Counting the Real Numbers

The real numbers as described by the decimals can be completely counted for each order of decimal place.  Take all the real numbers from zero through one.  For one decimal place, they are the numbers 0.0 through 0.9, there are ten of them or 10¹ real numbers and this list is exhaustive.  That is, there is no real number that can be generated from this list that would not be redundant. For two decimal places 10², there are 100 of them, the numbers 0.00, 0.01, through 0.99.  For each decimal place, there is a matrix 10ⁿ that completely lists all the real numbers to that many decimal places.  As n approaches infinity, the real numbers are completely listed.