English: Illustration of an example call of Kleene's T predicate T₁. The first argument gives the source code (in C rather than as a Gödel number) of a computable function, viz. the Collatz function f. The second argument gives the natural number i (in our example: 5) to which f is to be applied. The third argument gives a sequence x of computation steps simulating the evaluation of f on i (as an equation chain rather than a Gödel number). The predicate call evaluates to true since x is actually the correct computation sequence for the call f(5), and ends with an expression not involving f anymore. Kleene's function U, applied to the sequence x, will return its final expression, viz. 1.