Problems whose answers are independent of the framework you are using (The continuum hypothesis). [1]
Undecidable problems. [2]
These are pretty much the same thing. The continuum hypothesis is a case where you have a single formal system in mind ( ZFC ) and have proved that the continuum hypothesis is independent of the axioms.
In the case of the halting problem, you just have a couple of extra quantifiers. For all formal systems that don’t prove a contradiction, there exists a Turing machine, such that whether the Turing machine halts or not can’t be proved from the axioms of the formal system. (Technically, the formal system needs to be R.E., which means that there is a computer program that can tell if an arbitrary string is an axiom. )
These are pretty much the same thing. The continuum hypothesis is a case where you have a single formal system in mind ( ZFC ) and have proved that the continuum hypothesis is independent of the axioms.
In the case of the halting problem, you just have a couple of extra quantifiers. For all formal systems that don’t prove a contradiction, there exists a Turing machine, such that whether the Turing machine halts or not can’t be proved from the axioms of the formal system. (Technically, the formal system needs to be R.E., which means that there is a computer program that can tell if an arbitrary string is an axiom. )