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 formalist view of mathematics is untenable.
The existence of God is provable by a logical argument.
The ontological argument for the existence of God is valid.
The world is rational and comprehensible.
There is a realm of abstract objects that exists independently of human thought.
Mathematical intuition is a form of perception.
The development of mathematics is not arbitrary but guided by an objective reality.
The concept of a 'machine' is not sufficient to explain the human mind.
The future is not entirely determined by the past.
The concept of 'truth' is more fundamental than the concept of 'provability'.
The human mind is capable of understanding truths that cannot be formally proven.
The concept of 'set' is a primitive concept that cannot be reduced to more elementary notions.
There is a deeper reality behind the appearances.
The concept of 'number' is an objective concept.
The human mind is not limited by the laws of formal logic.
The concept of 'infinity' is a real concept.
The universe is not a meaningless chaos.
There is a purpose to the universe.
The concept of 'time' is not merely a subjective construct.
The human mind has access to a realm of abstract ideas.
Contemporaries of Kurt Gödel
Other Mathematicss born within 50 years of Kurt Gödel (1906–1978).