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Why is First Order Logic complete (as proved by Gödel in his Master thesis) while Second Order Logic is not (as implied by Gödel's PhD thesis)? What are the key differences between the two systems that make them differ on such a crucial property? Is it possible to explain the differences to a working mathematician that doesn't have a formal training in mathematical logic?

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    The title is now out of sync with the body of the question. Antd anyway, don't you really mean why is *first-order* logic complete and *second-order* logic incomplete?2012-11-26
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    You're right, @Peter, "First Order x Second Order" expresses my doubts more appropriately.2012-11-27
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    The title is now incomprehensible.2012-11-27
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    you should maybe precise that you are talking about proof systems, not to be confused with for instance incompleteness of first-order Peano arithmetic.2014-07-20

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