Kurt Gödel
Proved incompleteness theorems transforming mathematical logic
Most quoted
"Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F."
— from On Formally Undecidable Propositions of Principia Mathematica and Related Systems, 1931
"Either mathematics is incompletable in this sense, that its evident axioms can never be exhausted by a finite number of formal rules, or else there exist mathematical problems which are undecidable in principle."
— from On Formally Undecidable Propositions of Principia Mathematica and Related Systems I, 1931
"The incompleteness theorems are a profound statement about the limits of formal systems and the indispensable role of human intuition and insight in mathematics."
— from On Formally Undecidable Propositions of Principia Mathematica and Related Systems I, 1931
All quotes by Kurt Gödel (527)
The incompleteness results apply to all sufficiently powerful systems.
I have proved that Principia Mathematica is incomplete.
Mathematics deals with idealities.
The axiom of choice is consistent with ZF.
Reason cannot prove its own consistency.
The mind's ability to see truth beyond formalism is profound.
Gödel's theorem reveals the gap between truth and provability.
Formal systems are limited tools.
I am interested in the foundations of mathematics.
The infinite is the realm where mathematics truly shines.
Belief in God is rational.
Time is relative, but truth is absolute.
The second theorem shows that consistency proofs require stronger systems.
Mathematics is the science of the infinite.
I distrust formal systems after my discovery.
The world is mathematical.
Provability is not the same as truth.
Human reason has limits, but truth does not.
The diagonal argument is key to undecidability.
I prefer philosophy to pure mathematics now.
Contemporaries of Kurt Gödel
Other Mathematicss born within 50 years of Kurt Gödel (1906–1978).