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 formalization 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: there should be an algorithm for deciding the truth or falsity of any mathematical statement.
Read more about this topic: Hilbert's Program
Famous quotes containing the words statement of, statement and/or program:
“The honor my country shall never be stained by an apology from me for the statement of truth and the performance of duty; nor can I give any explanation of my official acts except such as is due to integrity and justice and consistent with the principles on which our institutions have been framed.”
—Andrew Jackson (1767–1845)
“Eroticism has its own moral justification because it says that pleasure is enough for me; it is a statement of the individual’s sovereignty.”
—Mario Vargas Llosa (b. 1936)
“In common with other rural regions much of the Iowa farm lore concerns the coming of company. When the rooster crows in the doorway, or the cat licks his fur, company is on the way.”
—For the State of Iowa, U.S. public relief program (1935-1943)