If a set is defined by its membership function, then we can see that there exist membership functions that will not define a set but a proper class.
If we try to define the class of set membership functions, we would need to exclude proper class membership functions.
This means that this class needs a membership function that can determine if other membership functions are set membership functions or else proper class membership functions.
We can turn functions into programs that accept input and return output.
Is it valid to say that because of Turing's Halting problem, it is impossible to design a program that will determine if these other programs will even halt, let alone determine if they represent valid set membership functions?
In that case, is it correct to say that it will never be possible to mechanically distinguish between set membership functions and proper class membership functions?