“It's just impossible to write a general-purpose algorithm to solve it in all cases.” That is the definition of an undecidable problem. Independence, as in Eucl

“It's just impossible to write a general-purpose algorithm to solve it in all cases.” That is the definition of an undecidable problem. Independence, as in Euclid’s fifth postulate, is not what we typically think of as undecidability, though can be referred to as such.