Meaning of Gödel's incompleteness theorem

Gö'del's incomplete'ness the"orem

Pronunciation: [key]
— Logic, Math. Logic,Math.
  1. the theorem that states that in a formal logical system incorporating the properties of the natural numbers, there exists at least one formula that can be neither proved nor disproved within the system.
  2. the corollary that the consistency of such a system cannot be proved within the system.
Random House Unabridged Dictionary, Copyright © 1997, by Random House, Inc., on Infoplease.