[bjourne]

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?