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 universe is not fundamentally chaotic.
The human being is capable of profound thought.
Truth is not subjective.
The mind is not a mere epiphenomenon.
There is an inherent order to the cosmos.
The human experience is rich with meaning.
For every formal system F, there are undecidable propositions.
The truth of an arithmetic proposition is not always decidable within the system itself.
The concept of objective mathematical truth is not exhausted by the concept of formal provability.
Mathematics is not an arbitrary game with symbols but deals with an objective reality.
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.
The development of mathematics has been, and still is, a process of ever-increasing abstraction.
The set-theoretic paradoxes are not due to a faulty logic, but to a faulty concept of set.
The axiom of choice is consistent with the other axioms of set theory.
There are objective facts about mathematics that are not provable from any finite set of axioms.
The universe is not a machine, and the human mind is not a machine.
The concept of time travel is consistent with the theory of general relativity.
There exist solutions to Einstein's field equations that allow for closed timelike curves.
The existence of an objective reality is a necessary presupposition for any meaningful scientific inquiry.
The human mind has an intuition for mathematical truth that goes beyond formal systems.
Contemporaries of Kurt Gödel
Other Mathematicss born within 50 years of Kurt Gödel (1906–1978).