[stcredzero]

I think Borel has confused names with things.

Pedantic positivist?

[TheOtherHobbes]

Or simply a pragmatist.

Reals are a useful sensory abstraction. Countability is another sensory abstraction, although for most people it's a rather less useful one.

There's no particular obligation for sensory abstractions to be logically watertight - if only because logic is an abstraction itself.