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A partial computable function is the function computed by a finite program, with an undefined value when its computation never terminates. A total computable function terminates on every input. Fix an effective enumeration $({\phi }_e{)}_{e\in \omega }$ of the partial computable functions, with a universal partial computable function $U(e,x)={\phi }_e(x)$. Composition, primitive recursion and unbounded minimization give a machine-independent description of this class. Unbounded minimization is allowed to diverge. The S-m-n theorem permits parameters in a program to be specialized effectively: an algorithm described uniformly in $e$ has an index obtained computably from $e$.

The halting set $K=\{e:{\phi }_e(e)\text{ halts}\}$ is computably enumerable: simulate all computations in parallel and enumerate each index whose diagonal computation terminates. It is not a computable set. Otherwise the program which terminates exactly when the proposed decision procedure says its own diagonal computation does not terminate gives a contradiction. There is also no effective enumeration consisting exactly of all total computable functions: from such an enumeration $(f_n)$ the total computable function $g(n)=f_n(n)+1$ differs from every listed function. Thus the effective enumeration of partial programs cannot be replaced by a decidable catalogue of total ones.

For the Rice theorem, let $C$ be a nontrivial property of partial computable functions, depending on the computed function rather than the program text. First suppose the nowhere-defined function is not in $C$, and choose a program computing some $g\in C$. Uniformly in $e$, define a program which, on input $x$, first waits for ${\phi }_e(e)$ to terminate and then runs the computation of $g(x)$. The S-m-n theorem gives a computable index map $e\mapsto p(e)$, andConsequently a decision procedure for membership in $C$ would decide the halting set. If the nowhere-defined function belongs to $C$, apply the same argument to its complement. Every nontrivial extensional property of partial computable functions has an undecidable index set. Syntactic properties of program descriptions are outside this assertion, and the theorem concerns unrestricted program indices, not merely an assumed list of terminating programs.

Here is an explicit Jockusch triple coloring computing the halting set. Choose computable increasing finite stages $K_s$ with union $K$. For $x<y<z$, putThis is a computable colouring. An infinite monochromatic set cannot have color $1$: fix its first element $x$ and choose two later elements beyond the stage at which $K_s\cap x$ stabilizes. Their triple has color $0$.

Suppose $H$ is infinite and homogeneous of color $0$. To decide whether $n\in K$ using $H$, find $x\in H$ with $x>n$, and then $y\in H$ with $y>x$. For every $z\in H$ above $y$, homogeneity gives $K_y\cap x=K_z\cap x$. Such $z$ are unbounded, so $K_y\cap x=K\cap x$. Testing $n\in K_y$ therefore decides $n\in K$. Hence every homogeneous set computes the halting set, and there is no infinite recursive monochromatic set. The infinity qualification is essential: finite homogeneous sets are recursive.

The arrow in the original PDF permits partial functions. A partial computable function ${\mathbb{N}}^k\rightharpoonup \mathbb{N}$ is computed by a finite program which halts with the specified value on every input in its domain and diverges on inputs outside the domain. A total computable function is the special case with full domain. Coding finite tuples by a primitive recursive pairing function identifies the different finite arities effectively, without identifying total and partial computability.

One precise machine model is a Turing machine. Its finite transition rules and configurations admit effective numerical coding. Bounded simulation is primitive recursive, and unbounded execution gives the partial result. Equivalently, start from zero, successor and projections and close under composition, primitive recursion and unbounded minimization. The Kleene normal form theorem makes the equivalence concrete:where $T$ is a primitive recursive predicate checking a coded halting computation and $U$ reads its output. If no computation exists, the minimum and hence the value are undefined. Checking a finite history is effective because every consecutive configuration must obey one of the finitely many transition rules.

The universal partial computable function evaluates the program indexed by $e$ on an input. The S-m-n theorem says that fixing finitely many input parameters can be done by a total computable operation on program Gödel numbers: a new program stores those parameters and calls the original universal evaluator. These facts explain both universal simulation and uniform reductions. A set is computably enumerable when it is the domain of a partial computable function, equivalently the range of an effective enumeration. It is recursive when its characteristic function is total computable. A set is recursive precisely when it and its complement are computably enumerable: dovetail the two enumerations to decide each input.

The diagonal halting problem $K=\{e:{\phi }_e(e)\downarrow \}$ is computably enumerable but not recursive. A total decider would yield a program which, on $e$, diverges if the decider says $e\in K$ and halts otherwise. Applying this program to its own index gives a contradiction. There is likewise no total computable universal listing of all total computable functions: the diagonal function $n\mapsto U(n,n)+1$ would be total and absent from the listing.

Here is a full proof of the Rice theorem. Let $\mathcal{P}$ be a nonempty proper class of partial computable functions whose property depends only on the function, not its program. First suppose the everywhere-undefined function $\perp$ is not in $\mathcal{P}$, and choose a program for a function $g\in \mathcal{P}$. Given an index $e$, construct uniformly a program which on $x$ first waits for ${\phi }_e(e)$ to halt and, only then, simulates $g(x)$. The S-m-n theorem gives a total computable index map $h$. If $e\notin K$, then ${\phi }_{h(e)}=\perp$; if $e\in K$, then ${\phi }_{h(e)}=g$. ThusA decision procedure for the property would decide $K$, impossible. If $\perp \in \mathcal{P}$, apply the same argument to its nontrivial complement. Every nontrivial extensional property of partial computable functions has an undecidable index set. Totality, having finite domain, and taking a particular value somewhere are examples. A syntactic property such as program length is not covered; nor does the theorem say that no individual program's behaviour can be established.

Self-reference can also be made effective. For a total computable index transformation $f$, let $d(x)$ be an index for the program which first computes ${\phi }_x(x)$ and then runs the program with that resulting index on its own input. The S-m-n theorem makes $d$ total computable. Choose an index $e$ for the total function $x\mapsto f(d(x))$, and put $n=d(e)$. ThenThis proves the recursion theorem: every effective transformation of programs has a fixed point at the level of the function computed, though not necessarily at the level of its numerical code.

Finally, consider the Jockusch triple coloring computing the halting set. Fix computable finite approximations $K_0\subseteq K_1\subseteq \cdots$ with union $K$. For $x<y<z$, defineThis is a total recursive two-coloring because both stages are finite computations. Suppose $H$ is infinite and homogeneous. Its color cannot be one: fix $x\in H$, wait until every enumeration below $x$ has stabilized, and choose later $y<z$ from $H$ beyond that stage. Their color is zero.

If $H$ has color zero, it computes $K$. On input $n$, use $H$ to find $x>n$ and $y>x$ in $H$. For arbitrarily large $z\in H$ with $z>y$, homogeneity gives $K_y\cap x=K_z\cap x$. Taking stages sufficiently large to reach the final finite approximation gives $K_y\cap x=K\cap x$. Consequently membership of $n$ in $K$ is decided by the finite stage $K_y$. A recursive $H$ would therefore make $K$ recursive, a contradiction. There is a recursive partition of triples with no infinite recursive homogeneous set. This is the nontrivial finite-color meaning of the partition of an infinite domain; if “infinite partition” meant infinitely many colors, the coloring by the least member would already give a trivial example. Ramsey's theorem still guarantees infinite homogeneous sets for the displayed two-coloring, but every such set is computationally nontrivial.