[zzless]

> it seems to me that the thing to do is to extend your language until it can.

This depends on what you mean by define. If you mean a unique model than this is impossible. The reason is compactness theorem (every theory in which every finite set of formulas has a model has a model). The basic idea is to add constants and introduce axioms stating they are different. This will allow models of, say reals where there are plenty of reals that are not real reals (sorry for the pun, I could not resist). Nonstandard analysis takes it a bit further and makes it a bit more precise and useable.

If you mean you want to deal with (naively) definable reals only then intuitively there are only countably many of those that you can define (by formulas, etc) and you are missing a huge chunk of the real line again.

[pdonis]

> 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.

[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.