[littlestymaar]

The error in your reasoning lies here :

> Without the decimal point these real numbers just become natural numbers

This is wrong because irrational numbers have an infinite number of digit, if you remove the «dot» you end up with a number infinitely long, which is not a natural number. □

Edit: the set of natural numbers is infinite, which means it can contain arbitrarily big numbers, yet it doesn't contain «numbers» with an infinite amount of digits.