Hilbert's Program was to formalize all existing theories to finite 'real' complete set of axioms, and provide a proof that these axioms were consistent. Hilbert's Program was proposed by German mathematician David Hilbert in 1920.
Hilbert proposed that the consistency of more complicated systems, such as real analysis, could be proven in terms of simpler systems. Ultimately, the consistency of all of mathematics could be reduced to basic arithmetic. However Gödel's second incompleteness theorem showed in 1931, that basic arithmetic cannot be used to prove its own consistency, so it certainly cannot be used to prove the consistency of anything stronger.
- Entry on Hilbert's program at the Stanford Encyclopaedia of Philosophy.