The Axiom of Choice, Goedel’s Theorem, and the Non-Aggression Principle.

APPENDIX TO THE APPENDIX

The Axiom of Choice and Goedel’s Incompleteness Theorem

It is almost certainly the case that you should skip this section. It had little to do with the rest of the book.

However, once upon a time I almost tried to become a professional mathematician. There is a specific issue that I find sufficiently annoying that I am going to discuss it here. You really want to turn ahead to the next Chapter now.

In short, one occasionally encounters assertions from some libertarians that all moral decisions can be ‘logically derived from the non-initiation principle’. My thesis here is that the phrase ‘logically derived’ as invoked in the previous sentence is a process of religious faith whose properties are fundamentally antilogical: They are indubitably logically inconsistent with the process ‘logically derived’ that most readers encountered in plane geometry.

Skip to the next Chapter. This is your last warning.

I am going to omit almost all mathematical details, so what I have to say reduced to four paragraphs.

First, we have the ‘Axiom of Choice’. What does the Axiom of Choice say? Suppose I form a collection of objects, all of which have some property. For example, I could make a list of all human women. According to the Axiom of Choice, I can then choose a representative person from that list, and we have agreement that ‘this person is a woman’. Now, given several interesting medical issues involving unusual chromosomal sorting, genetic defects, and the wonders of modern gender alteration surgery, there can be a range of opinions as to who belongs on that list. Sometimes one realizes that the definition of the list is incomplete or unambiguous. That doesn’t matter; definitions are in the end arbitrary. There is no claim that I can identify every single person on the list. The Axiom of Choice only claims that, for any list chosen to include all objects with a particular property, I can choose a representative object, which is prominent in no way except that it is an example of the objects on the list. Thus, I can choose a representative woman, who with respect to her membership on the list is distinguished only by being a human female. There is no implication that the object is typical in any sense. I may choose an average woman. I may choose the richest woman in the world. However, when discussing her, I am allowed only to refer to her as being female, not to her as being average or well-to-do. Similarly, when I choose a representative triangle, that triangle might or might not be a right triangle, but nothing in the proof can take advantage of the triangle’s being or not being a right triangle. The Axiom of Choice, once you understand it, sounds fairly obvious, except that it also applies to lists that have an infinite number of members.

Second, the Axiom of Choice is the basis of modern mathematical proofs. Modern proofs do not resemble the proofs that most readers saw in plane geometry. Modern proofs work by examining counterexamples. I give a simple case. Suppose we have a theorem ‘the sum of the internal angles of a triangle is 180 degrees’. A counterexample disproves the theorem. If I can show you a single triangle whose internal angles do not add to 180 degrees, I have disproven the theorem as stated. How in modern mathematics do I prove the theorem? I announce ‘consider a triangle whose angles do not add to 180 degrees’. I just invoked the Axiom of Choice. I selected a representative triangle with these particular properties and no others. Now I examine other properties of this odd triangle, and derive a contradiction, for example that the alleged triangle must have at least four corners. That’s a contradiction; by definition triangles only have three corners. Therefore, I have shown that any triangle that is a counterexample to the theorem ‘the sum of the internal angles of a triangle is 180 degrees’, is self-contradictory. It therefore does not exist. Ergo, all triangles must obey the theorem. Note that I have just proven the theorem without showing for even one triangle that the sum of the internal angles is 180 degrees.