Editing Hilbert's program (section)

==Statement of Hilbert's program==
The main goal of Hilbert's program was to provide secure foundations for all mathematics. In particular, this should include:
*A formulation of all mathematics; in other words all mathematical statements should be written in a precise [[formal language]], and manipulated according to well defined rules.
*Completeness: a proof that all true mathematical statements can be proved in the formalism.
*Consistency: a proof that no contradiction can be obtained in the formalism of mathematics. This consistency proof should preferably use only "finitistic" reasoning about finite mathematical objects.
*Conservation: a proof that any result about "real objects" obtained using reasoning about "ideal objects" (such as [[uncountable]] sets) can be proved without using ideal objects.
*[[Decidability (logic)|Decidability]]: there should be an algorithm for deciding the truth or falsity of any mathematical statement.