# First-order logic advantage over second-order logic

What is the advantage of using first-order logic over second-order logic? Second-order logic is more expressive and there is also a way to overcome Russell's paradox...

So what makes first-order logic standard for set theory?

Thanks.

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Completeness theorem. –  Asaf Karagila May 6 '12 at 13:45

I think some parts of this answer might be slightly misleading. It's only true that second order logic with full semantics lacks a compactness theorem. / It doesn't make sense to say "the second order theory proves that there is only one model". Of course we can prove in the metatheory that there's only one full model of PA2, but that can't even be stated in the second-order theory of $\mathbb{N}$, much less proved. Moreover, it isn't the second order induction axiom that causes categoricity, it's the restriction in the metatheory to only look at full models. –  Carl Mummert Feb 21 at 4:12