Gödel's Incompleteness Theorems/Second

Theorem
Let $T$ be the set of theorems of some recursive set of sentences in the language of arithmetic such that $T$ contains minimal arithmetic.

Let $\map {\mathrm {Cons} } T$ be the propositional function which states that $T$ is consistent.

Then it is not possible to prove $\map {\mathrm {Cons} } T$ by means of formal statements within $T$ itself.

Proof
https://en.wikipedia.org/wiki/Branch-decomposition

Also see

 * Gödel's First Incompleteness Theorem