What Made Gödel’s Incompleteness Theorem Hard to Prove: It’s about How You Say it, Not Just What You Say

Algorithm Soup

Roughly speaking, Gödel’s Incompleteness Theorem states that there are true mathematical statements that cannot be proven. When I was in 11-th grade, my geometry teacher Mr. Olsen, my friend Uma Roy, and I spent five weeks reading through Gödel’s original proof of the theorem. Why did it take so long? Partly because Uma and I were high-school students. Partly because Gödel was a less-than-talented writer. But mostly because the proof is actually pretty hard.

That might seem surprising, since it’s easy to present a one-paragraph summary of essentially how the proof works: Gödel begins by constructing a mathematical statement essentially equivalent to the sentence,

This statement cannot be proven.

Then Gödel considers what would happen if the statement were false. That would mean that the statement could be proven. But any statement that can be proven must be true, a contradiction. From this Gödel deduces that the statement must be…

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