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Kurt Gödel (1906-1978) was an Austrian-American logician, mathematician and philosopher. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when many, such as Bertrand Russell, A. N. Whitehead and David Hilbert, were pioneering the use of logic and set theory to understand the foundations of mathematics.
Both Ludwig Wittgenstein (1889-1951) and Kurt Gödel  advanced theories of ‘incompleteness,’ though their theories were  opposed.  Gödel had a Platonic view of numbers and mathematics, whereby  numbers have their own reality that humans attempt, in their imperfect way, to grasp.  Wittgenstein, on the other hand, had a linguistic, pragmatic view of numbers, and he saw mathematics simply as a highly formal language, with its own rigorous syntax.  For Gödel, the ‘incompleteness’ was an aspect of our ability to describe numbers and their relationships, in a systematic, consistent manner.  Wittgenstein, on the other hand, saw mathematical incompleteness as an absurd concept, like a language that would be somehow ‘incomplete’ (in Wittgenstein’s view, language says what we need it to say, to the extent our needs can be expressed).  Wittgenstein, however, felt that there was another, perhaps more fundamental kind of ‘incompleteness’—our inability to express in language, to reduce to linguistic form, those things that are most important to us.  
Here is how philosopher, essayist, and novelist Rebecca Goldstein succinctly compares Gödel and Wittgenstein: 
"Gödel would most likely not have known that, on some level, he and (the early) Wittgentstein shared a profound conviction of incompleteness, a shared rejection of the logical positivists’ endorsement of the Sophist’s “measure of all things.”.(…) Of course, Gödel and Wittgenstein located the escaped parts of reality in irreconcilably different ways.  Gödel’s conviction, the mathematical interpretation he gave his incompleteness theorems (as well as his work on the continuum hypothesis), was that it was aspects of mathematical reality that must escape our formal systematizing (although not our knowledge), and Wittgenstein’s view on the foundations of mathematics would not countenance this conviction.  For Wittgenstein, at least early Wittgenstein, all of knowledge, a fortiori mathematical knowledge, is systematizable; what systematically escapes our systems is the unsayable, which includes all that is important. Gödel believed our expressible knowledge, demonstrably our mathematical knowledge, is greater than our systems.  Whereof we cannot formalize, thereof we can still know, the mathematician might have said, had he any inclination toward the oracular.”
— Rebecca Goldstein, Incompleteness: The Proof and Paradox of  Kurt Gödel, (2005) (via mhsteger)

"All the limitative theorems of metamathematics and the theory of computation suggest that once the ability to represent your own structure has reached a certain critical point, that is the kiss of death: it guarantees that you can never represent yourself totally. Gödel’s Incompleteness Theorem, Church’s Undecidability Theorem, Turing’s Halting Theorem, Tarski’s Truth Theorem — all have the flavour of some ancient fairy tale which warns you that “To seek self-knowledge is to embark on a journey which … will always be incomplete, cannot be charted on any map, will never halt, cannot be described."
— Douglas Hofstadter, 1979 via Vinod K. Wadhawan, Complexity Explained: 17. Epilogue, Nirmukta

Kurt Gödel (1906-1978) was an Austrian-American logician, mathematician and philosopher. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when many, such as Bertrand Russell, A. N. Whitehead and David Hilbert, were pioneering the use of logic and set theory to understand the foundations of mathematics.

Both Ludwig Wittgenstein (1889-1951) and Kurt Gödel advanced theories of ‘incompleteness,’ though their theories were opposed.  Gödel had a Platonic view of numbers and mathematics, whereby numbers have their own reality that humans attempt, in their imperfect way, to grasp.  Wittgenstein, on the other hand, had a linguistic, pragmatic view of numbers, and he saw mathematics simply as a highly formal language, with its own rigorous syntax.  For Gödel, the ‘incompleteness’ was an aspect of our ability to describe numbers and their relationships, in a systematic, consistent manner.  Wittgenstein, on the other hand, saw mathematical incompleteness as an absurd concept, like a language that would be somehow ‘incomplete’ (in Wittgenstein’s view, language says what we need it to say, to the extent our needs can be expressed).  Wittgenstein, however, felt that there was another, perhaps more fundamental kind of ‘incompleteness’—our inability to express in language, to reduce to linguistic form, those things that are most important to us.  

Here is how philosopher, essayist, and novelist Rebecca Goldstein succinctly compares Gödel and Wittgenstein: 

"Gödel would most likely not have known that, on some level, he and (the early) Wittgentstein shared a profound conviction of incompleteness, a shared rejection of the logical positivists’ endorsement of the Sophist’s “measure of all things.”.
(…)
Of course, Gödel and Wittgenstein located the escaped parts of reality in irreconcilably different ways.  Gödel’s conviction, the mathematical interpretation he gave his incompleteness theorems (as well as his work on the continuum hypothesis), was that it was aspects of mathematical reality that must escape our formal systematizing (although not our knowledge), and Wittgenstein’s view on the foundations of mathematics would not countenance this conviction.  For Wittgenstein, at least early Wittgenstein, all of knowledge, a fortiori mathematical knowledge, is systematizable; what systematically escapes our systems is the unsayable, which includes all that is important. Gödel believed our expressible knowledge, demonstrably our mathematical knowledge, is greater than our systems.  Whereof we cannot formalize, thereof we can still know, the mathematician might have said, had he any inclination toward the oracular.”

Rebecca Goldstein, Incompleteness: The Proof and Paradox of Kurt Gödel, (2005) (via mhsteger)

"All the limitative theorems of metamathematics and the theory of computation suggest that once the ability to represent your own structure has reached a certain critical point, that is the kiss of death: it guarantees that you can never represent yourself totally. Gödel’s Incompleteness Theorem, Church’s Undecidability Theorem, Turing’s Halting Theorem, Tarski’s Truth Theorem — all have the flavour of some ancient fairy tale which warns you that “To seek self-knowledge is to embark on a journey which … will always be incomplete, cannot be charted on any map, will never halt, cannot be described."

Douglas Hofstadter, 1979 via Vinod K. Wadhawan, Complexity Explained: 17. Epilogue, Nirmukta