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**1 - 2**of**2**### Gödel's Incompleteness Theorems: A Revolutionary View of the Nature of Mathematical Pursuits

"... The work of the mathematician Kurt Gödel changed the face of mathematics forever. His famous incompleteness theorem proved that any formalized system of mathematics would always contain statements that were undecidable, showing that there are certain inherent limitations to the way many mathematicia ..."

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The work of the mathematician Kurt Gödel changed the face of mathematics forever. His famous incompleteness theorem proved that any formalized system of mathematics would always contain statements that were undecidable, showing that there are certain inherent limitations to the way many mathematicians studies mathematics. This paper provides a history of the mathematical developments that laid the foundation for Gödel's work, describes the unique method used by Gödel to prove his famous incompleteness theorem, and discusses the farreaching mathematical implications thereof. 2 I.

### Is Incompleteness A Serious Problem?

, 2006

"... In 1931 Kurt Gödel astonished the mathematical world by showing that no finite set of axioms can suffice to capture all of mathematical truth. He did this by constructing an assertion GF about the whole numbers that manages to assert that it itself is unprovable (from a given finite set F of axioms ..."

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In 1931 Kurt Gödel astonished the mathematical world by showing that no finite set of axioms can suffice to capture all of mathematical truth. He did this by constructing an assertion GF about the whole numbers that manages to assert that it itself is unprovable (from a given finite set F of axioms using formal logic). 1 GF: “GF cannot be proved from the finite set of axioms F.” This assertion GF is therefore true if and only if it is unprovable, and the formal axiomatic system F in question either proves falsehoods (because it enables us to prove GF) or fails to prove a true assertion (because it does not enable us to prove GF). If we assume that the former situation is impossible, we conclude that F is necessarily incomplete since it does not permit us to establish the true statement GF. Either GF is provable and F proves false statements, or GF is unprovable and therefore true, and F is incomplete. Today, a century after Gödel’s birth, the full implications of this “incompleteness ” result are still quite controversial. 2 An important step forward was achieved by Alan Turing in 1936. He showed that