Edward Nelson, Warning Signs of a Possible Collapse of Contemporary Mathematics (pdf)

Nelson arithmetic

In the same paper he defined a first order theory that was weaker than Peano arithmetic, which replaces the induction schema of induction on natural numbers with a predicate and two axioms that state that induction on natural numbers for that predicate is true:

Edward Nelson, Warning Signs of a Possible Collapse of Contemporary Mathematics (pdf)

He did not give a name to the theory, so it has been called “Nelson arithmetic” for lack of a better name.