I know that this is a pet peeve more than a serious language error (A Pet Peeve Warning is in Effect!), but when you have an appointment with a doctor, it should be called a doctor appointment, not a doctor’s appointment.
Think about it.
An apostrophe used in an example like this is generally used as a possessive, such as Bill’s car (the car belongs to Bill). It is also sometimes used in the genitive case, which is when a noun modifies another noun, such as Jack’s height (Jack doesn’t own his height; it’s a trait of his).
When you say that you have a doctor’s appointment, you’re unwittingly referring to an appointment that the doctor has; it’s parallel to saying that the car belongs to Bill when saying Bill’s car.
Maybe he has an appointment with his financial advisor, but that’s not what you mean. Technically, the doctor has an appointment with you, but you’re concerned with your appointment with him. In other words, it’s Your (Paul’s) appointment with the doctor, not a doctor’s appointment with you (Paul).
To further make my case, you never say that you have a dentist’s appointment. You probably say that you have a dental appointment or a dentist appointment instead.
Do the same with the doctor; it’s a medical appointment or doctor appointment.