The link davesque posted goes into a lot more detail, but here&#x27;s a short version. They created a Turing machine that encodes a set of axioms known as ZFC <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Zermelo%E2%80%93Fraenkel_set_theory" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Zermelo%E2%80%93Fraenkel_set_t...</a> The Turing machine just enumerates every single possible assertion using those axioms and tests them for consistency. If it ever proves, say, 1+1=3 then it will halt.<p>So here&#x27;s the point: According to Godel&#x27;s incompleteness theorem, ZFC can&#x27;t prove that ZFC is consistent. <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;G%C3%B6del%27s_incompleteness_theorems#First_incompleteness_theorem" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;G%C3%B6del%27s_incompleteness_...</a> But if ZFC could prove that this Turing machine never halts, it would prove its own consistency, so tada! we&#x27;ve put a hard upper bound on the complexity of problems that ZFC can deal with.