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- Ground all existing theories to a finite, complete set of axioms
- Provide a proof that these axioms were consistent
Ultimately, the consistency of all of mathematics could be reduced to basic arithmetic.
Statement
- Formulation of all mathematics -> statements should be written in a precise formal language
- Completeness -> proof that all true mathematical statements can be proved in the formalism
- Consistency -> proof that no contradiction can be obtained in the formalism of mathematics
- Conservation -> proof that any result about real objects obtained using reasoning about ideal objects can be proved without using ideal objects
- Decidability -> there should be an algorithm for deciding the truth or falsity of any mathematical statements
Godel showed that most of the goals of Hilbert program were impossible to achieve in the famous Gödel’s incompleteness theorems.
References
See also
- Gödel’s incompleteness theorems — the direct proof that Hilbert’s completeness and consistency goals are unreachable
- Axiom of Choice - for each family of non-empty sets it exists a function f(S) in S — an example of the kind of formal axiomatic foundation Hilbert’s program was trying to build on
- Modus ponens, P è vero quindi Q è vero — the basic inference rule that formal systems like Hilbert’s rely on