(For other posts in this series, see here.)
As I discussed in the previous post in this series, our inability to show that an axiomatic system is consistent (i.e., free of contradictions as would be evidenced by the ability to prove two theorems each of which contradicted the other) is not the only problem. Godel also showed that such systems are also necessarily incomplete. In other words, for all systems of interest, there will always be some truths of that system that cannot be proven as theorems using only the axioms and rules of that system. So the tantalizing goal that one day we might be able to develop a system in which every true statement can be proven to be true also turns out to be a mirage. Neither completeness nor consistency is attainable.
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