tot

$\text{[math]}$ [cite:;taken from @computability_soare_2016 definition 1.6.15]

there is no effective procedure for deciding, given any $\text{[math]}$ , whether or not $\text{[math]}$ is a total function. that is to say, there is no recursive function $\text{[math]}$ such that
$\text{[math]}$ [cite:;taken from @computability_rogers_1987 chapter 1.9 the halting problem; theorem VIII]

$\text{[math]}$ is not recursively enumerable.

consider the following reduction function: $\text{[math]}$ , where:
$\text{[math]}$ then we have
$\text{[math]}$ which gives us the reduction $\text{[math]}$ .