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$\begingroup$ The obvious motivation in incompleteness theorem that does not imply finite inaxiomatizability is that the latter condition severely reduces the class of theories to which the incompleteness theorem may be applicable: there are quite a lot of interesting finitely axiomatized theories, even extending PA, as you noted yourself. $\endgroup$
– Emil Jeřábek
Commented Aug 5, 2011 at 15:34

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