Lecture 18: The Constructive Real Numbers Continued

This lecture continues our discussion of the constructive reals and intuitionistic reals. We look at other options beyond the approach taken in the book Constructive Analysis by Bishop and Bridges which is a resource for this course available on the PRL project page (www.nuprl.org). Another resource is the calculator for the constructive reals provided in Nuprl as a public resource. It uses algorithms verified in Nuprl by Dr. Bickford. This calculator relies on an implementation of bignums and relies on the Nuprl software stack which includes Lisp at the lowest level. We have high confidence in this resource which we have been using for over a decade. It is likely that the very first verified implementation of the reals numbers was done in 1985 and reported on in Chapter 11 of the book Implementing Mathematics from 1986. In the next lecture we will look at nonstandard models of the reals which include elements called infinitessimals. We will consider them in a constructive setting as well.