Basic Model Theory

My Road to Unbelief

Interlude: Model Theory

Eventually I went to college. At this point I was still wavering between Deistic ideas and Atheistic ideas. I found out that many of the founding fathers were Deists (although most were Christians of various sects). If one could possibly prove that the universe is ordered in some way, I figured that would prove the existence of an almighty force, which would be labeled God. Hence Deism.

As a mathematics major I took a course in Set Theory and Model Theory, and to understand my arguments one needs a basic understanding of how Model Theory works. I found Model Theory to be one of the most effective tools to answering these kinds of questions.

In order to make statements, you have to start with initial assumptions about how things work. We call these initial assumptions Axioms. Once the Axioms are in place, they define your Model. Everything in the entire model can be derived from the initial Axioms. Usual mathematics that we are used to can be entirely derived from the Zermelo-Frankel Axioms. Just to get an idea of how basic the axioms are, I suggest you check them out (you can ignore all the complicated notation): ZF-Model Axioms. For instance, one axiom says that "the empty set exists" and another says that "two sets are equal if they contain the same elements."

These Axioms completely define the ZF-model. The model is unchanging, regardless of whether we discover more theorems and ideas inside of it. Those ideas were always in the model, we just hadn't discovered them yet. Often we want Calculus so we need the Axiom of Choice, which grants us the more powerful ZFC-model. The models have some important differences. For instance, in ZF there is no way to find an 'immeasurable set,' but in ZFC we can do it easily. It's not that immeasurable sets in ZF don't exist, it's just that they cannot be found in the model.

Now we can construct whatever Axioms we want, and create whatever model we want. Then prove things within those models. So in many ways we can essentially choose which model we are looking at. The models themselves are entirely objective, based on axioms, but the choice of model is subjective. It is not, as you will notice, arbitrary.

When considering models, there are some things one should be aware of. One is the Principle of Explosion. Essentially it states that if you allow for one statement to be both true and false in your model, all statements are rendered both true and false and your Model is trivial. In other words, contradictions and inconsistencies destroy the model.

The second and third major rules are Godel's Incompleteness Theorems. The first incompleteness theorem essentially says that within all models there are statements that cannot be proved (or disproved). You cannot have a single model that answers every question. The second incompleteness theorem says that it is impossible to prove the consistency of a model from within the model itself. So don't bother trying.

Anyway, using Models, I found that you make very powerful arguments about theology and morality. Simply construct some axioms and compare the models that are generated by them.