Gödel and Turing Kurt Gödel’s Incompleteness Theorems demonstrate that in any consistent formal system capable of elementary arithmetic, there are true statements that cannot be proven within that system,. Gödel achieved this by "arithmetizing" syntax, assigning numbers to formulas (Gödel numbering) to allow a system to make self-referential statements about its own provability,.

Alan Turing subsequently proved that the Halting Problem—determining whether an arbitrary computer program will stop or run forever—is undecidable,. These two results are fundamentally linked: Gödel’s incompleteness can be viewed as a consequence of the Halting Problem,. If a formal system were complete (able to prove or disprove every statement), one could solve the Halting Problem by systematically searching for a proof that a specific program halts or does not h ...