> n any system of mathematics with enough complexity to be interesting, there are statements that cannot be proven true or false.

Note that the article itself is not as careful as you are here about correctly stating what Godel's Incompleteness Theorem actually says. The article states it this way: "any set of rules powerful enough to handle basic arithmetic will contain true statements it cannot prove". But that's wrong, because, when you take into account Godel's Completeness Theorem, you realize that, for any formal system complex enough for the proof of Godel's Incompleteness Theorem to go through, there must be a semantic model of that system in which the Godel sentence for that system is false! The article's misstatement is unfortunately very common, but it's still a misstatement.