The modernity of Dedekind’s anticipations contained in What are numbers and what are they good for?

Abstract

We show that Dedekind, in his proof of the principle of definition by mathematical recursion, used implicitly both the concept of an inductive cone from an inductive system of sets and that of the inductive limit of an inductive system of sets. Moreover, we show that in Dedekind’s work on the foundations of mathematics one can also find specific occurrences of various profound mathematical ideas in the fields of universal algebra, category theory, the theory of primitive recursive mappings, and set theory, which undoubtedly point towards the mathematics of twentieth and twenty-first centuries.

Appendix A: A many-sorted algebraic model of the theory of primitive recursive mappings

We next provide—in the algebraic spirit of Dedekind—a new model, based on many-sorted universal algebra, of the theory of primitive recursive mappings which will allow us to diagrammatically prove that the three basic mathematical operations, addition, multiplication, and exponentiation, defined by Dedekind in $\S \S$ 11–13 of Dedekind (1888), are primitive recursive. We remark that the just-mentioned many-sorted algebraic model, appropriately modified, can be used to supply many-sorted algebraic models of the theories of general recursive mappings and of recursive partial mappings. Moreover, the subalgebra generating operators associated with these models make the notion of relative recursiveness, in each case, natural.

Since the set of all primitive recursive mappings will be the union of the underlying many-sorted set of the smallest many-sorted subalgebra of a convenient many-sorted algebra, we begin by defining the many-sorted signature of the many-sorted algebra at issue. Note that in what follows $\mathbb {N}^{\star }$ stands for the underlying set of the free monoid on $\mathbb {N}$, that $\curlywedge$ is the binary operation of concatenation on $\mathbb {N}^{\star }$, and that $\lambda$ denotes the empty word on $\mathbb {N}$. Moreover, given sets A, B, and C, and mappings and , we denote by $\left<f,g\right>$ the unique mapping from C to $A\times B$ such that $\mathrm {pr}_{A}\circ \left<f,g\right> = f$ and $\mathrm {pr}_{B}\circ \left<f,g\right> = g$, where $\mathrm {pr}_{A}$ is the canonical projection from $A\times B$ to A and $\mathrm {pr}_{B}$ the canonical projection from $A\times B$ to B.

Definition

We denote by $\varvec{{\varSigma }}^{\mathrm {pr}}$ the $\mathbb {N}$-sorted signature, for the primitive recursive mappings, whose (w, n)-th coordinate, where $(w,n)\in \mathbb {N}^{\star }\times \mathbb {N}$, is defined as follows:

$$\begin{aligned} \mathop {\varSigma }\nolimits ^{\mathrm {pr}}_{w,n} = {\left\{ \begin{array}{ll} \{\kappa _{0,0}\}, &{} \text {if }w =\lambda \text { and }n=0; \\ \{\mathrm {sc}\}\cup \{\mathrm {pr}_{1,0}\}, &{} \text {if }w=\lambda \text { and } n=1; \\ \{\,\mathrm {pr}_{n,i}\mid i\in n\,\}, &{} \text {if } w =\lambda \text { and }n\ge 2; \\ \left\{ {\varOmega }_{\mathrm {C}}^{m,n}\right\} , &{} \text {if } w=(m)\curlywedge (n\mid i\in m) \text { and }m\ge 1; \\ \left\{ {\varOmega }_{\mathrm {R}}^{m}\right\} , &{} \text {if }w=(m)\curlywedge (m+2)\text { and } n=m+1; \\ \varnothing , &{} \text {otherwise.} \end{array}\right. } \end{aligned}$$

Definition

We denote by the many-sorted $\varvec{{\varSigma }}^{\mathrm {pr}}$-algebra whose underlying $\mathbb {N}$-sorted set, , is $(\mathrm {Hom}(\mathbb {N}^{n},\mathbb {N}))_{n\in \mathbb {N}}$, so its n-th coordinate is the set of all mappings from $\mathbb {N}^{n}$ to $\mathbb {N}$, and where the structural operations are the following:

• 1. $\kappa _{0,0}$, the 0-ary constant mapping determined by 0, which is the mapping from $\mathbb {N}^{0}$ to $\mathbb {N}$ that sends the unique element of $\mathbb {N}^{0}$ to 0.

