The natural numbers are characterized by their induction principle (in second-order logic/in a higher universe/as an inductive type). If one only has a first order theory, then one cannot have an induction principle, and instead one has a entire category of models. Thus, the first order models of arithmetic typically found in classical logic and model theory do not define the natural numbers, and this is true even of first-order Peano arithmetic.
Real numbers
Lattices and -frames
A lattice is a set with terms , and functions , , such that