An accessible number, to Borel, is a number which can be described as a mathematical object. The problem is that we can only use some finite process to describe a real number so only such numbers are accessible. We can describe rationals easily enough, for example either as, say, one-seventh or by specifying the repeating decimal expansion 142857. Hence rationals are accessible. We can specify Liouville's transcendental number easily enough as having a 1 in place n! and 0 elsewhere. Provided we have some finite way of specifying the n-th term in a Cauchy sequence of rationals we have a finite description of the resulting real number. However, as Borel pointed out, there are a countable number of such descriptions. Hence, as Chaitin writes: "Pick a real at random, and the probability is zero that it's accessible - the probability is zero that it will ever be accessible to us as an individual mathematical object." [J.J. O'Connor and E.F. Robertson: "The real numbers: Attempts to understand"] http://www-history.mcs.st-and.ac.uk/HistTopics/Real_numbers_3.html
But how to pick this dark matter of numbers? Only accessible numbers can get picked. Unpickable numbers cannot appear anywhere, neither in mathematics nor in Cantor's lists. Therefore Cantor "proves" that the pickable numbers, for instance numbers that can appear as an antidiagonal of a defined list, i.e., the countable numbers, are uncountable.