Yes, in a way. What distinguishes irrational numbers from rational numbers is that all rational numbers can be represented by strings drawn from a regular language. For example, all strings generated by the regular language "-?\d+\.\d+?_\d+" (where "_" denotes the repeating decimal expansion as in 1/6 = 0.1_6) correspond to exactly one rational number and all rational numbers correspond to at least one string in this regular language. Irrational numbers (and other types of "numbers" such as +inf and -inf) cannot be represented by any such regular language.
Correct. Given the regular language I specified, each rational has an infinite number of matching strings: 1.0 = 1.00 = 1.000 = 0.9_99 = 1.0_0 etc. The point is that for every rational number you can think of, I can show you at least one string in my regular language to represent that number.
According to the finitists, this is a defining feature of a "number". Since the same can't be done for irrational numbers finitists conclude that irrational "numbers" aren't numbers. You probably agree that all numbers are (or can be represented by) symbols, but that not all symbols are numbers. So how do we distinguish symbols from numbers?
