From Wikipedia Indiscernibles are objects which cannot be distinguished by any property or relation defined by a formula. Usually only first-order formulas are considered.
Question: does the existence of indiscernibles mean that even in first order formal system for which all the set of true formulae are known, we will still have an infinite number of non-standard models? Those models would be the models in which the indiscernibles would have different second order properties but of course the same first order ones.
By Löwenheim–Skolem theorem, if we have any infinite model $M$ of a first-order theory $T$ (it can be anything, ranging from PA to $Th(\Bbb N)$ and things like that, the theory doesn't have to be recursive or even countable) then there are arbitrarily large (in terms of cardinality) models $M'$ satisfy precisely the same first-order statements as $M$, so in particular are also models of $T$.
Of course, models of different cardinalities can't be isomorphic, as isomorphism has to, in particulat, be a bijection between underlying sets.
Note that this result does not depend in any way on existence of indiscernibles in $M$.