Notes

Chapter 12: The Principle of Computational Equivalence

Section 9: Implications for Mathematics and Its Foundations


Consistency [in axiom systems]

Any axiom system that is universal can represent the statement that the system is consistent. But normally such a statement cannot be proved true or false within the system itself. And thus for example Kurt Gödel showed this in 1931 for Peano arithmetic (in his so-called second incompleteness theorem). In 1936, however, Gerhard Gentzen showed that the axioms of set theory imply the consistency of Peano arithmetic (see page 1160). In practical mathematics set theory is always taken to be consistent, but to set up a proof of this would require axioms beyond set theory.

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