Essential incompleteness [in axiom systems]

If a consistent axiom system is complete this means that any statement in the system can be proved true or false using its axioms, and the question of whether a statement is true can always be decided by a finite procedure. If an axiom system is incomplete then this means that there are statements that cannot be proved true or false using its axioms—and which must therefore be considered independent of those axioms. But even given this it is still possible that a finite procedure can exist which decides whether a given statement is true, and indeed this happens in the theory of commutative groups (see note below). But often an axiom system will not only be incomplete, but will also be what is called essentially incomplete. And what this means is that there is no finite set of axioms that can consistently be added to make the system complete. A consequence of this is that there can be no finite procedure that always decides whether a given statement is true—making the system what is known as essentially undecidable. (When I use the term "undecidable" I normally mean "essentially undecidable". Early work on mathematical logic sometimes referred to statements that are independent as being undecidable.)

One might think that adding rules to a system could never reduce its computational sophistication. And this is correct if with suitable input one can always avoid the new rules. But often these rules will allow transformations that in effect short-circuit any sophisticated computation. And in the context of axiom systems, adding axioms can be thought of as putting more constraints on a system—thus potentially in effect forcing it to be simpler. The result of all this is that an axiom system that is universal can stop being universal when more axioms are added to it. And indeed this happens when one goes from ordinary group theory to commutative group theory, and from general field theory to real algebra.