That’s not quite what Gödel’s Incompleteness Theorem says. It states that in any consistent, sufficiently expressive formal axiomatic system (such as one capabl

That’s not quite what Gödel’s Incompleteness Theorem says. It states that in any consistent, sufficiently expressive formal axiomatic system (such as one capable of expressing basic arithmetic), there exist true statements about the natural numbers that cannot be proven within the system itself. In other words, if you have a system where you can recursively enumerate all the theorems derivable from its axioms, there will still be statements that are true but not included in that set — meaning the set of such theorems is incomplete. These unprovable statements aren’t just guesses or philosophical curiosities — they’re considered true based on reasoning outside the system, such as meta-mathematical analysis or by interpreting them in the standard model of arithmetic. But because the system’s axioms aren’t strong enough, you can’t prove them from within. And this isn’t a rare edge case. Most of the systems we care about in mathematics — especially those involving arithmetic — are expressive enough to fall under Gödel’s Incompleteness Theorem. So incompleteness isn’t an exception; it’s a fundamental feature of formal systems that are powerful enough to be interesting.