Axiom System

Now any axiom system characterises at least two groups. There is more than one model for any set of axioms. Mathematicians are only interested in one system. They ignore the fact that other models are characterised by the axioms.

Mr Harvie finally asserted that mathematics is uncontainable. This allows mathematicians to hope their subject is alive after all. You are free to accept or to reject any axiom. Mathematics is an unbounded subject.

In the ensuing discussion various points inferring a relationship were raised. Professor Hughes said you cannot find the dividing line between logic and mathematics. Doctor Lundy said you can work in both fields at once without difficulty.

However the main point discussed was set theory. It was claimed that up to the "axiom of choice" set theory is logical as it is concerned with class membership. This axiom is objected to on logical not on mathematical grounds. Godel however has pointed out that the use of the axiom of choice makes mathematics no more inconsistent than before. This has done away with a lot of objections to using this axiom.

—D.F