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Which theorem states that for a powerful enough system to explain the arithmetic of natural numbers, some propositions can be neither proved nor disproved?
First incompleteness theorem
Second incompleteness theorem
Completeness theorem
Continuum hypothesis
Answer
Gödel's first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" (essentially, a computer program) is capable of proving all truths about the natural numbers. For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. This result has profound implications for the foundations of mathematics.
The Gödelicious Quiz: Unraveling the Genius of Kurt Gödel
More Questions
Which two well-known mathematicians were building on the work of Dedekind, Cantor, and Frege during Gödel's time?
Gödel's incompleteness theorems were published in what year?
The first incompleteness theorem deals with what kind of system?
The second incompleteness theorem follows from which other theorem?
What does the completeness theorem deal with?
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