Chapter 12: The Principle of Computational Equivalence

Section 9: Implications for Mathematics and Its Foundations

Forcing of operators [by axiom systems]

Given a particular set of forms for operators one can ask whether an axiom system can be found that will allow these but no others. As discussed in the note on operators on sets on page 1171 some straightforwardly equivalent forms will always be allowed. And unless one limits the number of elements k it is in general undecidable whether a given axiom system will allow no more than a given set of forms. But even with fixed k it is also often not possible to force a particular set of forms. And as an example of this one can consider commutative group theory. The basic axioms for this allow forms of operators corresponding to multiplication tables for all possible commutative groups (see note above). So to force particular forms of operators would require setting up axioms satisfied only by specific commutative groups. But it turns out that given the basic axioms for commutative group theory any non-trivial set of additional axioms can always be reduced to a single axiom of the form an==1 (where exponentiation is repeated application of ·). Yet even given a particular number of elements k, there can be several distinct groups satisfying an==1 for a given exponent n. (The groups can be written as products of cyclic ones whose orders correspond to the possible factors of n.) (Something similar is also known in principle to be true for general groups, though the hierarchy of axioms in this case is much more complicated.)

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