I'm curious: If you add a new axiom in order to make a undecidable proposition decidable, will this new axiom add more than one undecidable proposition, or is the number of new undecidable propositions unknown? Or are there infinitely undecidable propositions for every set of rules?
If the former is true, couldn't we theoretically just apply axioms in order to solve undecidable propositions which have a practical application if solved? Or will it practically be hard to find these axioms?
It's been a long time since I had my Theory of Computation class, but as I recall, the proof that there are computationally undecidable questions uses the same diagonalization technique[1] used to demonstrate that the real numbers are not countably infinite.
The same technique can be used to show there are still computationally undecidable questions if you have a Turing Machine with an Oracle.
This proof technique suggests not only are there undecidable questions, there are uncountably many of them. Even if you have an Oracle.
It's more like this. You can prove more theorems by adding axioms, but if you add too many, then you might get an inconsistent proof system (containing undiscovered contradictions). Because of this, mathematicians are extreme conservatives when it comes to adding axioms, and if they ever got to that point they would stop believing they were doing mathematics.
If the former is true, couldn't we theoretically just apply axioms in order to solve undecidable propositions which have a practical application if solved? Or will it practically be hard to find these axioms?