The works of Kurt Gödel, Alan Turing, and others shook this assumption, with the development of statements that are true but cannot be proven within the system.See, e.g., Chaitin, Gregory L., The Limits of Mathematics (1997) esp. 89 ff. Two examples of the latter can be found in Hilbert's problems. Work on Hilbert's 10th problem led in the late twentieth century to the construction of specific Diophantine equations for which it is undecidable whether they have a solution,M. Davis. "Hilbert's Tenth Problem is Unsolvable." American Mathematical Monthly 80, pp. 233–269, 1973 or even if they do, whether they have a finite or infinite number of solutions. More fundamentally, Hilbert's first problem was on the continuum hypothesis.Yandell, Benjamin H.. The Honors Class. Hilbert's Problems and Their Solvers (2002). Gödel and Paul Cohen showed that this hypothesis cannot be proved or disproved using the standard axioms of set theory.Chaitin, Gregory L., The Limits of Mathematics (1997) 1–28, 89 ff. In the view of some, then, it is equally reasonable to take either the continuum hypothesis or its negation as a new axiom.
the works of kurt gödel alan turing and others shook this assumption with the development of statements that are true but cannot be proven within the systemsee eg chaitin gregory l the limits of mathematics (1997) esp 89 ff two examples of the latter can be found in hilbert's problems work on hilbert's 10th problem led in the late twentieth century to the construction of specific diophantine equations for which it is undecidable whether they have a solutionm davis "hilbert's tenth problem is unsolvable" american mathematical monthly 80 pp 233–269 1973 or even if they do whether they have a finite or infinite number of solutions more fundamentally hilbert's first problem was on the continuum hypothesisyandell benjamin h the honors class hilbert's problems and their solvers (2002) gödel and paul cohen showed that this hypothesis cannot be proved or disproved using the standard axioms of set theorychaitin gregory l the limits of mathematics (1997) 1–28 89 ff in the view of some then it is equally reasonable to take either the continuum hypothesis or its negation as a new axiom