Gödel's results essentially conclude that there are True but Unprovable statements in arithmetic.
My thoughts are as follows:
Axioms form the foundation of mathematics -because we need to assume few things to get started. That is, in emptiness, where everything is possible, we assume few things, and then we create (or discover) new things -taking care that things stay consistent i.e. do not contradict each other.
What I wish to ask is: aren't the unprovable yet true statements (from Gödel's results) a consequence of the foundational assumptions (axioms) in the first place?
When we assumed the axioms, we did not prove them, but took them for granted. If a consistent system has been built on top of these axioms, only thing doubtful which remains are the axioms themselves. Anything then which, within theory, cannot be proved, but is true, should arise because of these axioms themselves. The hypothesis: Godel's statements are unprovable within the theory because they are the manifestations of the axioms themselves. They are true (within the theory) because they are a result of the axioms and the consistent theory developed upon them, but we find them to be unprovable. The source of this unprovability can only be attributed to the axioms we assumed, but did not prove. In effect, axioms, through the developed theory, only get translated to produce Gödel's statements.
These axioms, upon their assumption, should automatically set some constraints (because they impose constraints on what is allowed) in the empty space (which allows everything) -and these constraints can be said to be reflected through the existence of Gödel's unprovable (like our assumed axioms) but true statements (because our system, by incorporating axioms, assumes them to be true).
It is then also easy to see that mathematics is entirely a creation of mind, and not a platonic substance (as Gödel claimed).
The question therefore becomes:
If we could, hypothetically, prove all axioms perfectly, will there still exist Gödel's true but unprovable statements?