alonzo Church (June 14, 1903 – August 11, 1995) was an American mathematician, computer scientist, logician, philosopher, professor and editor who made major contributions to mathematical logic and the foundations of theoretical computer scien ...

and independently by

Alan Turing Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher, and theoretical biologist. Turing was highly influential in the development of theoretical co ...

in 1936. The latter was answered affirmatively by Kurt Gödel in 1929. In its description 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 ...

, mention is made of

Russell's paradox In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox discovered by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains ...

and the

Liar paradox In philosophy and logic, the classical liar paradox or liar's paradox or antinomy of the liar is the statement of a liar that they are lying: for instance, declaring that "I am lying". If the liar is indeed lying, then the liar is telling the truth ...

(page 145). Contemporary notation for logic owes more to this text than it does to the notation of ''

Principia Mathematica The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. ...

'', long popular in the English speaking world.

Notes

References

* David Hilbert and

(1928). ''Grundzüge der theoretischen Logik'' (''Principles of Mathematical Logic''). Springer-Verlag, . This text went into four subsequent German editions, the last in 1972. * Translators: Lewis M. Hammond, George G. Leckie & F. Steinhardt (1999) * Hendricks, Neuhaus, Petersen, Scheffler and Wansing (eds.) (2004). ''First-order logic revisited''. Logos Verlag, . Proceedings of a workshop, FOL-75, commemorating the 75th anniversary of the publication of Hilbert and Ackermann (1928). {{logic-stub 1928 non-fiction books 1938 non-fiction books Logic books Mathematics textbooks History of logic