Gödel's Incompleteness Theorems tell us about the limits of theories of arithmetic. More precisely, they tell us about the limits of effectively axiomatized formal theories of arithmetic. But what exactly does that mean?

Formalization as an ideal

Rather than just dive into a series of definitions, it is well worth pausing to remind ourselves of why we might care about formalizing theories.

So let's get back to basics. In elementary logic classes, beginners are drilled in translating arguments into an appropriate formal language and then constructing formal deductions of the stated conclusions from given premisses.

Why bother with formal languages? Because everyday language is replete with redundancies and ambiguities, not to mention sentences which simply lack clear truth-conditions. So, in assessing complex arguments, it helps to regiment them into a suitable artificial language which is expressly designed to be free from obscurities, and where surface form reveals logical structure.

Why bother with formal deductions? Because everyday arguments often involve suppressed premisses and inferential fallacies. It is only too easy to cheat. Setting out arguments as formal deductions in one style or another enforces honesty: we have to keep a tally of the premisses we invoke, and of exactly what inferential moves we are using. And honesty is the best policy. For suppose things go well with a particular formal deduction. Suppose we get from the given premisses to some target conclusion by small inference steps each one of which is obviously valid (no suppressed premisses are smuggled in, and there are no suspect inferential moves).