As the article explains (and the title alludes to), if your axioms are inconsistent, so you can prove a contradiction, you can prove anything. Literally any statement that can be formulated can be proven to be true (as can its negation!). So you'd be trying to model the universe with a maths in which 1 = 2. And also 1 = 3. And also 1 =/= 3.

Any useful set of axioms has to be consistent, because any inconsistency is catastrophic. It wrecks the entire thing. Of course that also means, thanks to Godel's theorem, that any useful set of axioms is incomplete, in that there are true statements that cannot be proved from them.