Hi MRN,
Concerning axiomatic systems and increasing complexity.
The following is a small historical review of some of the items that we are discussing. I wrote it as a jumping off point with the hopes that I could actually come to some reasonable conclusions. As you will notice I failed! However I did discover that my implication that Euclidean Geometry was a simple system compared to the Natural Numbers is probably unwarranted. They are simply different systems.
I am somewhat fearful that it will be redundant to you, as I view you to be well read. You can skip ahead to Body Here if you prefer.
There are many simple axiomatic systems, some examples of which are included in the book Godel Esher Bach and I assume the researchers in Formal languages have developed many others. Probably the most famous example of an axiomatic system is Euclidean Geometry. I believe that, until recently, Euclid’s Elements was the second highest selling book of all time.
Curiously enough, Godel offered a proof that Euclidean Geometry (technically the rigorously axiomatized form of Euclidean Geometry completed by David Hilbert) was both complete and consistent. Complete means that all true statements could be logically derived from the axioms or combinations of other statements (Theorems) that had previously been derived from the axioms. Consistent simply means that both proper mathematical statements A and Not A can be simultaneously true. This prompted Hilbert (significantly more important than most people realize), Russell and Whitehead to speculate that the Natural Numbers could be axiomatized and this system could be both complete and consistent. Additionally it should be noted that depending on the axioms in place there could be a number of different but consistent types of arithmetic.
But Godel in his famous theorem showed that an axiomatic approach to the Natural Numbers yielded a system that could not be both complete and consistent. Godel at the time did not realize it, but any larger system, such as the Whole Numbers, the Rational Numbers, and the Real Numbers would also have this same property.
So we have that Euclidean Geometry is complete and consistent, but an axiomatized Natural Numbers is not capable of being both complete and consistent.
Body Here:
I have wondered over the years what the consequences of adding and subtracting axioms from a given axiomatic system might be.
Very simple axiomatic systems appear to gain us very little information. For example if you assume that Descartes’ Cognito is true (a highly contentious statement), then you can only conclude that it is true. There is nothing else to discover.
With other more complex systems, we can even drop axioms and still have a meaningful system. Dropping the parallel postulate from Euclidean Geometry gives us the two standard non Euclidean Geometries.
It is clear that some systems can be greatly expanded with addional axioms. For example the Natural Numbers can be expanded to the Rationals. The Rationals can be expanded to the Algebraic (solutions to the polynomial equations can be added) and the Algebraic can be expanded to the Real by adding the transcendental numbers and finally the Complex Numbers can be added by adding i.
A confusing point about the Reals is that they are said to be complete. However the Reals are complete only in the technical sense that all sequences of Real Numbers, where the adjacent terms get arbitrarily close to one another (Cauchy sequences), actually converge to a Real Number. The Reals are not complete by Godel’s definition.
Aside from the obvious fact that useful axiomatic systems can give us interesting and useful tools by playing with the axioms (adding and subtracting axioms), I have been curious about the concept of adding a religious deity to the mix or at least developing a separate Ethical System with a presumption (axiom) of a religious deity.
I have also noted that different axiomatic systems can be combined. Mathematicians regularly study Analytic and Differential Geometry. I have never seen anyone explicitly mention the axioms used in these studies.
On how we develop axiomatic systems.
This is certainly getting into a contentious area. Do we have free will? Do we discover these systems? (Are they Platonic Ideals or Kantian synthetic a priori)? Do we invent systems? I will mention that you should take Godel with a grain of salt, because Godel did not create (if that is the right concept) any axiomatic systems. Like a good mathematician (or properly logician) he worked logically within a system.
Personally, I do not subscribe to any preexisting knowledge. For me the constructs that we develop are a process of abstraction drawn from the intellects’ (a genetic component should be considered) ability to differentiate and integrate. And of course the accumulation of data and experience.
No great insights. Sorry