Mathematics is an essentially creative activity
The incompleteness and undecidability results support the idea that mathematics is essentially creative, and that it's preferable to have multiple formal systems rather than a single universal system such as the Principia Mathematica.
The role of logic and mathematics is thereby shown to be one of revealing and developing the limitations of formal systems, not of revealing what one true formal system is.

Emil Post, 1941.
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Artificial Intelligence »Artificial Intelligence
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Are thinking computers mathematically possible? [7] »Are thinking computers mathematically possible? [7]
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No: computers are limited by Gödel's theorems »No: computers are limited by Gödel's theorems
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Theorems show limitations of machine thought »Theorems show limitations of machine thought
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Mathematics is an essentially creative activity
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