So "famous" undecidable statements, which is what you mention, are an example of statements that strongly opinionated people believe enough that are probably tr

So "famous" undecidable statements, which is what you mention, are an example of statements that strongly opinionated people believe enough that are probably true but unprovable. You are probably familiar with the Continuum Hypothesis or the Axiom of Choice, the latter being so "probably true" in the model that the axioms for set theory try to capture that it's usually added as another axiom. Kruskal's tree theorem is a more interesting example, you should look into that. You can prove it undecidable, so you can add either it or it's negation like you say, in Peano's arithmetic, but you can prove it ZFC or I think even less expressive axiomatic systems for set theory.