There are more structures in non-computable numbers. As an example, a non-computable number could be "definable", i.e. you can describe it. For example, a real number the nth digit in the binary representation of which equals to whether the nth Turing machine halts.
Since the set of all English sentences is countable, whereas there are uncountably many real numbers, it follows that there must be numbers that are NOT even definable.
Since the set of all English sentences is countable, whereas there are uncountably many real numbers, it follows that there must be numbers that are NOT even definable.
Think about that.