Calculus must consist of the study of real numbers in their decimal representation and not of the study of an abstract complete ordered field or nonstandard real numbers

Abstract

Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/tmes20Calculus must consist of the study of real numbers in their decimal representation and not of the study of an abstract complete ordered field or nonstandard real numbers

A tale of two algorithms

“…The classical VAA is a 'right-to-left' algorithm: we proceed from right to left, adding corresponding digits in the two expansions, having possibly a carryover -which is transferred to the sum of the adjacent digits to the left. As noted in [1], 'When the expansions are non-terminating the situation is quite challenging and puzzling'.…”

“…More specifically, the proof that every real number corresponds to a point relies on the following equivalent version (known as Cantor's axiom) of Hilbert's completeness axiom: Every rational net has a unique common point (for a proof of this basic equivalence see [2]). The need for Cantor's completeness axiom arises naturally in our presentation; furthermore, it is a simple natural assumption, unlike the standard incomprehensible 'least upper bound' assumption required in standard presentations of real numbers (see, for example, [1]). …”

“…is a non-partition point in [1,2] (interval #1). Applying the PLA to interval #1, it is easy to see (using arithmetic or geometric arguments, see figure 1) that, at step 1, Two important remarks.…”

“…Since the 'decimal notation is most practical, most simple and, in addition, it reflects most outstandingly the profound subtleties of the human analytical mind' [1], our algorithms are based on decimal notation. The algorithms could alternatively be defined in terms of 'binary expansions', or any other desired base.…”

Theorem Proving with the Real Numbers

“…However this is contrary to the spirit of HOL as set out in the introduction, where all new types are explicitly constructed and all new operations explicitly defined, an approach that can be guaranteed not to introduce inconsistency. There are also philosophical objections, vehemently expressed by Abian (1981): the reals are normally thought of intuitively using a concrete picture such as decimal expansions, so it's artificial to start from an abstract set of axioms. We chose to construct the reals in HOL.…”