A b s t r a c t. Let $\underline {N_n}=\left\langle N_n,=,1,\ldots ,n\right\rangle$ be the relational system of n - element set $N_n=\left\{1,2,\ldots ,n\right\}$, with identity and with fixed elements: $ 1,2,...,n$. We show that logic $L\left(\underline {N_ n}\right)$ determined by the system $\underline {N_n}$ is axiomatizable.