Formal system

[1] In 1921, David Hilbert proposed to use formal systems as the foundation of knowledge in mathematics.

[clarification needed] An example of a deductive system would be the rules of inference and axioms regarding equality used in first order logic.

Unlike the grammar for WFFs, there is no guarantee that there will be a decision procedure for deciding whether a given WFF is a theorem or not.

David Hilbert founded metamathematics as a discipline for discussing formal systems.

The standard model of arithmetic sets the domain of discourse to be the nonnegative integers and gives the symbols their usual meaning.

In more recent times, contributors include George Boole, Augustus De Morgan, and Gottlob Frege.

David Hilbert instigated a formalist movement called Hilbert’s program as a proposed solution to the foundational crisis of mathematics, that was eventually tempered by Gödel's incompleteness theorems.

[2] The QED manifesto represented a subsequent, as yet unsuccessful, effort at formalization of known mathematics.

This diagram shows the syntactic entities that may be constructed from formal languages . The symbols and strings of symbols may be broadly divided into nonsense and well-formed formulas . A formal language can be thought of as identical to the set of its well-formed formulas, which may be broadly divided into theorems and non-theorems.