Gödel's first incompleteness theorem just guarantees that for every system of axioms that is computable and "strong enough", an undecidable sentence exists. Fo

Gödel's first incompleteness theorem just guarantees that for every system of axioms that is computable and "strong enough", an undecidable sentence exists. For systems we actually use, such as ZFC or PA, we know a number of actually "meaningful" examples such as the Continuum Hypothesis, which is independent of ZFC. But of course, you can always just add this hypothesis (or its negation) as an axiom. Gödel's theorem has to rely on self-reference because we can't really give a "natural" example when we're not actually looking at a specific system, but trying to say something about all possible systems.