# Incompleteness thereom

There are a couple of non-universal notations here. What is a "Regular Φ axiom system", namely what does Φ mean? And $\Omega(\phi)$ is the formula stating consistency of $\phi$, something that is also denoted $\text{Con}(\phi)$ or $\text{Consis}(\phi)$?
Presumably, $\Omega(\phi)$ says something "there is no $\phi$-derivation of 0=1". If it is false (in the standard model $\mathbb{N}$), then there exists, in fact, a $\phi$-derivation of 0=1. Then $\phi$ is contradictory and therefore derives everything. However, it is proved that $\phi$ does not prove $\Omega(\phi)$.