Definition:Hilbert's Program

Definition
Hilbert's program was an attempt to place the foundations of mathematics on a firm logical footing, by providing:


 * $(1): \quad$ A formulation of all mathematics: all mathematical statements should be written in a formal language, and manipulated according to well defined rules.
 * $(2): \quad$ Completeness: a proof that all true mathematical statements can be proved in the formalism.
 * $(3): \quad$ 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.
 * $(4): \quad$ 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.
 * $(5): \quad$ Decidability: there should be an algorithm for deciding the truth or falsity of any mathematical statement.