Epimenides paradox | Hilbert's second problem | Hilbert's fourth problem | Hilbert's fifth problem | Hilbert's eleventh problem | Hilbert's thirteenth problem (total: 6)
The Epimenides paradox is the self-contradiction inherent in the statement, "All Cretans are liars; one of their own poets has said so."
Hilbert's second problem asks for proof that the axioms of arithmetic are consistent.
Hilbert's fourth problem asks for the construction of all metrics where lines are geodesics.
Hilbert's fifth problem asks if continuous groups are automatically differential groups.
Hilbert's eleventh problem asks for solution of quadratic forms with algebraic numerical coefficients.
Hilbert's thirteenth problem asks for proof of the impossibility of solving the general seventh degree equation by functions of two variables.
Epimenides was a Cretan who made one immortal statement: "All Cretans are liars." One of Crete's own prophets has said it: "Cretans are always liars, evil brutes, lazy gluttons." He has surely told the truth. For this reason correct them sternly, that they may be sound in faith instead of paying attention to Jewish fables and to commandments of people who turn their backs on the truth.
Sometimes interpreted as: "prove that Peano arithmetic is consistent."
Sometimes interpreted as: "find geometries whose axioms are closest to those of Euclidean geometry if the ordering and incidence axioms are retained, the congruence axioms weakened, and the equivalent of the parallel postulate omitted."
Can the assumption of differentiability for functions defining a continuous transformation group be avoided? (This is a generalization of the Cauchy functional equation.)
Extend the results obtained for quadratic fields to arbitrary integer algebraic fields.
Solve all 7th degree equations using functions of two parameters.
| formulation date | formulators | status Epimenides paradox | 550 BC (2574 years ago) | Epimenides | not necessarily a true paradox Hilbert's second problem | 1900 (125 years ago) | David Hilbert | ambiguous Hilbert's fourth problem | 1900 (125 years ago) | David Hilbert | ambiguous Hilbert's fifth problem | 1900 (125 years ago) | David Hilbert | ambiguous Hilbert's eleventh problem | 1900 (125 years ago) | David Hilbert | ambiguous Hilbert's thirteenth problem | 1900 (125 years ago) | David Hilbert | ambiguous | proof date | provers Hilbert's second problem | 1934 (34 years later) (91 years ago) | Gerhard Gentzen | Kurt Gödel Hilbert's fifth problem | 1950 (50 years later) (75 years ago) | Andrew Mattei Gleason Hilbert's eleventh problem | 1923 (23 years later) (102 years ago) | Helmut Hasse Hilbert's thirteenth problem | 1957 (57 years later) (68 years ago) | Vladimir Arnold | Andrey Nikolaevich Kolmogorov | additional people involved Epimenides paradox | Bertrand Russell Hilbert's fourth problem | Georg Karl Wilhelm Hamel Hilbert's fifth problem | John von Neumann | Deane Montgomery | Hidehiko Yamabe | Leo Zippin
Not a true paradox since the poet may have knowledge that at least one Cretan is, in fact, honest, and so is lying when he says that all Cretans are liars. There therefore need be no self-contradiction in what could simply be a false statement by a person who is himself a liar.
No consensus on whether results of Gödel and Gentzen gave a solution to the problem as originally stated. Gentzen proved the consistency of Peano arithmetic using the principle of transfinite induction up to ϵ_0. This principle is widely regarded as being true. Nevertheless, Gentzen's proof is not convincing to individuals who doubt the consistency of Peano arithmetic. This state of affairs is unavoidable as a consequence of Gödel's second incompleteness theorem.
Too vague to be able to give a definite solution. Hamel found examples of geometries requested in the alternate statement.
Solved by John von Neumann in 1930 for bicompact groups. Solved for the Abelian and solvable cases in 1952 with complementary results by Montgomery and Zippin; subsequently combined by Yamabe in 1953. By one possible interpretation, resolved by Andrew Gleason for all locally bicompact groups. If understood as equivalent to the Hilbert-Smith conjecture, still unsolved.
Sometimes considered solved by the Hasse (local-global) principle.
Kolmogorov had shown in 1956 that any function of several variables can be constructed with a finite number of three-variable functions. Arnold showed in 1957 that any function of several variables can be constructed with a finite number of two-variable functions. It has been argued however that Hilbert intended for a solution in terms of multi-valued algebraic functions, making it an extension of Galois theory, an intepretation which would leave the problem unsolved.
ambiguous problem
