Third, until early in the last century the objective of mathematics was to reduce all results to logical derivations from a few axioms. Along came the German mathematician Kurt Goedel. Goedel proved, using the Axiom of Choice, that except in trivially simple logical systems you cannot produce a simple set of axioms that describe all mathematical results. That is, in any complicated mathematical system there are an infinite number of statements that are true, but that cannot be derived from any simple set of axioms using orthodox logic.* Alternatively, Goedel showed that excepting truly trivial logical systems all complete logical systems have an infinite number of independent axioms.
Fourth, we now return to the statement that all moral decisions can be logically derived from the non-initiation principle, which may be phrased as an axiom: ‘any act that violates the non-initiation principle is immoral’. Any moral question may be phrased as a theorem ‘this action does not have the property immoral.’ Morality is not mathematically trivial. Ergo, from the Axiom of Choice and the Goedel Incompleteness Theorem there are actions that are moral or that are immoral that cannot be logically proven to be moral or immoral from any short set of axioms. Claims that one can logically derive all moral conclusions from the Non-Initiation Principle are claims that an entire nontrivial logical system can be derived from a single axiom. Such claims are mathematically incompatible with the properties of mathematical logic, and must be recognized as non-logical statements of faith.
FOOTNOTE TO THE APPENDIX TO THE APPENDIX
*The Axiom of Choice could be in error. There are several truly remarkable mathematical results that do not look entirely reasonable that have been derived using the axiom. There is an alternative to the axiom that I have seen given several names, e.g., constructivism, which holds that proof by showing the falsity of counterexamples is invalid. Valid proofs must advance by positive calculation. For example, suppose you want to claim that you can take a sphere, cut it into five parts, and reassemble your five pieces into two new spheres, each, by the way, having the same volume as the initial sphere. To do so via constructivism you must specify the cuts. [No, that is not an arbitrary example; it is a very important example.] With the Axiom of Choice you can prove that you can slice a sphere into five parts and reassemble them into two spheres, each having the volume of the original sphere.
See, I told you that you should skip to the next Chapter.
