what the author means to allude to, through some nonsensical rambling, are the incomputable numbers [1]
the cardinality of all real numbers that can be described by a terminating computer program to some accuracy is a countable set (since the number of such programs is countable) however, the cardinality of the reals is uncountable.
hence most real numbers cannot be computed beyond a certain accuracy.
Edit (additionally): the set of computable numbers forms a field (if a,b are computable, so is their sum, etc). and there are several movements in "constructive" mathematics, to work exclusively in this field, instead of the field of real numbers. however, many cornerstone theorems in analysis fail in this context, such as, the least upper bound of a bounded increasing computable sequence of computable numbers need not be a computable number [1].
Yeah, but you can describe many incomputable numbers. Indescribable numbers are not the incomputable numbers- Chaitin's constant's a nice one, it's the proportion of Turing machines that halt. Described. Now, compute it...
the cardinality of all real numbers that can be described by a terminating computer program to some accuracy is a countable set (since the number of such programs is countable) however, the cardinality of the reals is uncountable.
hence most real numbers cannot be computed beyond a certain accuracy.
Edit (additionally): the set of computable numbers forms a field (if a,b are computable, so is their sum, etc). and there are several movements in "constructive" mathematics, to work exclusively in this field, instead of the field of real numbers. however, many cornerstone theorems in analysis fail in this context, such as, the least upper bound of a bounded increasing computable sequence of computable numbers need not be a computable number [1].
[1] http://en.wikipedia.org/wiki/Computable_number