The Gödel completeness theorem establishes that the set of all valid formulas of any first-order language L can be generated from a simple set of axioms via straightforward inference rules.
Gödel(as a historical figure in mathematical logic)
Kurt Gödel was a 20th-century mathematician and logician who proved that any consistent formal system (a set of logical rules) is incomplete—meaning there are true statements it can't prove.
Valid formulas(the focus of what the theorem proves)
Statements written in the precise language of logic that are correctly formed and logically true—they follow the rules and make sense.
axioms(Stumpf, 1891)
Propositions that we assume to be true and necessary, originating in the content of judgments.
inference rules(As used in the DIRT system by Lin and Pantel)
Rules expressing approximate equivalence between relational phrases, such as 'X finds a solution to Y ≈ X solves Y', derived statistically from text corpora.
Probably the most important result about first-order languages is the Gödel completeness theorem which of course says that the set of all valid formulas of any first-order language L can be generated from a simple set of axioms by means of a few straightforward rules of inference. A major consequence of this theorem is that, if the formulas of L are coded as natural numbers in some constructive way, then the set of (codes of) valid sentences is recursively enumerable. Thus, the completeness