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.

Sources

• 1931: D. Hilbert: Die Grundlegung der elementaren Zahlenlehre (Math. Ann. Vol. 104: pp. 485 – 494)
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Hilbert's programme
• 1993: M. Ben-Ari: Mathematical Logic for Computer Science ... (previous) ... (next): Chapter $1$: Introduction: $\S 1.1$: The origins of mathematical logic