Gödel's incompleteness theorem

American  

noun

Logic, Mathematics.

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.

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Etymology

Origin of Gödel's incompleteness theorem

After K. Gödel ( def. ), who formulated it

Example Sentences

By applying advanced mathematical principles, including Gödel's incompleteness theorem, they proved that any consistent and complete model of existence requires what they call "non-algorithmic understanding."

From Science Daily • Nov. 10, 2025

Hofstadter is obsessed, I’d say it’s fair to say, with things that refer to, talk about or otherwise interact with themselves—notably Gödel’s incompleteness theorem, a proof about the limits of proofs.

From Scientific American • Sep. 20, 2021

Gödel’s incompleteness theorem implies that both mathematics and physical reality will challenge us with “inexhaustible” problems.

From Scientific American • Jun. 16, 2021

Does Gödel’s incompleteness theorem have implications beyond mathematics?

From Scientific American • Nov. 4, 2018

In 1963 a mathematician, Paul Cohen, proved that this puzzle, the so-called continuum hypothesis, was neither provable nor disprovable, thanks to Gödel’s incompleteness theorem.

From "Zero: The Biography of a Dangerous Idea" by Charles Seife