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A function in intension is a finite description of a computation, such as a program or a formal construction expression. A function in extension is the resulting input-output map, including its domain if it is partial. Different descriptions may have the same extension: the programs that return $x$ directly and that compute $x+0$ describe the same function. Finite descriptions have Gödel numbers; equality of the resulting partial computable functions is a separate semantic question.

The primitive recursive functions are the smallest class of numerical functions containing the zero functions, the successor function and all projection functions, and closed under function composition in recursion theory and primitive recursion. More explicitly, from $g:{\mathbb{N}}^k\to \mathbb{N}$ and $h:{\mathbb{N}}^{k+2}\to \mathbb{N}$ the latter operation formsFor function composition in recursion theory, an $m$-ary $g$ and $k$-ary $h_1,\dots ,h_m$ give $g(h_1(\mathbf{x}),\dots ,h_m(\mathbf{x}))$. Zero functions of all allowed arities can be included, with nullary zero included if arity $0$ is part of the coding convention. Every resulting function is total: the initial functions are total, function composition in recursion theory preserves totality, and mathematical induction on $n$ verifies totality of primitive recursion. Applying structural induction for primitive recursive functions then covers all construction expressions.

Use a fixed effective Gödel numbering of finite tagged construction trees with decidable decoding. A node is tagged as zero, successor, projection, composition, or recursion. The primitive recursive syntax and arity checking algorithm first checks that the input decodes as a finite tree, then works upwards from its leaves. A projection tag $(k,j)$ is legal exactly when $1\le j\le k$, with output arity $k$. A composition node declares its output arity $k$ and is legal when its outer child has arity $m$, there are exactly $m$ inner children, and each has arity $k$; its arity is $k$. The declared arity also handles the case $m=0$. A recursion node is legal when its base child has arity $k$ and its step child has arity $k+2$; its arity is $k+1$. Zero and successor tags have their declared arities. Ill-formed nodes are rejected. The process terminates because there are only finitely many nodes. Thus, for each fixed $i$,This is a syntactic assertion about functions in intension. It does not assert that arbitrary machine indices computing primitive recursive functions form a decidable set. That extensional property is nontrivial, hence undecidable by Rice theorem. For an explicit reduction, given a program $e$ construct a program that, on every input, waits for ${\phi }_e(e)$ to halt and then returns $0$. Its extension is a primitive recursive function if $e\in K$, and otherwise is nowhere defined and so is not a primitive recursive function. A decision procedure for this latter semantic set would decide the diagonal halting set.

Enumerate the decidable syntax set $D_1$ in increasing code order as $p_0,p_1,\dots$. There are infinitely many such descriptions; iterating successor after a unary zero expression already supplies infinitely many. Let $F_p$ be the extension of a valid construction and defineTo compute $U$, find $p_n$ by the syntax test and interpret its finite construction tree. Evaluate function composition in recursion theory by evaluating the children, and evaluate primitive recursion by the prescribed finite loop of length equal to its final input. Termination follows from the totality argument above, so $U$ and $d$ are total computable functions.

If $d$ were a unary primitive recursive function, some construction $p_j$ would have extension $d$. Then the diagonal argument giveswhich is impossible. The displayed $d$ is total and computable but not primitive recursive. Repetitions of extensions in the enumeration cause no problem: every possible extension is represented, which is all the diagonal argument needs. The same argument shows that the two-variable interpreter $U$ cannot itself be primitive recursive.

For (1)(a), a primitive recursive function is obtained by a finite construction from the zero function, successor function and projection functions, using function composition in recursion theory and primitive recursion. The recursion rule, with parameters $\mathbf{x}$, isHere $g,h$ have already been constructed. All these functions are total on nonnegative integers: the initial functions are total, composition preserves totality, and ordinary mathematical induction proves totality of each recursion.

For (1)(b), use a shifted hyperoperation sequence, beginning with addition. Defineand, for each fixed $r\ge 0$, defineThe first three members are therefore addition, multiplication and exponentiation:The exponentiation convention includes $a^0=1$, hence $0^0=1$. The next member iterates exponentiation, with $H_3(a,0)=1$; there is no need for an exceptional domain restriction at $a=0$.

For (1)(c), $H_0$ is a primitive recursive function by its displayed recursion. If $H_r$ is a primitive recursive function, thenare primitive recursive functions, using constants, projection functions and function composition in recursion theory. Primitive recursion applied to $g_r,h_r$ gives $H_{r+1}$. Mathematical induction on the fixed rank $r$ proves every member of the sequence is primitive recursive. This does not assert that the joint three-variable evaluator $(r,a,b)\mapsto H_r(a,b)$ is a primitive recursive function.

