Chapter 12: The Principle of Computational Equivalence

Section 9: Implications for Mathematics and Its Foundations

Axiom schemas

An axiom like a + 0 == a is a single well-formed formula in the sense of page 1150. But sometimes one needs infinite collections of such individual axioms, and in the main text these are represented by axiom schemas given as Mathematica patterns involving objects like x_. Such schemas are taken to stand for all individual axioms that match the patterns and are well-formed formulas. The induction axiom in arithmetic is an example of a schema. (See the note on finite axiomatizability on page 1176.) Note that as mentioned on page 1150 all the axioms given for basic logic should really be thought of as schemas.

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