Chapter 12: The Principle of Computational Equivalence

Section 9: Implications for Mathematics and Its Foundations

Properties [of operators from axioms]

There are kk2 possible forms for binary operators with k possible values for each argument. There is always at least some operator that satisfies the constraints of any given axiom system—though in a case like a == b it has k=1. Of the 274,499 axiom systems of the form {… == a} where involves up to 6 times, 32,004 allow only operators {6,9}, while 964 allow only {1,7}. The only cases of 2 or less operators that appear with k=2 are {{}, {10}, {12}, {1, 7}, {3, 12}, {5, 10}, {6, 9}, {10, 12}}. (See page 1174.)

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