# 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.

## Source of Name

This entry was named for David Hilbert.

## Linguistic Note

Hilbert's program is presented in British English sources as Hilbert's programme.