# Understanding importance of Godel's incompleteness theorem

I have been reading Chapter 42 of Kleene's "Introduction to Metamathematics" where the following result is proven:

(Rosser's form of Godel's theorem) If the number-theoretic formal system is simply consistent then there is a formula $$A$$ such that neither $$\vdash A$$ nor $$\vdash \lnot A$$.

I know that Godel's theorems are famous for their influence in mathematics. I have trouble understanding what is so influential here.

Consider pure propositional calculus. There, the only provable formulas are those which are identically true (under certain evaluation scheme using truth tables). Consider a formula that is not identically true and not identically false (call it $$B$$). Then, in pure propositional calculus neither $$\vdash B$$ nor $$\vdash \lnot B$$. Somehow, we do not think that this result is influential. What is the reasoning here?

• Godel's statement is actually much stronger, it is the claim that there is a statement $A$ such that $\models A$ but not $\vdash A$. That's what makes it so frustrating, it creates a clear formal example of a distinguishable case of "true" and "provable". – DanielV Oct 14 '18 at 0:26
• @DanielV That notation is highly misleading, since there it uses "$\models$" with respect to a fixed structure and "$\vdash$" with respect to a theory, and this makes it look like it contradicts the completeness theorem. It should instead read something like "For any (reasonable) theory $T$ there is some $A$ such that $\mathbb{N}\models A$ but $T\not\vdash A$." – Noah Schweber Oct 14 '18 at 0:33
• See the post How is the Gödel's Completeness Theorem not a tautology ? for the link between the "obvious" completeness of the calculus (e.g. of predicate logic) and the unexpected incompleteness of first-order arithmetic. – Mauro ALLEGRANZA Oct 14 '18 at 9:36

Each hypothesis is necessary. There are plenty of natural, interesting theories which are complete, consistent, and reasonably simple - they don't let us "interpret arithmetic," though. Similarly, and more seriously, this implies that for example the true theory of arithmetic - that is, the set of all true statements about natural numbers that can be expressed using $$+,\times$$, Boolean operations, quantifiers, variables, and parentheses - is not computable (it's obviously complete and consistent and contains basic arithmetic). This may not seem as surprising now, but it means that there are "simple number-theoretic claims" which we cannot hope to prove in whatever axiom system we're using. Put another way: no matter what axiom system we decide to use to study arithmetic, there will be some very concrete statements which we'll never be able to prove or disprove.