User blog comment:Vel!/Formal logic challenge/@comment-32876686-20171023182635

Here is my plain English answer for the first one:

Let $$P$$ be a function that is computable, or can be calculated in a finite number of steps within a finite amount of time.

$$P$$ will be a function such that it's outputs are mapped to by at least one input. This means different input can return either different or the same outputs, but there will always be an input for a given output.

The function $$P$$ assigns a natural number, (that is, a whole number greater than 0), to any conceivable computable function that takes in one input, yet may have undefined outputs. Let us call this computable function which is being assigned a natural number, $$Q$$.

Given a set $$(\{A})\$$ which is contained with the set of all $$Q$$'s, let $$D$$ be the problem of whether the function $$P$$ is an element of the set $$(\{A})\$$ given an input to the function $$P$$.

A theorem states that the problem $$D$$ can be solved if and only if the set $$(\{A})\$$ is empty, or is exactly the same as the $$Q$$'s.