# (In)Completeness: First Order x Second Order

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? –  Peter Smith Nov 26 '12 at 8:24
You're right, @Peter, "First Order x Second Order" expresses my doubts more appropriately. –  Ari Nov 27 '12 at 21:56
The title is now incomprehensible. –  Chris Eagle Nov 27 '12 at 22:06
you should maybe precise that you are talking about proof systems, not to be confused with for instance incompleteness of first-order Peano arithmetic. –  Denis Jul 20 '14 at 23:47

In particular, if $T$ is a type, the condition that the interpretation of the type $\mathcal{P}(T)$ must be the power set of the interpretation of the type $T$.