[zzless]

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.