# From the Four-Colors Theorem to a Generalizing “Four-Letters Theorem”: a Sketch for “Human Proof” and the Philosophical Interpretation

### EasyChair Preprint no. 3194

7 pages•Date: April 18, 2020### Abstract

The “four-color” theorem seems to be generalizable as follows. The four-letter alphabet is sufficient to encode unambiguously any set of well-orderings including a geographical map or the “map” of any logic and thus that of all logics or the DNA (RNA) plan(s) of any (all) alive being(s).

Then the corresponding maximally generalizing conjecture would state: anything in the universe or mind can be encoded unambiguously by four letters.

That admits to be formulated as a “four-letters theorem”, and thus one can search for a properly mathematical proof of the statement.

It would imply the “four colour theorem”, the proof of which many philosophers and mathematicians believe not to be entirely satisfactory for it is not a “human proof”, but intermediated by computers unavoidably since the necessary calculations exceed the human capabilities fundamentally. It is furthermore rather unsatisfactory because it consists in enumerating and proving all cases one by one.

Sometimes, a more general theorem turns out to be much easier for proving including a general “human” method, and the particular and too difficult for proving theorem to be implied as a corollary in certain simple conditions.

The same approach will be followed as to the four colour theorem, i.e. to be deduced more or less trivially from the “four-letter theorem” if the latter is proved. References are only classical and thus very well-known papers: their complete bibliographic description is omitted.

**Keyphrases**: completeness, Four-Color Theorem, Gentzen arithmetic, incompleteness, Peano arithmetic, set theory, Skolem's paradox