Undecidable Statements
A statement that is neither provable nor disprovable from a set of axioms is called undecidable (from those axioms). One example is the parallel postulate, which is neither provable nor refutable from the remaining axioms of Euclidean geometry.
Mathematicians have shown there are many statements that are neither provable nor disprovable in Zermelo-Fraenkel set theory with the axiom of choice (ZFC), the standard system of set theory in mathematics (assuming that ZFC is consistent); see list of statements undecidable in ZFC.
Gödel's (first) incompleteness theorem shows that many axiom systems of mathematical interest will have undecidable statements.
Read more about this topic: Mathematical Proof
Famous quotes containing the word statements:
“The true critic is a scrupulous avoider of formulae; he refrains from statements which pretend to be literally true; he finds fact nowhere and approximation always.”
—T.S. (Thomas Stearns)