We define the super-normal-irreducible-irrational numbers from some irreducible-irrational numbers and with the help of the $n$-irreducible sequents (see my previous articles). Instead of taking some integer part of the irreducible-irrational number (or from its inverse), we add a super-normal-irreducible formula which give the position of the first digit breaking some super-normal number definition. From $84$ irreducible-irrational numbers, we deduce from the axiom at second-order of logic that they are all super-normal numbers as well. Moreover, with some random digits, the probability that the super-normal-irreducible formula holds for the $84$ ones is about $9.0\times 10^{-10}$ and we have taken in account that some irreducible-irrational numbers are only some different functions of the same irreducible-irrational number. From this large coincidence, we introduce the axiom at third-order of logic which states that every irreducible-irrational numbers are super-normal numbers as well. From that new axiom at third-order of logic, we deduce the none-existence of an exotic $4$-sphere. Finally, we conclude about the finitude of the total number of $n$-irreducible sequents.