I don't think there is a contradiction, but someone please correct me if I'm wrong. Godel's theories are something I've recently begun trying to wrap my head around.
Godel's completeness theorem proves the equivalence of logical implication and deducibility. Logical implication being the formal definition of what it means for a collection of sentences (axioms) to logically imply another sentence (theorem); deducibility being the method with which a working mathematician would like to use to show logical implication (i.e. any proof you've ever read in a mathematical textbook).
Incompleteness, on the other hand, shows that there are some true sentences that cannot be deduced from any set of axioms (and therefore, by completeness, are not logically implied by any set of axioms).
This is my understanding after having spent a few weeks reading Enderton's "A Mathematical Introduction to Logic." If I am misunderstanding this, I would love input (so please comment). Godel's theorems are something I'm very anxious to wrap my head around.
Godel's completeness theorem proves the equivalence of logical implication and deducibility. Logical implication being the formal definition of what it means for a collection of sentences (axioms) to logically imply another sentence (theorem); deducibility being the method with which a working mathematician would like to use to show logical implication (i.e. any proof you've ever read in a mathematical textbook).
Incompleteness, on the other hand, shows that there are some true sentences that cannot be deduced from any set of axioms (and therefore, by completeness, are not logically implied by any set of axioms).
This is my understanding after having spent a few weeks reading Enderton's "A Mathematical Introduction to Logic." If I am misunderstanding this, I would love input (so please comment). Godel's theorems are something I'm very anxious to wrap my head around.