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> The reason is compactness theorem

As I understand it, there is no compactness theorem in second-order logic, only in first-order logic. So your objection would not apply to extending one's language by using second-order logic.

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True, the first order logic is unique in that it satisfies compactness and L-S. Extending the language to second order language (although this is not quite standard terminology) is a whole new ball of wax since the concept of a model changes as well. You can introduce a quantifier over 'unique' reals but this is a rather hollow extension since the nature of that quantifier remains just as vague. I also fail to see why the uniqueness of reals is so important. Using second order language you would have to forcefully / 'declare' every such object.

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