• 2. $\mathrm {sc}$, the successor mapping.

• 3. $\mathrm {pr}_{1,0}$, the identity mapping at $\mathbb {N}$.

• 4. For every $n\ge 2$ and every $i\in n$, $\mathrm {pr}_{n,i}$, the i-th canonical projection from $\mathbb {N}^{n}$ to $\mathbb {N}$.

• 5. For every $m\in \mathbb {N}-1$ and every $n\in \mathbb {N}$, ${\varOmega }_{\mathrm {C}}^{m,n}$, the operator of (generalized) composition of arity $(m)\curlywedge (n\mid i\in m)$ and co-arity n, which is the mapping from $\mathrm {Hom}(\mathbb {N}^{m},\mathbb {N})\times (\mathrm {Hom}(\mathbb {N}^{n},\mathbb {N}))^{m}$ to $\mathrm {Hom}(\mathbb {N}^{n},\mathbb {N})$ that sends a pair $(f,(g_{i})_{i\in m})$ in $\mathrm {Hom}(\mathbb {N}^{m},\mathbb {N})\times (\mathrm {Hom}(\mathbb {N}^{n},\mathbb {N}))^{m}$ to the mapping ${\varOmega }_{\mathrm {C}}^{m,n}(f,(g_{i})_{i\in m})$ from $\mathbb {N}^{n}$ to $\mathbb {N}$ obtained by composing $\left<g_{i}\right>_{i\in m}$ and f, where $\left<g_{i}\right>_{i\in m}$ is the unique mapping from $\mathbb {N}^{n}$ to $\mathbb {N}^{m}$ such that, for every $i\in m$, $g_{i} = \mathrm {pr}_{m,i}\circ \left<g_{i}\right>_{i\in m}$.

• 6. For every $m\in \mathbb {N}$, ${\varOmega }_{\mathrm {R}}^{m}$, the operator of primitive recursion (also known as the parameterized operation of primitive recursion) of arity $(m)\curlywedge (m+2)$ and co-arity $m+1$, which is the mapping from $\mathrm {Hom}(\mathbb {N}^{m},\mathbb {N})\times \mathrm {Hom}(\mathbb {N}^{m+2},\mathbb {N})$ to $\mathrm {Hom}(\mathbb {N}^{m+1},\mathbb {N})$ that sends a pair (f, g) in $\mathrm {Hom}(\mathbb {N}^{m},\mathbb {N})\times \mathrm {Hom}(\mathbb {N}^{m+2},\mathbb {N})$ to the mapping ${\varOmega }_{\mathrm {R}}^{m}(f,g)$ from $\mathbb {N}^{m+1}$ to $\mathbb {N}$ obtained from f and g by primitive recursion. Let us recall that, for every $m\in \mathbb {N}$, if and , then, from the principle of definition by mathematical recursion, it follows immediately that there exists a unique mapping such that the following diagram:

  

  commutes (where $\omega _{\mathbb {N}^{m}}$ is the unique mapping from $\mathbb {N}^{m}$ to $1 = \{0\}$ and $\kappa _{0}$ the mapping from 1 to $\mathbb {N}$ that sends 0 to 0), i.e., such that:

  • (a) $\forall a\in \mathbb {N}^{m}\,(h(a,0) = f(a)).$

  • (b) $\forall a\in \mathbb {N}^{m}\,\,\forall n\in \mathbb {N}\,(h(a,\mathrm {sc}(n)) = g(a,n,h(a,n))).$

In the above definition, we have identified the set $\mathrm {Hom}(\mathbb {N}^{1},\mathbb {N})$ with the set $\mathrm {End}(\mathbb {N})$, of all endomorphisms of the set $\mathbb {N}$. Moreover, to simplify the notation, no distinction has been made between the symbols of operation and their realizations on the $\mathbb {N}$-sorted set $(\mathrm {Hom}(\mathbb {N}^{n},\mathbb {N}))_{n\in \mathbb {N}}$.

