>Likewise we cannot just redefine the set of real numbers to be restricted to the set of computable reals. Analysis rapidly breaks down when you do so and it fo

>Likewise we cannot just redefine the set of real numbers to be restricted to the set of computable reals. Analysis rapidly breaks down when you do so and it follows that the real field is no longer closed (by definition, and therefore no longer subject to the field axioms). I was with you until you wrote that Analysis would break down. We can easily do analysis with dense subsets of the reals and also extensions of the reals, like in Non-Standard Analysis.