Hello. This is a question for the philosophers of mathematics or the logicians. I have heard that first order logic is complete, and that second order logic is incomplete. The completeness of first order logic I have seen characterized as the fact that every true proposition (in the semantic sense) is also provable (in the syntactic sense). I've also heard that the completeness at stake in both cases is not the same, but it has never been clear to me in what they differ. Supposedly second order logic, having more expressive power, has enough resources to express arithmetic and thus the first incompleteness theorem applies to it, but that theorem says of such systems that they are incomplete. But I also have heard some people (or maybe I have misheard them) discussing such incompleteness in the same terms, that is, as saying that not every true theorem of such systems is provable, though the converse is true (they are sound). I am no logician, so I would appreciate firstly, if someone can point out any mistakes in my question, and secondly if someone could clarify the different senses of completeness. Thanks a lot.
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