Since it will be used below to define the set of all primitive recursive mappings, we will next recall under what conditions an $\mathbb {N}$-sorted subset of the underlying $\mathbb {N}$-sorted set of is a subalgebra of the many-sorted $\varvec{{\varSigma }}^{\mathrm {pr}}$-algebra .

Definition

An $\mathbb {N}$-sorted subset $\mathcal {F}=(\mathcal {F}_{n})_{n\in \mathbb {N}}$ of the underlying $\mathbb {N}$-sorted set of is a subalgebra of exactly if it satisfies the following conditions:

• 1. $\kappa _{0,0}\in \mathcal {F}_{0}$.

• 2. $\mathrm {sc}\in \mathcal {F}_{1}$.

• 3. $\mathrm {pr}_{1,0}\in \mathcal {F}_{1}$.

• 4. For every $n\ge 2$ and every $i\in n$, $\mathrm {pr}_{n,i}\in \mathcal {F}_{n}$.

• 5. For every $m\in \mathbb {N}-1$, every $n\in \mathbb {N}$, every $f\in \mathcal {F}_{m}$, and every $(g_{i})_{i\in m}\in (\mathcal {F}_{n})^{m}$, ${\varOmega }_{\mathrm {C}}^{m,n}(f,(g_{i})_{i\in m})\in \mathcal {F}_{n}$.

• 6. For every $m\in \mathbb {N}$, every $f\in \mathcal {F}_{m}$, and every $g\in \mathcal {F}_{m+2}$, ${\varOmega }_{\mathrm {R}}^{m}(f,g)\in \mathcal {F}_{m+1}$.

We denote by the set of all subalgebras of the many-sorted $\varvec{{\varSigma }}^{\mathrm {pr}}$-algebra .

We next state the fundamental properties of the set of all subalgebras of .

Proposition

The set , of all subalgebra of the many-sorted $\varvec{{\varSigma }}^{\mathrm {pr}}$-algebra , is an algebraic closure system on , i.e., satisfies the following conditions:

• 1. $(\mathrm {Hom}(\mathbb {N}^{n},\mathbb {N}))_{n\in \mathbb {N}}$ is a subalgebra de .

• 2. If $(\mathcal {F}^{i})_{i\in I}$ is a nonempty family of subalgebras of , then $\bigcap _{i\in I}\mathcal {F}^{i}$ is a subalgebra of .

• 3. If $(\mathcal {F}^{i})_{i\in I}$ is a nonempty family of subalgebras of , and if, for every $i,j\in I$, there exist $k\in I$ such that $\mathcal {F}^{i}\cup \mathcal {F}^{j}\subseteq \mathcal {F}^{k}$, then $\bigcup _{i\in I}\mathcal {F}^{i}$ is a subalgebra of .

From the algebraic closure system on we obtain the algebraic closure operator $\mathrm {Sg}_{\mathrm {pr}}$ on induced by it, as stated in the following corollary.

Corollary

For the many-sorted $\varvec{{\varSigma }}^{\mathrm {pr}}$-algebra , the endomorphism $\mathrm {Sg}_{\mathrm {pr}}$ of the set , of all $\mathbb {N}$-sorted subsets of , defined as:

has the following properties:

• 1. , i.e., for every , $\mathrm {Sg}_{\mathrm {pr}}(\mathcal {F})$ is a subalgebra of .

• 2. , i.e., the fixed points of $\mathrm {Sg}_{\mathrm {pr}}$ are exactly the subalgebras of .

