So consider this system.
Axiom 1. A point exists.
Axiom 2. From any point, you may draw a 1-unit arrow down and to the left. The end of the arrow is a point.
Axiom 3. From any point, you may draw a 1-unit arrow down and to the right. The end of the arrow is a point.
The metalogic of the system is that two diagrams are equal iff they have the same arrow structure, and one diagram is larger than another iff the first contains all the arrows of the second but the two are not equal.
So "/\" > "/" > "" and "/\" > "\" > "", but not ("/" > "\") and not ("\" > "/"). The system can make many different theorems ("diagrams" in its own terminology) but they don't work like the natural numbers, or any sensible number line at all.