I'm reading these independent notes on the Lectures on the Geometric Anatomy of Theoretical Physics, by Dr. Frederic P. Schuller. The first chapter is a brief summary of axiomatic systems and set theory. In page 6 the following definition is presented:
An axiomatic system $a_1, a_2, . . . , a_n$ is said to be consistent if there exists a proposition $q$ which cannot be proven from the axioms.
The motivation for such definition is that since within an axiomatic system which contains a contradiction every proposition is provable, a sufficient condition for a consistent system is that there exist propositions which cannot be deduced from the axioms. Then the following theorem is announced:
Any axiomatic system powerful enough to encode elementary arithmetic is either inconsistent or contains an undecidable proposition, i.e. a proposition that can be neither proven nor disproven within the system.
The Continuum hypothesis is then given as an example of an undecidable proposition within ZF.
Why isn't the undecidability of CH proof that ZF is consistent, if, after all, it should be provable if ZF was inconsistent?