• 3. $\mathrm {Sg}_{\mathrm {pr}}$ is extensive, i.e., for every , it happens that $\mathcal {F}\subseteq \mathrm {Sg}_{\mathrm {pr}}(\mathcal {F})$.

• 4. $\mathrm {Sg}_{\mathrm {pr}}$ is isotone, i.e., for every , if $\mathcal {F}\subseteq \mathcal {G}$, then $\mathrm {Sg}_{\mathrm {pr}}(\mathcal {F})\subseteq \mathrm {Sg}_{\mathrm {pr}}(\mathcal {G})$.

• 5. $\mathrm {Sg}_{\mathrm {pr}}$ is idempotent, i.e., for every , it happens that $\mathrm {Sg}_{\mathrm {pr}}(\mathcal {F}) = \mathrm {Sg}_{\mathrm {pr}}(\mathrm {Sg}_{\mathrm {pr}}(\mathcal {F}))$.

• 6. $\mathrm {Sg}_{\mathrm {pr}}$ is algebraic, i.e., for every nonempty family $(\mathcal {F}^{i})_{i\in I}$ in , if, for every $i,j\in I$, there exists a $k\in I$ such that $\mathcal {F}^{i}\cup \mathcal {F}^{j}\subseteq \mathcal {F}^{k}$, then $\mathrm {Sg}_{\mathrm {pr}}(\bigcup _{i\in I}\mathcal {F}^{i}) = \bigcup _{i\in I}\mathrm {Sg}_{\mathrm {pr}}(\mathcal {F}^{i})$.

Thus, for any , $\mathrm {Sg}_{\mathrm {pr}}(\mathcal {F})$ is the smallest subalgebra of that includes $\mathcal {F}$. $\mathrm {Sg}_{\mathrm {pr}}(\mathcal {F})$ is called the subalgebra of generated by $\mathcal {F}$ and $\mathrm {Sg}_{\mathrm {pr}}$ the subalgebra generating operator on .

Definition

Let $\mathcal {F} = (\mathcal {F}_{n})_{n\in \mathbb {N}}$ be a countable $\mathbb {N}$-sorted subset of , i.e., an $\mathbb {N}$-sorted subset of such that $\mathrm {card}(\coprod \mathcal {F})\le \aleph _{0}$. Then we call the mappings in $\bigcup \mathrm {Sg}_{\mathrm {pr}}(\mathcal {F})$, the union of the subalgebra of generated by $\mathcal {F}$, the $\mathcal {F}$-primitive recursive mappings or the primitive recursive mappings relative to $\mathcal {F}$, and, to shorten the notation, we will write $\mathrm {PRM}(\mathcal {F})$ instead of $\bigcup \mathrm {Sg}_{\mathrm {pr}}(\mathcal {F})$. In particular, the set of all primitive recursive mappings, denoted by $\mathrm {PRM}$, is $\bigcup \mathrm {Sg}_{\mathrm {pr}}((\varnothing )_{n\in \mathbb {N}})$, the union of the subalgebra of generated by the $\mathbb {N}$-sorted set $(\varnothing )_{n\in \mathbb {N}}$ (which is constantly empty).

We next provide, for a countable $\mathbb {N}$-sorted subset $\mathcal {F}$ of , a constructive characterization of $\mathrm {PRM}(\mathcal {F})$ (which is also valid when $\mathcal {F}$ is arbitrary).

