Request for clarification of what is a non first-order definable set.

Hi:

I recently put together a proof that states for every predicate p(x)
there is a set { x : p(x) }. Thus, a set of predicates P defines a set
of sets S. A reviewer made a comment:

"If we have a set S why must it be definable? That is, why does P
exist? There are sets which are not first-order definable."

I am unclear on how there can be sets which are not first-order
definable when the definition of a set requires a predicate. Can
someone provide an explaination or provide an example? Your help would
be appreciated. Thanks.

Scott

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