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*To*: srfi-70@xxxxxxxxxxxxxxxxx*Subject*: Re: inexactness vs. exactness*From*: William D Clinger <will@xxxxxxxxxxx>*Date*: Sun, 24 Jul 2005 10:46:18 +0200*Delivered-to*: srfi-70@xxxxxxxxxxxxxxxxx*User-agent*: Gnus/5.110003 (No Gnus v0.3) XEmacs/21.5-b20 (berkeley-unix)

Aubrey Jaffer claims to have proved that the language of the R5RS not only regards inexact numbers as neighborhoods, but that no other interpretations of the R5RS are tenable. Jaffer's alleged proof contains many errors of logic, which I will happily enumerate if anyone claims to remain convinced by the alleged proof. Below I merely offer a simple argument that the "inexact numbers denote neighborhoods" interpretation is itself untenable. Suppose (for a contradiction) that inexact numbers do denote neighborhoods. Then let [x, y] be the neighborhood denoted by the inexact number 1.0. If 0 < x <= y, then the inexact number (* 1.0 1.0) denotes [x*x, y*y]. If (* 1.0 1.0) evaluates to 1.0, then 1.0 denotes both [x, y] and [x*x, y*y], hence x = x*x and y = y*y. Therefore x = 1.0 = y, so under our assumptions, the inexact number 1.0 really denotes only itself. (Had we considered an open neighborhood (x, y) with 0 < x <= y, we'd have concluded that the neighborhood denoted by 1.0 is empty, which is even less satisfactory.) The argument above generalizes to arbitrary positive inexact numbers z by considering the expression (* (/ z z) (/ z z)), which probably evaluates to the inexact number 1.0. Therefore, under the assumption that every positive inexact real number denotes a neighborhood [x, y] with 0 < x, we can conclude that every positive inexact real number really denotes only itself. That argument generalizes to arbitrary inexact real numbers by considering the expression (- z). The generalization to inexact complex numbers is left as an exercise. Hence, under reasonable assumptions, every inexact number really denotes only itself. There are only two ways to avoid this conclusion. One way is to conclude that every inexact real number denotes the neighborhood consisting of all real numbers, and that every inexact complex number denotes the neighborhood consisting of all complex numbers. The other is to abandon the idea that inexact real numbers denote neighborhoods. Will

**Follow-Ups**:**Re: inexactness vs. exactness***From:*Alex Shinn

**Re: inexactness vs. exactness***From:*Aubrey Jaffer

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