Let $A$ be any set of natural numbers that is computably enumerable, but not decidable. There are numerous natural instances of such sets, such as the set of all finite presentations of the trivial group, the set of all finite tile families that cannot tile the plane, and so on.

Every such set $A$ is saturated with infinitely many instances of the type you request. That is, there must be infinitely many candidates $a$, which are not in $A$, but such that this assertion is neither provable nor refutable in PA, or in whichever fixed background theory you wish to work, such as ZFC or ZFC + large cardinals. The reason is that otherwise, we would be able to decide membership in $A$ as follows: given a candidate $a$, wait for $a$ to show up in the enumeration of elements of $A$ and simultaneously search for a proof that $a$ is not in $A$. If non-membership was always provable for large candidates, then this would show that $A$ is decidable, contrary to assumption.

Thus, for every such $A$, there must be infinitely many particular non-members $a$ such that $a$ is not in $A$, but this is neither provable nor refutable from the fixed axioms.

To anticipate an objection, let me say that when $A$ is a "naturally" defined set, and $a$ is a particular candidate instance, then the assertion that $a$ is not in $A$ would seem to be a perfectly clear statement, not directly involving any logical "tricks". For example, we would have the assertion that a particular finite group presentation is not a presentation of the trivial group, or the assertion that a particular finite set of tiles does not tile the plane.

The logical ~~trickery~~ magic comes into play, of course, in the details of the explanation that a particular set $A$ is not decidable, where one might show a reduction from the halting problem or something similar. My opinion, however, is that consistency statements themselves are highly natural, and should not be considered a form of trickery. There is a sense in which every $\Pi^0_1$ statement is a consistency statement.