If a system is complete, must it be inconsistent?

Question

Godel's incompleteness theorem basically says that a set of axioms cannot prove everything. But you can add those unprovable truths to your set of axioms to expand it. Suppose you keep expanding your set of axioms until it contained all truths (which I think isn't possible in finite time). Or suppose somehow, you had a complete set of axioms. Would it necessarily be inconsistent, because if it wasn't, then it would be consistent and compete?

Answer 1

Yes, you could have a system $S$ in which all propositions that are true in $S$ are axioms of $S$. The system $S$ is then, by definition, complete.

If $S$ is powerful enough to represent all the truths of the arithmetic of natural numbers then we know there are an infinite number of such true statements, so $S$ must have an infinite number of axioms.

But then you cannot be sure that $S$ is consistent because there is no effective decision procedure (a procedure that always halts eventually) that can show that $p$ and $\lnot p$ are not both axioms of $S$.

Answer 2

Goedel's first incompleteness theorem states that every theory that is strong enough (containing the peano axioms) is either inconsistent or incomplete. If it is is inconsistent, it can prove everything ! If it is consistent, there are necessarily statements that cannoe be proven within this theory. This does not mean that it can prove nothing.

Goedel's second incompleteness theorem states that no system that is strong enough (containing the peano axioms) can prove its own consistency.

Goedel's completeness theorem states that a statement is provable within a theory if and only if it is true under every interpretation within this theory.