In the early 1920s, the German mathematician David Hilbert put forward a new proposal to formalise mathematical theory. This has come to be known as Hilbert’s program, also called the formalism. A formulation of all mathematics; in other words all mathematical statements should be written in a precise formal language, and manipulated according to well defined rules.
Kurt Gödel’s Incompleteness Theorem first disproved this program. Later, Alan Turing and Alonzo Church independently came up with the Church-Turing Thesis that disproved the idea of decidability,
He wanted to create a finite set of axioms, that can be used to prove everything in maths. He also wanted to create a proof that this set was complete & consistent
Statement
- Completeness: a proof that all true mathematical statements can be proved in the formalism. Every mathematical statement that is true can be proved
- 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. There is no paradoxes in maths
- 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. If we need to prove something that actually exists, then we need to use existing things as part of the proof
- Decidability: there should be an algorithm for deciding the truth or falsity of any mathematical statement. Determine whether a program/statement is true/false
Source: ‘Hilbert’s program’, Wikipedia. https://en.wikipedia.org
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