Notes on Recursive Analysis by R. L. Goodstein

"
Recursive analysis is a free variable theory of functions in a rational field, 
founded on recursive arithmetic.
It involves no logical presuppositions and proceeds from definition to theory 
by means of derivation schemata alone.

The elementary formulae of recursive arithmetic are equations between terms, 
and the class of formulae is constructed from the elementary formulae by the 
operations of propositional calculus.
The terms are the free numeral variables, the sign 0 and the signs for 
functions.
The function signs include the sign S(x) for the successor function (so that 
S(x) plays the part of x+1 in elementary algebra) and signs for functions 
introduced by recursion.
The derivation rules are taken to be sufficient to establish the universally 
valid sentences of the propositional calculus, and include a schema permitting 
the substitution of terms for variables, the schema of equality

a = b -> {A(a) -> A(b)},

and the induction schema

A(0), A(n) -> A(S(n))
---------------------
        A(n)

the schemata for explicit definition of functions for any number of arguments, 
and finally schemata for definition by recursion.
The simplest definition schema for recursion, the schema of primitive 
recursion, is 

f(0, a) = g(a), f(S(n), a) = h(n, a, f(n, a))

Specifically this schema defines f(n, a) by primitive recursion from the 
functions g and h.
We take as initial primitive recursive functions the successor function S(x), 
the identity function I(x), defined explicitly by the equation I(x)=x, and the 
zero function Z(x) defined by Z(x)=0.
A function is said to be primitive recursive if it is an initial function or is 
defined from primitive recursion functions by substitution or by primitive 
recursion.
" Goodstein Recursive Analysis Chapter 1 Section 1 pg. 1 and 2