The intrinsic enumerability of linear orders

We study into the question of which linearly ordered sets are intrinsically enumerable. In particular, it is proved that every countable ordinal lacks this property. To do this, we state a criterion for hereditarily finite admissible sets being existentially equivalent, which is interesting in its own right. Previously, Yu. L. Ershov presented the criterion for elements $h_0$, $h_1$ in $HF(\mathfrak M)$ to realize a same type as applied to suficiently saturated models $\mathfrak M$. Incidentally, that criterion fits with every model $\mathfrak M$ on the condition that we limit ourselves to 1-types.