Logic and axioms

From: Andrew Barton <AndrewBarton_at_compuserve.com>
Date: Wed, 14 Mar 2001 13:47:14 -0500

> So if "because..." is an axiom the logical analysis stops here? What
about
such basic axioms as "If two straight lines are parallel..." They are unproveable despite being pretty obvious and actually observable.

You can't 'prove' an axiom, no. You can test by observation whether it seems to apply to the real world - once you know the world is round you know that there are -no- parallel lines for the purposes of navigating the ocean.

You can apply logic to an axiom (or set of axioms) and discover that it leads to a contradiction. The best-known example is Russell's 'Spanish Barber' paradox which showed that the set theory of his day was inconsistent.

Sometimes you can prove that a set of axioms -is- consistent.

These statements all derive from progress in mathematical logic and model theory as developed in the twentieth century. It's unlikely that anyone in Glorantha would know of them.

Andrew

---