1. W. Ackermann, Zum Hilbertschen Aufbau der reellen Zahlen, Math. Annalen99 (1928), S. 118–133; Rózsa Péter, Konstruktion nichtrekursiver Funktionen, Math. Annalen111 (1935), S. 42–60.
2. In the “functions” which we consider, the arguments are understood to range over the natural numbers (i. e. non-negative integers) and the values to be natural numbers. Also, for abbreviation, we use propositional functions of natural numbers, calling them “relations” (alternatively “classes”, when there is only one variable) and employing the following notations:(x) A (x) [for all natural numbers,A (x)], (E x) A (x) [there is a natural numberx such thatA (x)], εx [A (x)] [the least natural numberx such thatA (x), or 0 if there is no such number], — [not], ∀ [or], & [and],→[implies], ≡[is equivalent to].
3. Kurt Gödel, Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatsh. für Math. u. Physik38 (1931), S. 173–198.
4. This form of the definition was introduced by Gōdel to avoid the necessity of providing for omissions of arguments on the right in schemas (1) and (2). The operations in the construction of primitive recursive functions can be further restricted. See Rózsa Péter, Über den Zusammenhang der verschiedenen Begriffe der rekursiven Funktionen, Math. Annalen110 (1934), S. 612–632.
5. In these operations we do not require thatA andB=C be equations and that σ be a functional variable, since R1−R3 as stated when applied to equations generate equations. Thereby, our proof of IV is simplified.