In this work, we introduce the $n$-irreducible sequents and the $n$-irreducible numbers defined with the help of the second order logic. We give many concrete examples of $n$-irreducible numbers and $n$-irreducible sequents with the Peano's axioms and the axioms of the real numbers. Shortly, a sequent is $n$-irreducible iff the sequent is composed by some closed hypotheses and a $n$-irreducible formula (a close formula with one internal variable such that the formula is only true when we set that variable to the unique natural number $n$), and it does not exist some strict sub-sequent which are composed by some closed sub-hypotheses and some sub-$m$-irreducible formula with $m>1$. The definition is motivated by the intuition that \Nathypo do not carry natural numbers or "hidden natural numbers" except for the numbers $0$ and $1$, i.e., they can be used in a $n$-irreducible sequent. Moreover, we postulate at second order of logic that \Nathypo are not chosen randomly: \Nathypo has the propriety to give the largest $n$-irreducible number $N_Z \NZ$ among a finite number of $n$-irreducible sequents. The Collatz conjecture, the Goldbach's conjecture, the Polignac's conjecture, the Firoozbakht's conjecture, the Oppermann's conjecture, the Agoh-Giuga conjecture, the generalized Fermat's conjecture and the Schinzel's hypothesis H are reviewed with this new (second order logic) $n$-irreducible axiom. Finally, two open questions remain: Can we prove that a natural number is not $n$-irreducible? If a $n$-irreducible number $n$ is found with a function symbol $f$ where its outputs values are only $0$ and $1$, can we always replace the function symbol $f$ by a another function symbol $\tilde{f}$ such that $\tilde{f}=1-f$ and the new sequent is still $n$-irreducible?