Known Unknowns 205 spread through the entire body. Think about it. If 1 am allowed to prove anything either way, of course, my system is complete. It can say anything it wants, but the proofs I make are worthless. Let us imagine, for a moment, we created a new system of mathematics where all the numbers in our new theory behave as we expect, except for the numbers 5 and 6. You may use them to count, but they are also equal to each other! This feels bad and it certainly breaks the Peano axioms. In my new system | plus 5 and 0 plus 5 are the same, so I can equate 0 to 1. Because 0 and | are the basis of binary arithmetic, all numbers can be equated. Numbers now have no guaranteed meaning in my system and, what is worse, since logic uses 1 and 0 to represents true and false, all of logic falls apart as well. Whenever we allow inconsistency into mathematics it rapidly brings the whole pack of cards down. The example I gave was glaring; an inconsistency right in the middle of the counting numbers! Maybe I was too aggressive and a subtle and less damaging inconsistency might be tolerable. However, any inconsistency allows me to make zero equal one somewhere in my system and, therefore, any theorem based on proof by counterexample will be suspect. There might be systems where inconsistency could be a legitimate part of a mathematical system, but I would always need positive corroboration for each proof. IfI tried hard enough, I could always prove something either way. I would need to formulate a new mathematical rule — something like “I will believe short, sensible-looking proofs to be right and circuitous proofs to be wrong.” Mathematics would be a bit like a court of law. You would have to weigh up the evidence from a variety of sources and the verdict would be a matter of subjective opinion rather than objective fact. Inconsistency is very bad in mathematics. The Lucas Argument J.R. Lucas of Oxford University believes Gédel’s theorem says something fundamental abou