A Difficult, But Fascinating Riddle

It's true. There was a talk between me and Karl. Karl also said that - about the computers. (I'm not close to this, so I cannot bring quite examples.)

Also can't say I tried to step by step investigated these theorems (or at least first of them), but I read some books that helped me to. The theorem of Godel has at least two tricks that helps the systems itself to try to recreate a liar paradox statement about the system:
    a) digitalization
    b) diagonalization
The first demonstrates how to rewrite any formula into a number formula. So this step is simple. Another one is indeed difficult. Hardly I understand it. This step allow to get a dull or a free formula from an ordinary one. It's like firstly you have something like this:


And then a function lambda is added:


If the first formula is indeed a formula, then the 2nd is just a set that looks like a formula. So, the final step for Godel - as far as I can comprehend or to realize - is to digitize the 2nd formula, and to entail something funny to achieve the self-contradiction formula. 

But here is another task. It doesn't suppose any abstractions as in Godel's theorem (by the way, I don't trust Godel. By intuition it doesn't seem to be true. He used his method to describe finite systems only. Besides some of his steps go far from the usual Predicate Calculus. So, I don't know). This time only what we've got is - Cartesian coordinates, and we want to draw a picture/function (a message) that tells (mathematically) exactly that function. In other words, that curved line we'd like to see drawn on Cartesian coordinates must be a function, and if to apply this function - we will see this function appear on the Cartesian coordinates.
∀ ∃ □ ◊ ¬ ∧ → ⊃ ⊢ ⊨ ≡ ⊆ ⊂ ⊄ ∈ ∉ ⋂ ∅ ϕ 𝓕 ξ