Chapter 12: The Principle of Computational Equivalence

Section 9: Implications for Mathematics and Its Foundations

Model theory

In model theory each form of operator that satisfies the constraints of a given axiom system is called a model of that axiom system. If there is only one inequivalent model the axiom system is said to be categorical—a notion discussed for example by Richard Dedekind in 1887. The Löwenheim-Skolem theorem from 1915 implies that any axiom system must always have a countable model. (For an operator system such a model can have elements which are simply equivalence classes of expressions equal according to the axioms.) So this means that even if one tries to set up an axiom system to describe an uncountable set—such as real numbers—there will inevitably always be extra countable models. Any axiom system that is incomplete must always allow more than one model. The model intended when the axiom system was originally set up is usually called the standard model; others are called non-standard. In arithmetic non-standard models are obscure, as discussed on page 1169. In analysis, however, Abraham Robinson pointed out in 1960 that one can potentially have useful non-standard models, in which there are explicit infinitesimals—much along the lines suggested in the original work on calculus in the late 1600s.

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