# Definition:Hilbert's Program

Jump to navigation
Jump to search

## 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