Kurt Godel's Logical System

Kurt Godel successfully demonstrated how these standards are in fact paradoxical. At the core of mathematics there is a logical system of rules that enable us to count. Can this logical system be used to prove that it is itself without contradiction? Godel's answer was yes but only if the logical system is in fact contradictory. It seems then that a logical system is always in a catch 22. If a system can prove its own consistency then it has in fact proved the opposite. In other words a logical system cannot be both consistent and complete. Godel's incompleteness theorem revealed two things: 1) Not all mathematical truths are axiomatically grounded. 2) No consistent mathematical system can prove its own consistency and is thus incomplete. Godel's