I know that there are two different notions of completeness and that we shouldn't be surprised that a theory might be complete (in the sense of the theorem of completeness) but undecidable (so incomplete in the sense of the Incompleteness theorem), but I still have a hard time wrapping my head around those two different notions. Could anyone explain clearly how in particular first order logic is complete but undecidable ? And how in general this makes perfect sense for a theory to be complete but undecidable ? Thanks a lot
Asked
Active
Viewed 72 times
0
-
4Do you know the subtle differences between the completeness theorem and the incompleteness theorem and why they do not contradict each other ? In short : the completeness theorem states that a statement can be proven if and only if it is true no matter which intepretation we use. The incompleteness theorem states that a sufficient powerful consistent theory must be incomplete that is there must be true but unprovable statements. – Peter Aug 30 '22 at 17:17
-
1You can aee this post – Mauro ALLEGRANZA Aug 30 '22 at 18:23
-
1And also this one – Mauro ALLEGRANZA Aug 30 '22 at 19:10
-
Thank you very much – Yannis A Aug 31 '22 at 09:27