Chapter 12: The Principle of Computational Equivalence

Section 9: Implications for Mathematics and Its Foundations

Finite axiomatizability

It is known that the axiom systems (such as Peano arithmetic and set theory) given with axiom schemas on pages 773 and 774 can be set up only with an infinite number of individual axioms. But because such axioms can be described by schemas they must all have similar forms, so that even though the definition in the main text suggests that each corresponds to an interesting theorem these theorems are not in a sense independently interesting. (Note that for example the theory of specifically finite groups cannot be set up with a finite number even of schemas—or with any finite procedure for checking whether a given candidate axiom should be included.)

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