Proposition

Let $\mathcal {F} = (\mathcal {F}_{n})_{n\in \mathbb {N}}$ be an $\mathbb {N}$-sorted subset of such that $\mathrm {card}(\coprod \mathcal {F})\le \aleph _{0}$ and $f\in \bigcup _{n\in \mathbb {N}}\mathrm {Hom}(\mathbb {N}^{n},\mathbb {N})$. Then $f\in \mathrm {PRM}(\mathcal {F})$ if, and only if, there exists a formative construction for f relative to $\varvec{{\varSigma }}^{\mathrm {pr}}$ and $\mathcal {F}$, i.e., if, and only if, there exists a $p\in \mathbb {N}-1$ and a family $(f_{i})_{i\in p}$ in $\bigcup _{n\in \mathbb {N}}\mathrm {Hom}(\mathbb {N}^{n},\mathbb {N})$ such that $f = f_{p-1}$ and, for every $i\in p$, it happens that:

• 1. $f_{i}\in \mathcal {F}_{n}$, for some $n\in \mathbb {N}$, or

• 2. $f_{i} = \kappa _{0,0}$, or

• 3. $f_{i} = \mathrm {sc}$, or

• 4. $f_{i} = \mathrm {pr}_{1,0}$, or

• 5. $f_{i} = \mathrm {pr}_{n,j}$, for some $n\ge 2$ and some $j\in n$, or

• 6. $f_{i}$ is $m+1$-ary and $f_{i} = {\varOmega }_{\mathrm {R}}^{m}(f_{j},f_{k})$, for a j and a $k\in i$ such that $f_{j}$ is m-ary and $f_{k}$ is $m+2$-ary, or

• 7. $f_{i}$ is n-ary and $f_{i} = {\varOmega }_{\mathrm {C}}^{m,n}(f_{j},(f_{k_{\alpha }})_{\alpha \in m})$, for an $m\in \mathbb {N}-1$, a $j\in i$ and a family $(k_{\alpha })_{\alpha \in m}\in i^{m}$ such that $f_{j}$ is m-ary and, for every $\alpha \in m$, $f_{k_{\alpha }}$ is n-ary.

Proof

Let $\mathcal {L}$ denote the $\mathbb {N}$-sorted subset of whose n-th coordinate, $\mathcal {L}_{n}$, for $n\in \mathbb {N}$, has as elements all mappings $f\in \mathrm {Hom}(\mathbb {N}^{n},\mathbb {N})$ for which there exists a formative construction for f relative to $\varvec{{\varSigma }}^{\mathrm {pr}}$ and $\mathcal {F}$. Since $\mathrm {PRM}(\mathcal {F})$ is the union of the underlying $\mathbb {N}$-sorted set of the least subalgebra of that contains $\mathcal {F}$, to show that $\mathrm {PRM}(\mathcal {F})\subseteq \bigcup _{n\in \mathbb {N}}\mathcal {L}_{n}$ it suffices to verify that $\mathcal {L}$ contains $\mathcal {F}$ and that $\mathcal {L}$ is a subalgebra of . $\square$

We have that $\mathcal {F}\subseteq \mathcal {L}$, since, for $n\in \mathbb {N}$ and $f\in \mathcal {F}_{n}$, the family $(f_{i})_{i\in 1}$ with $f_{0} = f$, is a formative construction for f. It is obvious that $\kappa _{0,0}\in \mathcal {L}_{0}$, that $\mathrm {sc}$ and $\mathrm {pr}_{1,0}\in \mathcal {L}_{1}$ and that, for every $n\ge 2$ and every $j\in n$, $\mathrm {pr}_{n,j}\in \mathcal {L}_{n}$. Moreover, given an $m\in \mathbb {N}-1$, an $n\in \mathbb {N}$, an $f\in \mathcal {L}_{m}$, and an m-indexed family $(g_{j})_{j\in m}$ in $\mathcal {L}_{n}$, then, by virtue of the definition of $\mathcal {L}$, there exists a formative construction $(f_{i})_{i\in n_{f}}$ for f and, for every $j\in m$, there exists a formative construction $(f_{j,i})_{i\in n_{j}}$ for $g_{j}$. Since it may be helpful for the sake of understanding, let us represent the situation just described by the following figure:

$$\begin{aligned} \begin{matrix} f_{0} &{} f_{1} &{} ... &{} f_{n_{f}-1}= f\\ f_{0,0} &{} f_{0,1} &{} ... &{} f_{0,n_{0}-1}= g_{0}\\ f_{1,0} &{} f_{1,1} &{} ... &{} f_{1,n_{1}-1}= g_{1}\\ \vdots &{} \vdots &{}\ddots &{} \vdots \\ f_{m-1,0} &{} f_{m-1,1} &{} ... &{} f_{m-1,n_{m-1}-1}= g_{m-1} \end{matrix} \end{aligned}$$

