2).

W. Ackermann, Zum Hilbertschen Aufbau der reellen Zahlen, Math. Annalen

**99** (1928), S. 118–133; Rózsa Péter, Konstruktion nichtrekursiver Funktionen, Math. Annalen

**111** (1935), S. 42–60.

Google Scholar3).

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 number*x* such that*A (x)*], ε*x* [*A (x)*] [the least natural number*x* such that*A (x)*, or 0 if there is no such number], — [not], ∀ [or], & [and],→[implies], ≡[is equivalent to].

4).

Kurt Gödel, Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatsh. für Math. u. Physik

**38** (1931), S. 173–198.

Google Scholar5).

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. Annalen

**110** (1934), S. 612–632.

Google Scholar7).

In these operations we do not require that*A* and*B=C* be equations and that σ be a functional variable, since R_{1}−R_{3} as stated when applied to equations generate equations. Thereby, our proof of IV is simplified.

8).

In what follows, the word “recursive” (when not qualified by the adjective “primitive”) will mean recursive under any one of the definitions 2a, 2b and 2c, except when the definition involved is mentioned explicitly (as is necessary in the course of establishing the theorems VI and IX on their equivalence).

9).

A more general definition would not be obtained by allowing under R_{3} also the substitution of*B* for*C*, since*E* may be chosen to include*b*=*a* whenever*a*=*b* is included.

9a).

Similarly, the equations of the system*E* of Def. 2b are verifiable by use of the values under Def. 2b, if they are of the form σ(*a*
_{1}, ...,*a*
_{8})=*b*.

9b).

Also see Th. Skolem, Begründung der elementaren Arithmetik durch die rekurrierende Denkweise ohne Anwendung scheinbarer Veränderlichen mit unendlichem Ausdehnungsbereich, Videnskapsselskapets Skrifter 1923. I. Mat. naturv. Kl., Nr. 6, S. 1–38.

10).

Note that*l*(I)=0 and*x**1=1**x*=*x*[l(*x*)>0].

12).

If*k*=0, replace “λ(0,*z*)=*l(z)*” by “λ(0,*z*)=1” in the definition of θ(*z, m*).

13).

Besides the method, for demonstrating that a function is not primitive recursive (or not definable by given additional means, such as recursions with respect to*n* variables simultaneously), which consists in finding a lower bound for the values, we have the method, for demonstrating relationships of the opposite kind, which consists in finding an upper bound for the number of steps in the computation algorithm.

16).

In XV below is given an example (*Ey*)*T*
_{1} (*x, x, y*) of a non-recursive class which by III is recursively enumerable.

18).

Since the means given for passing from definitions under Def. 2a (2b) to definitions under Def. 2c, and vice versa, are effective, the problem which we now study (which numbers define functions recursively) is equivalent to the one first proposed (which systems of equations define functions recursively).

20).

The undecidable proposition*A*
_{f} can be effectively constructed for a given logic, whenever the number*a*
_{k}, recursive definitions of*A (x); x, y B z*; β*(y)* and*C (n)*, and effective means of constructing*A*
_{f} from*f*, are given. Whenever the supposition in this proof, that there is a*k* such that*A*
_{k} is provable, is not realized, the theorem holds trivially.

23).

From the great generality of the problems, which*e*'s define recursively functions of one variable, and which*e*'s “determine recursively” the*e*
^{th} value of a function of one variable, as displayed by this theorem, the result, that they are not “effectively” soluble, could have been anticipated.

24).

E. g. (*x*
_{1},*x*
_{2},*x*
_{3})*R (x*
_{1},*x*
_{2},*x*
_{3}) ≡*(x) (Ey) [R*(1*Gl x*, 2*Gl x*, 3*Gl x*)≡&*y=y*].

25).

XV, XVI, and XVII are similar, respectively, to results obtained in a different connection by Prof. Alonzo Church (An unsolvable problem of elementary number theory, see Bull. Amer. Math. Soc. Abstract**41**-5-205), Dr. J. B. Rosser (unpublished), and the present writer (A theory of positive integers in formal logic, Part II, Amer. Jour. Math.**57** No. 2, pp. 230 ff.).