Presburger arithmetic is an axiom system for the natural numbers under addition.

However, we do have a limited tool for capturing some metaphors: axiom systems.

A more modern notion of an axiom system is slightly different.

For our notion of an axiom system, we actually have two components:

In other words, an axiom system is a set of true statements about words that have no meaning.

If you have never seen an axiom system before, that probably sounds ridiculous.

For example, suppose we wanted to study geometry with this type of axiom system.

As this example illustrates, axiom systems capture the idea of abstraction.

The adjustments to be made depend upon the axiom system being used.

He also argued that there is no shorter axiom system.