Given a sorted array and a value we know is in it, why can we know for sure that if we search it in linear order starting at the beginning, we will eventually always find the value at some index? It sounds very basic, and it's not difficult to prove, but the mythical algebra could be built on simple theorems like this, much like Eucledian geometry, and then expanded to cover deeper and more meaningful theorems.

Just a gut feeling, but I suspect most questions like these are going to map into set theory and number theory, in much the same way steady state kinetics equations map into the mathematics of graph theory.