Chapter 12: The Principle of Computational Equivalence

Section 9: Implications for Mathematics and Its Foundations

Proofs of axiom systems

One way to prove that an axiom system can reproduce all equivalences for a given operator is to show that its axioms can be used to transform any expression to and from a unique standard form. For then one can start with an expression, convert it to standard form, then convert back to any expression that is equivalent. We saw on page 616 that in ordinary logic there is a unique DNF representation in terms of And, Or and Not for any expression, and in 1921 Emil Post used essentially this to give the first proof that an axiom system like the first one on page 773 can completely reproduce all theorems of logic. A standard form in terms of Nand can be constructed essentially by direct translation of DNF; other methods can be used for the various other operators shown. (See also page 1175.)

Given a particular axiom system that one knows reproduces all equivalences for a given operator one can tell whether a new axiom system will also work by seeing whether it can be used to derive the axioms in the original system. But often the derivations needed may be very long—as on page 810. And in fact in 1948 Samuel Linial and Emil Post showed that in general the problem is undecidable. They did this in effect by arguing (much as on page 1169) that any multiway system can be emulated by an axiom system of the form on page 803, then knowing that in general it is undecidable whether a multiway system will ever reach some given result. (Note that if an axiom system does manage to reproduce logic in full then as indicated on page 814 its consequences can always be derived by proofs of limited length, if nothing else by using truth tables.)

Since before 1920 it has been known that one way to disprove the validity of a particular axiom system is to show that with k > 2 truth values it allows additional operators (see page 805). (Note that even if it works for all finite k this does not establish its validity.) Another way to do this is to look for invariants that should not be present—seeing if there are features that differ between equivalent expressions, yet are always left the same by transformations in the axiom system. (Examples for logic are axiom systems which never change the size of an expression, or which are of the form {expr a} where Flatten[expr] begins or ends with a.)

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