You're constructing a paradox where none exists. The completeness and the incompleteness theorems don't contradict each other, they are about fundamentally diff

You're constructing a paradox where none exists. The completeness and the incompleteness theorems don't contradict each other, they are about fundamentally different things (and it's really a shame we use the same terminology, because so many people are confused by it). > there are truths which, though existent within the system, cannot be proven by it This isn't what the first incompleteness theorem shows - rather, it shows that such "truths" don't exist within the system in the first place (in other words, there are some models in which a certain statement is true and others in which it is false - at least for first-order logic). Otherwise, this would indeed contradict the completeness theorem, but it doesn't. I don't want to stop you from making your own metaphysical conclusions, but I'm not sure they're actually supported by the theorems themselves.