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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.
- The completeness theorem, mathematically, is indeed an almost trivial consequence of Skolem 1923a. However, the fact is that, at that time, nobody (including Skolem himself) drew this conclusion (neither from Skolem 1923a nor, as I did, from similar considerations of his own).
- as quoted, from a 1967 letter to Hao Wang, by Hao Wang in From Mathematics to Philosophy, 1974, p. 8
- But every error is due to extraneous factors (such as emotion and education); reason itself does not err.
- Attributed as a remark of 29th November 1972, in Incompleteness (2005) by Rebecca Goldstein
- Either mathematics is too big for the human mind, or the human mind is more than a machine.
- As quoted in Topoi : The Categorial Analysis of Logic (1979) by Robert Goldblatt, p. 13
- 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].
- As quoted in "On 'computabilism’ and physicalism: Some Problems" by Hao Wang, in Nature’s Imagination (1995), edited by J. Cornwall, p.161-189
- I like Islam, it is a consistent idea of religion and open-minded.
- As quoted in A Logical Journey: From Gödel to Philosophy (1996) by Hao Wang
- Ninety percent of [contemporary philosophers] see their principal 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.
- As quoted in Logical Dilemmas : The Life and Work of Kurt Gödel (1997) by John W. Dawson Jr.
- The meaning of the world is the separation of wish and fact. Wish is a force as applied to thinking beings, to realize something. A fulfilled wish is a union of wish and fact. The meaning of the whole world is the separation and the union of fact and wish.
- As quoted in The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us (MIT Press) 2013 by Yanofsky, Noson S
- Secondly, even disregarding the intrinsic necessity of some new axiom, and even in case it has no intrinsic necessity at all, a probable decision about its truth is possible also in another way, namely, inductively by studying its "success." Success here means fruitfulness in consequences, in particular in "veriﬁable" consequences, i.e. consequences veriﬁable without the new axiom, whose proofs with the help of the new axiom, however, are considerably simpler and easier to discover, and make it possible to contract into one proof many different proofs. The axioms for the system of real numbers, rejected by the intuitionists, have in this sense been veriﬁed to some extent, owing to the fact that analytic number theory frequently allows one to prove number-theoretical theorems which, in a more cumbersome way, can subsequently be veriﬁed by elementary methods. A much higher degree of veriﬁcation than that, however, is conceivable. There might exists axioms so abundant in their veriﬁable consequences, shedding so much light upon a whole ﬁeld, and yielding such powerful methods for solving problems, (and even solving them constructively, as far as that is possible) that, no matter whether or not they are intrinsically necessary, they would have to be accepted at least in the same sense as any well-established physical theory.
- Kurt Godel Collected Works: Volume II: Publications 1938-1974, S. Feferman et al., editors (1990)
Quotes about Gödel
- If a 'religion' is defined to be a system of ideas that contains unprovable statements, then Gödel taught us that mathematics is not only a religion, it is the only religion that can prove itself to be one.
- John D. Barrow, The Artful Universe (1995)
- Fifty years ago Kurt Gödel... proved that the world of pure mathematics is inexhaustible. No finite set of axioms and rules of inference can ever encompass the whole of mathematics. Given any finite set of axioms, we can find meaningful mathematical questions which the axioms leave unanswered. This discovery... came at first as an unwelcome shock to many mathematicians. It destroyed... the hope that they could solve the problem of deciding by a systematic procedure the truth or falsehood of any mathematical statement. ...Gödel's theorem, in denying ...the possibility of a universal algorithm to settle all questions, gave... instead, a guarantee that mathematics can never die. ...there will always be, thanks to Gödel, fresh questions to ask and fresh ideas to discover.
- Freeman Dyson, Infinite in All Directions (1988)
- Gödel published comparatively little, but almost always to maximum effect; his papers are models of precision and incisive presentation.
- Solomon Feferman, in "Gödel's Life and Work", Kurt Gödel: Collected Works, Volume I : Publications 1929-1936 (1986), p. 1
- In the end we search out the beginnings. Established, beyond comparison, as the most important logician of our times by his remarkable results of the 1930s, Kurt Gödel was also most unusual in the ways of his life and mind. Deeply private and reserved, he had a superb all embracing rationality, which could descend into a maddening attention to detail in matters of everyday life.
- Solomon Feferman, in "Gödel's Life and Work", Kurt Gödel: Collected Works, Volume I : Publications 1929-1936 (1986), p. 2
- The progenitor of information theory, and perhaps the pivotal figure in the recent history of human thought, was Kurt Gödel, the eccentric Austriac genius and intimate of Einstein who drove determinism from its strongest and most indispensable redoubt; the coherence, consistency, and self-sufficiency of mathematics.
Gödel demonstrated that every logical scheme, including mathematics, is dependent upon axioms that it cannot prove and that cannot be reduced to the scheme itself. In an elegant mathematical proof, introduced to the world by the great mathematician and computer scientist John von Neumann in September 1930, Gödel demonstrated that mathematics was intrinsically incomplete. Gödel was reportedly concerned that he might have inadvertently proved the existence of God, a faux pas in his Viennese and Princeton circle. It was one of the famously paranoid Gödel's more reasonable fears.
- George Gilder, in Knowledge and Power : The Information Theory of Capitalism and How it is Revolutionizing our World (2013), Ch. 10: Romer's Recipes and Their Limits
- After Einstein's death, Gödel's sense of exile must have deepened enormously. When Einstein had been ordered by his doctor to take a rest cure, there had been nobody, as Gödel complained to his mother, for him to speak to. Now there would permanently be nobody.
- Toward the end of his life, Gödel feared that he was being poisoned, and he starved himself to death. His theorem is one of the most extraordinary results in mathematics, or in any intellectual field in this century. If ever potential mental instability is detectable by genetic analysis, an embryo of someone with Kurt Gödel's gifts might be aborted.
- Brian L. Silver, in The Ascent of Science (1998)
- The one man who was, during the last years, certainly by far Einstein's best friend, and in some ways strangely resembled him most, was Kurt Gödel, The great logician. They were very different in almost every personal way — Einstein gregarious, happy, full of laughter and common sense, and Gödel extremely solemn, very serious, quite solitary, and distrustful of common sense as a means of arriving at the truth. But they shared a fundamental quality: both went directly and wholeheartedly to the questions at the very center of things.
- "Kurt Gödel", at the MacTutor History of Mathematics archive
- "Kurt Gödel" by Juliette Kennedy in Stanford Encyclopedia of Philosophy
- Profile at Wolfram Research
- "Time Bandits" an article about the relationship between Gödel and Einstein by Jim Holt
- "Gödel and the limits of logic" by John W Dawson Jr. (June 2006)
- Notices of the AMS, April 2006, Volume 53, Number 4 · Kurt Gödel Centenary Issue
- Paul Davies and Freeman Dyson discuss Kurt Godel
- "Gödel and the Nature of Mathematical Truth" Edge: A Talk with Rebecca Goldstein on Kurt Gödel
- It's Not All In The Numbers: Gregory Chaitin Explains Gödel's Mathematical Complexities
- National Academy of Sciences Biographical Memoir (PDF)