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It has sqrt(2), for starters? Not sure what do you mean by useful.

It is not "useful" in the sense that reals are most "famous" for: it is not complete. Cauchy sequences can diverge in the useful reals field.

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Completeness in the "full" reals is a useless feature, though. All is gives you is an emotional crutch to pretend your cauchy sequences can be mapped to regular numbers. But it doesn't give you anything you didn't already have in the cauchy sequences and useful reals.

You are of course right, reals are isomorphic to equivalence classes of Cauchy sequences on Q. But once you are dealing with equivalence classes of Cauchy sequences on Q you might as well give it a name. Maybe call it R.

His point is different. You cannot (by definition) ever write a "name", a formula, a rule, a lim expression, anything really, for a real that is not in the useful reals.

Ah, I was curious if there are any interesting properties.

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