Going back to this topic from long ago, not being myself a fan of Wittgenstein, I would refer all to Carnap, whose position on ontology I think clear and sensible.
It is articulated most concisely perhaps in his paper "Empiricism, Semantics and Ontology", though the internal/external distinction for ontological questions which he introduced here was not well-received.
Anyway, its relevance here is that it provides a basis for affirming the objectivity of set theoretic truths (internal) without accepting that questions about what sets exists have any absolute truth value (these would be external questions).
More on this belongs perhaps at Carnap Corner.
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