"A system encompasses all facts derivable within its framework" is a very vague sentence, but the way it's written I would consider it false. Second order logic is incomplete, for example.
The fact that different set of axioms lead to different conclusions is... kind of obvious? And not at all what logicians (or probably anybody) mean by "contradiction". Moreover, incompleteness doesn't at all prevent that. You can take ZFC + CH and ZFC + not(CH), both are incomplete, but they obviously entail different conclusions.
In more standard terminology, there are systems that are free of contradictions, such as Presburger arithmetic or propositional calculus.
Gödel's completeness and incompleteness theorems are really about entirely different things, completeness is about first-order logic as a system, incompleteness is about consistent, effectively axiomatisable theories of sufficient strength.
You are partly right, I made a mistake, I meant to say that contradiction is when we can derive both a statement and its negation, indicative of inconsistency. You are correct to point out that incompleteness is not a flaw, but a natural property of sufficiently complex systems and does not detract from their validity or usefulness. I must admit, I don't know enough about ZFC and the Continuum Hypothesis, but I did not mean to imply that all systems were incompatible.
The fact that different set of axioms lead to different conclusions is... kind of obvious? And not at all what logicians (or probably anybody) mean by "contradiction". Moreover, incompleteness doesn't at all prevent that. You can take ZFC + CH and ZFC + not(CH), both are incomplete, but they obviously entail different conclusions.
In more standard terminology, there are systems that are free of contradictions, such as Presburger arithmetic or propositional calculus.
Gödel's completeness and incompleteness theorems are really about entirely different things, completeness is about first-order logic as a system, incompleteness is about consistent, effectively axiomatisable theories of sufficient strength.