A set of (originally) unsolved problems in mathematics proposed by Hilbert. Of the 23 total, ten were presented at the Second International Congress in Paris in 1900. These problems were designed to serve as examples for the kinds of problems whose solutions would lead to the furthering of disciplines in mathematics.

- 1a. Is there a transfinite number between that of a Denumerable Set and the numbers of the Continuum? This question was answered by Gödel and Cohen to the effect that the answer depends on the particular version of Set Theory assumed.
- 1b. Can the Continuum of numbers be considered a Well-Ordered Set? This question is related to Zermelo's Axiom of Choice. In 1963, the Axiom of Choice was demonstrated to be independent of all other Axioms in Set Theory, so there appears to be no universally valid solution to this question either.
- 2. Can it be proven that the Axioms of logic are consistent? Gödel's Incompleteness Theorem indicated that the answer is ``no,'' in the sense that any formal system interesting enough to formulate its own consistency can prove its own consistency Iff it is inconsistent.
- 3. Give two Tetrahedra which cannot be decomposed into congruent Tetrahedra directly or by adjoining congruent Tetrahedra. Max Dehn showed this could not be done in 1902. W. F. Kagon obtained the same result independently in 1903.
- 4. 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. This problem was solved by G. Hamel.
- 5. Can the assumption of differentiability for functions defining a continuous transformation Group be avoided? (This is a generalization of the Cauchy Functional Equation.) Solved by John von Neumann in 1930 for bicompact groups. Also solved for the Abelian case, and for the solvable case in 1952 with complementary results by Montgomery and Zipin (subsequently combined by Yamabe in 1953). Andrew Glean showed in 1952 that the answer is also ``yes'' for all locally bicompact groups.
- 6. Can physics be axiomized?
- 7. Let be Algebraic and Irrational. Is then Transcendental? Proved true in 1934 by Aleksander Gelfond (Gelfond's Theorem; Courant and Robins 1996).
- 8. Prove the Riemann Hypothesis. The Conjecture has still been neither proved nor disproved.
- 9. Construct generalizations of the Reciprocity Theorem of Number Theory.
- 10. Does there exist a universal algorithm for solving Diophantine Equations? The impossibility of obtaining a general solution was proven by Julia Robinson and Martin Davis in 1970, following proof of the result that the equation (where is a Fibonacci Number) is Diophantine by Yuri Matijasevich (Matijasevic 1970, Davis 1973, Davis and Hersh 1973, Matijasevic 1993).
- 11. Extend the results obtained for quadratic fields to arbitrary Integer algebraic fields.
- 12. Extend a theorem of Kronecker to arbitrary algebraic fields by explicitly constructing Hilbert class fields using special values. This calls for the construction of Holomorphic Functions in several variables which have properties analogous to the exponential function and elliptic modular functions (Holzapfel 1995).
- 13. Show the impossibility of solving the general seventh degree equation by functions of two variables.
- 14. Show the finiteness of systems of relatively integral functions.
- 15. Justify Schubert's Enumerative Geometry (Bell 1945).
- 16. Develop a topology of Real algebraic curves and surfaces. The Shimura-Taniyama Conjecture postulates just this connection. See Ilyashenko and Yakovenko (1995) and Gudkov and Utkin (1978).
- 17. Find a representation of definite form by Squares.
- 18. Build spaces with congruent Polyhedra.
- 19. Analyze the analytic character of solutions to variational problems.
- 20. Solve general Boundary Value Problems.
- 21. Solve differential equations given a Monodromy Group. More technically, prove that there always exists a Fuchsian System with given singularities and a given Monodromy Group. Several special cases had been solved, but a Negative solution was found in 1989 by B. Bolibruch (Anasov and Bolibruch 1994).
- 22. Uniformization.
- 23. Extend the methods of Calculus of Variations.

**References**

Anasov, D. V. and Bolibruch, A. A. *The Riemann-Hilbert Problem.* Braunschweig, Germany: Vieweg, 1994.

Bell, E. T. *The Development of Mathematics, 2nd ed.* New York: McGraw-Hill, p. 340,
1945.

Borowski, E. J. and Borwein, J. M. (Eds.). ``Hilbert Problems.'' Appendix 3 in
*The Harper Collins Dictionary of Mathematics.* New York: Harper-Collins, p. 659, 1991.

Boyer, C. and Merzbach, U. ``The Hilbert Problems.'' *History of Mathematics, 2nd ed.* New York:
Wiley, pp. 610-614, 1991.

Browder, Felix E. (Ed.). *Mathematical Developments Arising from Hilbert Problems.*
Providence, RI: Amer. Math. Soc., 1976.

Courant, R. and Robbins, H. *What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed.*
Oxford, England: Oxford University Press, p. 107, 1996.

Davis, M. ``Hilbert's Tenth Problem is Unsolvable.'' *Amer. Math. Monthly* **80**, 233-269, 1973.

Davis, M. and Hersh, R. ``Hilbert's 10th Problem.'' *Sci. Amer.*, pp. 84-91, Nov. 1973.

Gudkov, D. and Utkin, G. A. *Nine Papers on Hilbert's 16th Problem.*
Providence, RI: Amer. Math. Soc., 1978.

Holzapfel, R.-P. *The Ball and Some Hilbert Problems.* Boston, MA: Birkhäuser, 1995.

Ilyashenko, Yu. and Yakovenko, S. (Eds.). *Concerning the Hilbert 16th Problem.*
Providence, RI: Amer. Math. Soc., 1995.

Matijasevic, Yu. V. ``Solution to of the Tenth Problem of Hilbert.'' *Mat. Lapok* **21**, 83-87, 1970.

Matijasevich, Yu. V. *Hilbert's Tenth Problem.* Cambridge, MA: MIT Press, 1993.

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