In mathematics, Hilbert's program, prepared by German mathematician David Hilbert in the 1920s, was a proposed solution to the foundation crisis in mathematics, when earlier attempts to clarify the basics of mathematics were found to be under paradoxes and contradictions.
As a solution, Hilbert proposed to base all existing theories on a finite, complete set of axioms, and to provide evidence that these axioms were consistent. Hilbert suggested that the consistency of more complicated systems, such as real analysis, could be proven in terms of simpler systems. Ultimately, the question of the consistency of the entire mathematics could be reduced to a question about the consistency of elementary arithmetic.
The incompleteness of Gödel, however, showed in 1931 that Hilbert's program was not feasible. In his first statement, Kurt Gödel demonstrated that any consistent system with a computable set capable of expressing the arithmetic can never be complete: it is possible to construct an expression that can be proven to be true but which can not be deduced from the formal rules of the system. In his second statement, he showed that such a system could not prove its own consistency so that it could certainly not be used to prove the consistency of a more complicated system. This refuted Hilberts assumption that a finitarian system could be used to prove the consistency of a more complex theory. Externe link
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