Gödel's theorem

Here is a short and naive description of Gödel's theorem that I draw from R. Penrose, The emperor's new mind. From what I understand, if you lay down precisely all axioms necessary to define a logical system, you will find a contradiction very similar to the following statement:

this sentence is false.

This infinite indecision, similar to the image reflected onto two facing mirrors, is one of the driving argument in The moment of complexity by M. C. Taylor.

One important consequence of Gödel's theorem is that you find, again within the axioms and rules you have yourself constructed, propositions that are true even if you cannot find any proof of it within the system. Said more formally

"What Gödel showed was that any such precise ('formal') mathematical system of axioms and rules of procedure whatever, provided that it is broad enough to contain descriptions of simple arithmetical propositions [...] and provided that it is free from contradiction, must contain some statements which are neither provable nor disprovable by the means allowed within the system."

Thus even when playing with a mathematical system, will we find that 'truth' has not an absolute meaning and even R. Penrose conceives that

"[m]athematical truth is something that goes beyond mere formalism."