For (2), as written, the answer is no: the empty set $X=\varnothing$ is semidecidable, whereas a total computable function from $\mathbb{N}$ to $\mathbb{N}$ always has a nonempty range. This is the only obstruction. To prove the stronger useful statement, suppose $X$ is a nonempty computably enumerable set and choose $x_0\in X$. Fix a Turing machine which halts exactly on the inputs in $X$. Its bounded halting predicateis primitive recursive: encode configurations arithmetically, iterate its total single-step operation $t$ times by primitive recursion, and check the halt state. A halted configuration is kept fixed. This is a bounded simulation, not unbounded waiting.

Fix a primitive recursive pairing function $⟨x,t⟩$ with primitive recursive inverse coordinate functions. For example, the Cantor pairing function has inverses obtained by bounded minimization. DefineThe two coordinate functions, the bounded predicate, and the finite case distinction are primitive recursive functions, so $f$ is a primitive recursive function. Its values always lie in $X$. Conversely, if $x\in X$, its computation halts within some finite $t$, and $f(⟨x,t⟩)=x$. ThusThe range need not be a computable set; deciding whether a value occurs anywhere in this total enumeration is an unbounded search.

Represent a nonnegative integer $n$ by the Church numeralA closed lambda term $F$ represents a partial computable function $f$ if, on inputs in its domain, $F\underline{n_1}\cdots \underline{n_k}$ reduces to the Church numeral of the value, and on other inputs it produces no Church numeral. We use untyped lambda calculus and normal-order beta reduction.

First every primitive recursive function has such a representation on all inputs. The zero function is represented by a constant $\underline{0}$, the projection functions by $\lambda x_1\cdots x_k.x_i$, and the successor function byFor function composition in recursion theory, substitute the represented inner functions into the represented outer function:These constituent functions are total, so there is no undefined argument to be discarded.

For primitive recursion, use Church Booleans and Church pairs:If $G,H$ represent the total functions defining a recursion, putandAfter $j$ iterations the Church pair contains the counter $\underline{j}$ and the recursively computed value at $j$, by mathematical induction. Thus this lambda definition of primitive recursion by pair iteration represents the recursion. A structural induction for primitive recursive functions now represents every primitive recursive function.

To pass to arbitrary partial computable functions, use the Kleene normal form theorem in the following form: for every $f:{\mathbb{N}}^k\rightharpoonup \mathbb{N}$, there are total primitive recursive functions $C,U$ such thatwith $f$ undefined precisely when no such $t$ exists. The predicate checks codes of valid finite halting histories for the chosen program; $U$ extracts their common output. This expresses unbounded minimization of a total predicate, so no earlier undefined test can interfere with the search.

Let $\hat{C},\hat{U}$ be the represented total functions. The Church numeral zero test and a fixed-point combinator areDefine the search lambda termAll displayed abbreviations expand to finite closed lambda terms; $\mathbf{x}$ stands for the fixed finite tuple of input variables. Because $YH{\equiv }_{\beta }H(YH)$, normal-order beta reduction tests $t=0,1,2,\dots$. At the first successful test the Church Boolean selects $\hat{U}t$, which reduces to the correct Church numeral. The unused recursive branch is discarded.

If no test succeeds, every finite search step returns to another head search, never exposing a numeral. The term has no head normal form and hence cannot be beta equivalent to a Church numeral. This last point is why the output operation occurs inside the successful branch: simply applying a possibly argument-discarding outer function to a divergent search would not justify undefinedness. Therefore every computable function is representable, including partial computable functions with their correct undefined inputs:

Use untyped lambda calculus and normal-order beta reduction. Write $c_n=\lambda fx.f^{n}x$ for the Church numeral of $n$. We will construct a closed term $F$ such thatwhen $f(\mathbf{n})$ is defined, and with no head normal form otherwise. The latter condition implies that the result is not in beta equivalence with any Church numeral, so both definedness and the output are represented.

First represent all total primitive recursive functions. The zero function is $\lambda \mathbf{x}.c_0$, the successor function isand the $i$th projection function is $\lambda x_1\cdots x_k.x_i$. Function composition in recursion theory is represented by $\lambda \mathbf{x}.G(H_1\mathbf{x})\cdots (H_r\mathbf{x})$. This composition claim is used here for total functions, whose inner terms all reduce to numerals.

