Not all transadental numbers exist Does a tree falling in the woods make a sound if no one hears it fall? For sake of arguement let's assume the answer is - No. The tree does not make a sound. Using the above reasoning I would argue that transadental numbers that can not be described or constructed do not exist either. If you can't describe the number or represent the number in any finite way then it is no different than a tree falling in the woods that nobody can hear. Transadental numbers such as Pi and e certainly exists. This is because they can either be described or represented. We can also describe an infinite number of transadental numbers. For example, we could say something like "replace every 5th digit in Pi with the number 3." There is an infinite number of ways of modifiying Pi in this manner, so we could easily come up with an infinite number of transadental numbers. However, here is the cool part. We can count every single one of these transadental numbers. All we have to do is order their English descriptions by length and alphabetical order and start counting. Therefore, I have intuitively proved that transadental numbers are either countable, or I have proved that even if no one hears a tree fall that it still makes a sound.