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- To every ω-consistent recursive class κ of formulae there correspond recursive class signs r, such that neither v Gen r nor Neg (v Gen r) belongs to Flg (κ) (where v is the free variable of r).
- Proposition VI, On Formally Undecidable Propositions in Principia Mathematica and Related Systems I (1931)
- Informally, recursive systems of axioms cannot be complete.
- "I like Islam, it is a consistent idea of religion and open-minded" (Kurt Godel as quoted by Hao Wang in "A Logical Journey: From Gödel to Philosophy", Hao Wang. The MIT Press. 1996.)
- Ninety percent of [contemporary philosophers] see their principle task as that of beating religion out of men's heads. ... We are far from being able to provide scientific basis for the theological world view. (in "Logical Dilemmas: The Life and Work of Kurt Gödel" by John W. Dawson Jr.)
- The formation in geological time of the human body by the laws of physics (or any other laws of similar nature), starting from a random distribution of elementary particles and the field is as unlikely as the separation of the atmosphere into its components. The complexity of the living things has to be present within the material [from which they are derived] or in the laws [governing their formation]. (Kurt Gödel, quoted in H. Wang. “On `computabilism’ and physicalism: Some Problems.” in Nature’s Imagination, J. Cornwall, Ed, pp.161-189, Oxford University Press (1995).)
- But every error is due to extraneous factors (such as emotion and education); reason itself does not err.
- Attributed in Rebecca Goldstein (2005), Incompleteness, as a remark on 29 November 1972