Savy, you're wrong, one set isn't bigger than the other, both are infinite. Now, if you decide to suddenly stop the infinite count of natural numbers then yes at the point you stop counting there are more real numbers, but the point you stop counting means you stopped that set being infinite. As long as both sets are infinite then neither is larger than the other as neither has a size, you need to stop them being infinite in order to give them a size and do the count.