For primitive recursion use Church pairs and iteration. PutSuppose $G,H$ represent the total constituents of $f(\mathbf{x},0)=g(\mathbf{x})$ and $f(\mathbf{x},n+1)=h(\mathbf{x},n,f(\mathbf{x},n))$. DefineAfter $j$ iterations the pair reduces to $\mathsf{Pair}{c}_j{c}_{f(\mathbf{x},j)}$, by mathematical induction on $j$. Its second projection is the required result. This lambda definition of primitive recursion by pair iteration and structural induction on the primitive recursive construction give terms for all primitive recursive functions.

For an arbitrary partial computable function use the Kleene normal form theorem with its program index fixed. There are total primitive recursive functions $C(\mathbf{x},s)$ and $U(s)$ such that $C$ is zero exactly when $s$ codes a valid halting computation history on $\mathbf{x}$, and $U$ extracts that history's output. Checking finite coded configurations and each transition is primitive recursive. Determinism ensures that every accepted history on an input has the same output. HenceIf the input computation never halts, no history is accepted. Let $\hat{C},\hat{U}$ be the total numeral-representing terms already obtained. The Church numeral zero test is $\mathsf{Zero}=\lambda n.n(\lambda z.\mathsf{F})\mathsf{T}$, returning $\mathsf{T}$ exactly on zero. A Church Boolean applied to two arguments selects the appropriate branch without evaluating the other.

Take the fixed-point combinator $Y=\lambda h.(\lambda z.h(zz))(\lambda z.h(zz))$. For the input tuple defineExpanding the displayed abbreviations gives a genuine finite combinator. At a numeral code $c_j$, the total predicate computation terminates. If it returns a positive numeral, the false Church Boolean selects the next search code. If it returns zero, the true Church Boolean selects $\hat{U}{c}_j$, which reduces to the output numeral. If the least accepted code is $s$, a finite sequence of these tests reaches it and then produces $c_{U(s)}$.

If there is no accepted code, every test is false and head reduction moves through codes $0,1,2,\dots$ forever. At no finite point can a head variable or an outer numeral abstraction be produced: the applied search term must first unfold the fixed point and evaluate the next total test, and the decoder branch is always discarded. Thus it has no head normal form. The standard head-normalization property, or the Church-Rosser theorem together with normalization of normal-order reduction, rules out beta equivalence to a numeral. In particular, we have avoided a lazy-composition pitfall: an outer function that discards an argument need not force a divergent inner computation. The decoder is reached only after the guarded search succeeds. This proves lambda representation of partial computable functions:

The partial recursive functions form the smallest class containing the zero function, successor function, and projection functions and closed under composition, primitive recursion, and unbounded minimization.

Fix a register machine $P$ with $k$ inputs. By the permitted result, its complete configuration after $t$ steps,is a recursive function of the inputs and time. The predicate $H(\mathbf{x},t)$ saying that this configuration is halted is therefore recursive. Its halting time is the partial recursive functionwhere unbounded minimization is undefined when no halting time exists. Extracting the output register from a configuration is recursive, sois partial recursive. Hence every partial computable function is partial recursive.

Let $g(n,k)=r(n)$, where $r$ is the total primitive-recursive parity function from part (ii), and apply unbounded minimization in $k$:If $n$ is even, then $g(n,0)=0$ and $p(n)=0$. If $n$ is odd, then $g(n,k)=1$ for every $k$, so the search never terminates and $p(n)$ is undefined. Hence $p$ is partial recursive. Because every function built without minimization is total, $p$ cannot be defined without minimization; the functions $s$ and $r$ can.

It remains to characterize the functions built using only the initial functions and composition. They are exactly the functionsfor some $c\in \mathbb{N}$ and some input coordinate $i$.

Indeed, the zero function is the first form, the successor function and each projection function have the second form, and composing functions of these forms preserves the classification: a constant outer function remains constant, while an outer function $y_j+c$ selects one inner function and adds $c$. This proves necessity by structural induction. Conversely, applying the successor function $c$ times to zero produces the constant $c$, and applying it $c$ times to ${\pi }_i^k$ produces $x_i+c$.

Every function in this class is directly computable: an algorithm either writes the fixed constant $c$, or copies the $i$th input and performs $c$ successor steps.

A partial recursive function is obtained from the three kinds of initial function of recursion theory—
the zero function, successor function, and projection function—by finitely many applications of function composition in recursion theory, primitive recursion, and unbounded minimization. Explicitly, primitive recursion has the formwhile minimization takes the least $y$ for which $g(\mathbf{x},y)=0$ and is undefined if no such $y$ is found.

For the function in part (i), defineThe initial value $1=S(Z(0))$ and the identically zero recursion step are primitive recursive, so this is a primitive-recursive definition of the required zero test. It uses no minimization.