Then, for $n = n_{f}+\left( \sum _{j\in m}n_{j}\right) +1$ and taking as $(h_{i})_{i\in n}$ the family of mappings whose last term is ${\varOmega }_{\mathrm {C}}^{m,n}(f,(g_{j})_{j\in m})$ and being the other terms those formed by the terms of the previous figure, going from left to right and from top to bottom, we have that $(h_{i})_{i\in n}$ is a formative construction for ${\varOmega }_{\mathrm {C}}^{m,n}(f,(g_{j})_{j\in m})$. Hence ${\varOmega }_{\mathrm {C}}^{m,n}(f,(g_{j})_{j\in m})\in \mathcal {L}_{n}$. In the same way one proves that $\mathcal {L}$ is closed under ${\varOmega }_{\mathrm {R}}^{m}$. Therefore $\mathcal {L}$ is a subalgebra of . From all this we conclude that $\mathrm {PRM}(\mathcal {F}) \subseteq \bigcup _{n\in \mathbb {N}}\mathcal {L}_{n}$.

We next show, by induction limited to an initial segment of $\mathbb {N}$, that $\bigcup _{n\in \mathbb {N}}\mathcal {L}_{n}$ is a subset of $\mathrm {PRM}(\mathcal {F})$.

Let $n\in \mathbb {N}$ and $f\in \mathcal {L}_{n}$. Then, by definition, there exists a $p\in \mathbb {N}-1$ and a family $(f_{i})_{i\in p}$ in $\bigcup _{n\in \mathbb {N}}\mathrm {Hom}(\mathbb {N}^{n},\mathbb {N})$ such that $f = f_{p-1}$ and, for every $i\in p$, we have that $f_{i}\in \mathcal {F}_{n}$, for some $n\in \mathbb {N}$, or $f_{i} = \kappa _{0,0}$, or $f_{i} = \mathrm {sc}$, or $f_{i} = \mathrm {pr}_{1,0}$, or $f_{i} = \mathrm {pr}_{n,j}$, for some $n\ge 2$ and some $j\in n$, or $f_{i}$ is $m+1$-ary and $f_{i} = {\varOmega }_{\mathrm {R}}^{m}(f_{j},f_{k})$, for a j and a $k\in i$ such that $f_{j}$ is m-ary and $f_{k}$ is $m+2$-ary, or $f_{i}$ is n-ary and $f_{i} = {\varOmega }_{\mathrm {C}}^{m,n}(f_{j},(f_{k_{\alpha }})_{\alpha \in m})$, for an $m\in \mathbb {N}-1$, a $j\in i$ and a family $(k_{\alpha })_{\alpha \in m}\in i^{m}$ such that $f_{j}$ is m-ary and, for every $\alpha \in m$, $f_{k_{\alpha }}$ is n-ary.

