User blog:Vel!/Second-order arithmetic

I recently made this diagram to help visualize some of the important first- and second-order arithmetic theories. Maybe it will help some of you out in understanding how they all fit together.




 * Q is.
 * EFA is.
 * PRA is.
 * RCA0 is recursive comprehension axiom.
 * WKL0 is weak König's lemma.
 * PA is Peano arithmetic.
 * ACA0 is arithmetical comprehension axiom.
 * ATR0 is arithmetical transfinite recursion.
 * Pi11-CA0 is \(\Pi_1^1\) comprehension axiom.
 * Z2 is second-order arithmetic.

The vertical division represents the languages used by these theories. Intuitively, theories on the left side can only understand statements about the natural numbers, while theories on the right side can understand statements about the real numbers.

To help make this scale more familiar, we'll compare it to some familiar functions: