Chapter 12: The Principle of Computational Equivalence

Section 9: Implications for Mathematics and Its Foundations

[Universality of] algebraic axioms

How universality works with algebraic axioms depends on how those axioms are being used (compare page 1153). What is said in the main text here assumes that they are being used as on page 773—with each variable in effect standing for any object (compare page 1169), and with the axioms being added to predicate logic. The first of these points means that one is concerned with so-called pure group theory—and with finding results valid for all possible groups. The second means that the statements one considers need not just be of the form …==…, but can explicitly involve logic; an example is Cayley's theorem

(a·x == a·y \[Implies] x == y && Exists[z, a·z == x]) && (a·x == b·x \[Implies] a == b) && ((a·b)·x == a·(b·x))

With this setup, Alfred Tarski showed in 1946 that any statement in Peano arithmetic can be encoded as a statement in group theory—thus demonstrating that group theory is universal, and that questions about it can be undecidable. This then also immediately follows for semigroup theory and monoid theory. It was shown for ring theory and field theory by Julia Robinson in 1949. But for commutative group theory it is not the case, as shown by Wanda Szmielew around 1950. And indeed there is a procedure based on quantifier elimination for determining in a finite number of steps whether any statement in commutative group theory can be proved. (Commutative group theory is thus a decidable theory. But as mentioned in the note above, it is not complete—since for example it cannot establish the theorem a==b which states that a group has just one element. It is nevertheless not essentially incomplete—and for example adding the axiom a==b makes it complete.) Real algebra is also not universal (see page 1153), and the same is for example true for finite fields—but not for arbitrary fields.

As discussed on page 1141, word problems for systems such as groups are undecidable. But to set up a word problem in general formally requires going beyond predicate logic, and including axioms from set theory. For a word problem relates not, say, to groups in general, but to a particular group, specified by relations between generators. Within predicate logic one can give the relations as statements, but in effect one cannot specify that no other relations hold. It turns out, however, that undecidability for word problems occurs in essentially the same places as universality for axioms with predicate logic. Thus, for example, the word problem is undecidable for groups and semigroups, but is decidable for commutative groups.

One can also consider using algebraic axioms without predicate logic—as in basic logic or in the operator systems of page 801. And one can now ask whether there is then universality. In the case of semigroup theory there is not. But certainly systems of this type can be universal—since for example they can be set up to emulate any multiway system. And it seems likely that the axioms of ordinary group theory are sufficient to achieve universality.

From Stephen Wolfram: A New Kind of Science [citation]