Now we will prove, by induction on $i\in p$, that $f = f_{p-1}\in \mathrm {PRM}(\mathcal {F})$. For $i = 0$, $f_{0}\in \mathrm {PRM}(\mathcal {F})$, because, in this case, $f_{0}$ either belongs to $\mathcal {F}_{n}$, for some $n\in \mathbb {N}$, or is of the form $\kappa _{0,0}$, or $\mathrm {sc}$, or $\mathrm {pr}_{1,0}$, or $\mathrm {pr}_{n,j}$, for some $n\ge 2$ and some $j\in n$, and then $f_{0}\in \mathrm {PRM}(\mathcal {F})$, because $\mathrm {PRM}(\mathcal {F})$ is the union of the underlying $\mathbb {N}$-sorted set of the least subalgebra of that contains $\mathcal {F}$. Let $k\in p$ and let us suppose that, for every $i\in k$, $f_{i}\in \mathrm {PRM}(\mathcal {F})$. Then, by definition, $f_{k}\in \mathcal {F}_{n}$, for some $n\in \mathbb {N}$, or $f_{k} = \kappa _{0,0}$, or $f_{k} = \mathrm {sc}$, or $f_{k} = \mathrm {pr}_{1,0}$, or $f_{k} = \mathrm {pr}_{n,j}$, for some $n\ge 2$ and some $j\in n$, or $f_{k}$ is $m+1$-ary and $f_{k} = {\varOmega }_{\mathrm {R}}^{m}(f_{u},f_{v})$, for a u and a $v\in k$ such that $f_{u}$ is m-ary and $f_{v}$ is $m+2$-ary, or $f_{k}$ is n-ary and $f_{k} = {\varOmega }_{\mathrm {C}}^{m,n}(f_{j},(f_{k_{\alpha }})_{\alpha \in m})$, for an $m\in \mathbb {N}-1$, a $j\in k$ and a family $(k_{\alpha } )_{\alpha \in m}\in k^{m}$ such that $f_{j}$ is m-ary and, for every $\alpha \in m$, $f_{k_{\alpha }}$ is n-ary. It is evident that in the first five cases $f_{k}\in \mathrm {PRM}(\mathcal {F})$. In the last two cases too $f_{k}\in \mathrm {PRM}(\mathcal {F})$, because being, by hypothesis, $f_{0},\dots ,f_{k-1}\in \mathrm {PRM}(\mathcal {F})$, also $f_{u}$, $f_{v}$ and $f_{k_{0}},\dots ,f_{k_{m-1}}\in \mathrm {PRM}(\mathcal {F})$, hence, since $\mathrm {PRM}(\mathcal {F})$ is the union of the underlying $\mathbb {N}$-sorted set of the least subalgebra of that contains $\mathcal {F}$, $f_{k} = {\varOmega }_{\mathrm {R}}^{m}(f_{u},f_{v})\in \mathrm {PRM}(\mathcal {F})$ and $f_{k} = {\varOmega }_{\mathrm {C}}^{m,n}(f_{j},(f_{k_{\alpha }})_{\alpha \in m})\in \mathrm {PRM}(\mathcal {F})$. Therefore, for every $k\in p$, $f_{k}\in \mathrm {PRM}(\mathcal {F})$. Hence, for $k = p-1$, $f = f_{p-1}\in \mathrm {PRM}(\mathcal {F})$. From this it follows that $\bigcup _{n\in \mathbb {N}}\mathcal {L}_{n}\subseteq \mathrm {PRM}(\mathcal {F})$. Hereby completing our proof.

From the above proposition, for $\mathcal {F} = (\varnothing )_{n\in \mathbb {N}}$ one obtains, immediately, a constructive characterization of the set $\mathrm {PRM}$.

Corollary

Let f be an element of $\bigcup _{n\in \mathbb {N}}\mathrm {Hom}(\mathbb {N}^{n},\mathbb {N})$. Then $f\in \mathrm {PRM}$ if, and only if, there exists a $p\in \mathbb {N}-1$ and a family $(f_{i})_{i\in p}$ in $\bigcup _{n\in \mathbb {N}}\mathrm {Hom}(\mathbb {N}^{n},\mathbb {N})$ such that $f = f_{p-1}$ and, for every $i\in p$, it happens that:

