Is it true that the following definitions of completeness are equivalent?(adsbygoogle = window.adsbygoogle || []).push({});

[itex]\mbox{For theory } \Sigma \mbox{ and for any sentence } A[/itex].

[itex]\mbox{ Either } \Sigma \vdash A \mbox{ or } \Sigma \vdash \lnot A [/itex]

and

[itex]\mbox{ Either } A \in \Sigma \mbox{ or } (\lnot A) \in \Sigma[/itex].

(The second clearly implies the first.)

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# Completeness notions in logic

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