If it consistent, then it is incomplete.

Let PM be a formal deductive system.

Godel show that a formale G in PM is formally undecidable, if PM is consistent.
( this is plausible since for any deductive system that is consistent, there is at least one statement that is not deducible)

How did Godel show PM to be incomplete, if PM is consistent?
Here is how: We already know that G is undecidable. We need to show that G is true. How do we know that G is true? By 'metamathematical' thinking. Here is how:

(G) ' the formula G (with godel number g) is not derivable from PM'

Note that G is a metamathematical statement about itself.

Godel already show that G is not derivable from PM, and that G is formally undecidable. So, it is easy to see that G is true!


Thus:

If PM is consistent , then PM is incomplete.