• 1. $f_{i} = \kappa _{0,0}$, or

• 2. $f_{i} = \mathrm {sc}$, or

• 3. $f_{i} = \mathrm {pr}_{1,0}$, or

• 4. $f_{i} = \mathrm {pr}_{n,j}$, for some $n\ge 2$ and some $j\in n$, or

• 5. $f_{i}$ is $m+1$-ary and $f_{i} = {\varOmega }_{\mathrm {R}}^{m}(f_{j},f_{k})$, for a j and a $k\in i$ such that $f_{j}$ is m-ary and $f_{k}$ is $m+2$-ary, or

• 6. $f_{i}$ is n-ary and $f_{i} = {\varOmega }_{\mathrm {C}}^{m,n}(f_{j},(f_{k_{\alpha }})_{\alpha \in m})$, for an $m\in \mathbb {N}-1$, a $j\in i$ and a family $(k_{\alpha })_{\alpha \in m}\in i^{m}$ such that $f_{j}$ is m-ary and, for every $\alpha \in p$, $f_{k_{\alpha }}$ is n-ary.

Since it will be used afterwards, to prove that the multiplication and exponentiation are primitive recursive mappings, we notice that, for every $n\in \mathbb {N}$ and every $k\in \mathbb {N}$, the constant mapping , which sends every element of $\mathbb {N}^{n}$ to k, is a primitive recursive mapping.

Proposition

The addition , defined as:

$$\begin{aligned} {\left\{ \begin{array}{ll} x+0 = x, \\ x+\mathrm {sc}(y) = \mathrm {sc}(x+y), &{} \text {if }y\ge 0, \end{array}\right. } \end{aligned}$$

is a primitive recursive mapping.

Proof

This is so because $+ = {\varOmega }_{\mathrm {R}}^{1}(\mathrm {pr}_{1,0},{\varOmega }_{\mathrm {C}}^{1,3}(\mathrm {sc},(\mathrm {pr}_{3,2})))$, or in a diagrammatic manner:

where ${\varOmega }_{\mathrm {C}}^{1,3}(\mathrm {sc},(\mathrm {pr}_{3,2}))$ is the mapping from $\mathbb {N}^{3}$ to $\mathbb {N}$ obtained as:

$\square$

Proposition

The multiplication , defined as:

$$\begin{aligned} {\left\{ \begin{array}{ll} x\cdot 0 = 0, \\ x\cdot \mathrm {sc}(y) = x\cdot y+x, &{} \text {if }y\ge 0, \end{array}\right. } \end{aligned}$$

is a primitive recursive mapping.

Proof

This is so because $\cdot = {\varOmega }_{\mathrm {R}}^{1}(\kappa _{1,0}, {\varOmega }_{\mathrm {C}}^{2,3}(+,(\mathrm {pr}_{3,2},\mathrm {pr}_{3,0})))$, or in a diagrammatic manner:

where ${\varOmega }_{\mathrm {C}}^{2,3}(+,(\mathrm {pr}_{3,2},\mathrm {pr}_{3,0}))$ is the mapping from $\mathbb {N}^{3}$ to $\mathbb {N}$ obtained as:

$\square$

Proposition

The exponentiation , defined as:

$$\begin{aligned} {\left\{ \begin{array}{ll} x^{0} = 1, \\ x^{\mathrm {sc}(y)} = x^{y}\cdot x, &{} \text {if }y\ge 0, \end{array}\right. } \end{aligned}$$

is a primitive recursive mapping.

Proof

This is so because $\mathrm {exp} = {\varOmega }_{\mathrm {R}}^{1}(\kappa _{1,1}, {\varOmega }_{\mathrm {C}}^{2,3}(\cdot ,(\mathrm {pr}_{3,2},\mathrm {pr}_{3,0})))$, or in a diagrammatic manner:

where ${\varOmega }_{\mathrm {C}}^{2,3}(\cdot ,(\mathrm {pr}_{3,2},\mathrm {pr}_{3,0}))$ is the mapping from $\mathbb {N}^{3}$ to $\mathbb {N}$ obtained as:

$\square$