universality

for each $\text{[math]}$ , we define
$\text{[math]}$ it may also be written as $\text{[math]}$ , see standard numbering of turing programs.

one of the key tools in computability theory is

for each $\text{[math]}$ , the function $\text{[math]}$ is partially computable.
[cite:;taken from @computability_davis_1994 theorem 3.1 univesrality theorem]

[cite:;taken from @computability_davis_1994 chapter 4.3 universality]