Well, with Godel's Incompleteness Theorems, such behavior appears to be an unavoidable property of any mathematical or computational system that can do anything of note. We may or may not be able to build systems where such properties are further removed from the 'practical' aspects... although such work would necessarily build upon the work in the linked post, which is finding just how far removed it is in the current system.
Not quite "anything of note" it has to be able to do arithmetic. There are very simple systems that are useful but not powerful enough for Gödel's trick and so we think those would be safe.
